Download VTU (Visvesvaraya Technological University) MBA 2nd Semester (Second Semester) 17MBA23-Research Methodology RM Module 5 Important Lecture Notes (MBA Study Material Notes)
Hypothesis
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1:
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ?
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration:
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis Multivariate analysis
? Multivariate analysis is essentially the statistical process of
simultaneously analysing multiple independent (or predictor)
variables with multiple dependent (outcome or criterion) variables
using matrix algebra (most multivariate analyses are correlational).
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis Multivariate analysis
? Multivariate analysis is essentially the statistical process of
simultaneously analysing multiple independent (or predictor)
variables with multiple dependent (outcome or criterion) variables
using matrix algebra (most multivariate analyses are correlational).
Purpose.
? Behaviors, emotions, cognitions, and attitudes can rarely be
described in terms of one or two variables.
? Furthermore, these traits cannot be measured directly, as say running
speed, but must be inferred from constructs which in turn are
measured by multiple factors or variables.
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis Multivariate analysis
? Multivariate analysis is essentially the statistical process of
simultaneously analysing multiple independent (or predictor)
variables with multiple dependent (outcome or criterion) variables
using matrix algebra (most multivariate analyses are correlational).
Purpose.
? Behaviors, emotions, cognitions, and attitudes can rarely be
described in terms of one or two variables.
? Furthermore, these traits cannot be measured directly, as say running
speed, but must be inferred from constructs which in turn are
measured by multiple factors or variables.
? Importance is usually based upon how much common or shared
variance can be extracted from the data.
? Variance is a numerical representation of the distribution of a trait
(behavior, emotion, cognition, etc.) in the population.
? We assume it represents how much of that trait is present in each
individual.
? If two variables are associated or correlated with one another, then
they share some common underlying trait/factor that causes some
equality in how they vary on the scores in the data set.
? That underlying trait is causing them to co-vary together.
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Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis Multivariate analysis
? Multivariate analysis is essentially the statistical process of
simultaneously analysing multiple independent (or predictor)
variables with multiple dependent (outcome or criterion) variables
using matrix algebra (most multivariate analyses are correlational).
Purpose.
? Behaviors, emotions, cognitions, and attitudes can rarely be
described in terms of one or two variables.
? Furthermore, these traits cannot be measured directly, as say running
speed, but must be inferred from constructs which in turn are
measured by multiple factors or variables.
? Importance is usually based upon how much common or shared
variance can be extracted from the data.
? Variance is a numerical representation of the distribution of a trait
(behavior, emotion, cognition, etc.) in the population.
? We assume it represents how much of that trait is present in each
individual.
? If two variables are associated or correlated with one another, then
they share some common underlying trait/factor that causes some
equality in how they vary on the scores in the data set.
? That underlying trait is causing them to co-vary together.
Why the multivariate approach?
With univariate analyses we have just one dependent variable of interest
Although any analysis of data involving more than one variable could be seen
as ?multivariate?, we typically reserve the term for multiple dependent
variables
So MV analysis is an extension of UV ones, or conversely, many of the UV
analyses are special cases of MV ones
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis Multivariate analysis
? Multivariate analysis is essentially the statistical process of
simultaneously analysing multiple independent (or predictor)
variables with multiple dependent (outcome or criterion) variables
using matrix algebra (most multivariate analyses are correlational).
Purpose.
? Behaviors, emotions, cognitions, and attitudes can rarely be
described in terms of one or two variables.
? Furthermore, these traits cannot be measured directly, as say running
speed, but must be inferred from constructs which in turn are
measured by multiple factors or variables.
? Importance is usually based upon how much common or shared
variance can be extracted from the data.
? Variance is a numerical representation of the distribution of a trait
(behavior, emotion, cognition, etc.) in the population.
? We assume it represents how much of that trait is present in each
individual.
? If two variables are associated or correlated with one another, then
they share some common underlying trait/factor that causes some
equality in how they vary on the scores in the data set.
? That underlying trait is causing them to co-vary together.
Why the multivariate approach?
With univariate analyses we have just one dependent variable of interest
Although any analysis of data involving more than one variable could be seen
as ?multivariate?, we typically reserve the term for multiple dependent
variables
So MV analysis is an extension of UV ones, or conversely, many of the UV
analyses are special cases of MV ones
Multivariate Pros and Cons Summary
Advantages of using a multivariate statistic
? Richer realistic design
? Looks at phenomena in an overarching way (provides multiple levels of
analysis)
? Each method differs in amount or type of Independent Variables (IVs) and
DVs
? Can help control for Type I Error
Disadvantages
? Larger Ns are often required
? More difficult to interpret
? Less known about the robustness of assumptions
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis Multivariate analysis
? Multivariate analysis is essentially the statistical process of
simultaneously analysing multiple independent (or predictor)
variables with multiple dependent (outcome or criterion) variables
using matrix algebra (most multivariate analyses are correlational).
Purpose.
? Behaviors, emotions, cognitions, and attitudes can rarely be
described in terms of one or two variables.
? Furthermore, these traits cannot be measured directly, as say running
speed, but must be inferred from constructs which in turn are
measured by multiple factors or variables.
? Importance is usually based upon how much common or shared
variance can be extracted from the data.
? Variance is a numerical representation of the distribution of a trait
(behavior, emotion, cognition, etc.) in the population.
? We assume it represents how much of that trait is present in each
individual.
? If two variables are associated or correlated with one another, then
they share some common underlying trait/factor that causes some
equality in how they vary on the scores in the data set.
? That underlying trait is causing them to co-vary together.
Why the multivariate approach?
With univariate analyses we have just one dependent variable of interest
Although any analysis of data involving more than one variable could be seen
as ?multivariate?, we typically reserve the term for multiple dependent
variables
So MV analysis is an extension of UV ones, or conversely, many of the UV
analyses are special cases of MV ones
Multivariate Pros and Cons Summary
Advantages of using a multivariate statistic
? Richer realistic design
? Looks at phenomena in an overarching way (provides multiple levels of
analysis)
? Each method differs in amount or type of Independent Variables (IVs) and
DVs
? Can help control for Type I Error
Disadvantages
? Larger Ns are often required
? More difficult to interpret
? Less known about the robustness of assumptions
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis Multivariate analysis
? Multivariate analysis is essentially the statistical process of
simultaneously analysing multiple independent (or predictor)
variables with multiple dependent (outcome or criterion) variables
using matrix algebra (most multivariate analyses are correlational).
Purpose.
? Behaviors, emotions, cognitions, and attitudes can rarely be
described in terms of one or two variables.
? Furthermore, these traits cannot be measured directly, as say running
speed, but must be inferred from constructs which in turn are
measured by multiple factors or variables.
? Importance is usually based upon how much common or shared
variance can be extracted from the data.
? Variance is a numerical representation of the distribution of a trait
(behavior, emotion, cognition, etc.) in the population.
? We assume it represents how much of that trait is present in each
individual.
? If two variables are associated or correlated with one another, then
they share some common underlying trait/factor that causes some
equality in how they vary on the scores in the data set.
? That underlying trait is causing them to co-vary together.
Why the multivariate approach?
With univariate analyses we have just one dependent variable of interest
Although any analysis of data involving more than one variable could be seen
as ?multivariate?, we typically reserve the term for multiple dependent
variables
So MV analysis is an extension of UV ones, or conversely, many of the UV
analyses are special cases of MV ones
Multivariate Pros and Cons Summary
Advantages of using a multivariate statistic
? Richer realistic design
? Looks at phenomena in an overarching way (provides multiple levels of
analysis)
? Each method differs in amount or type of Independent Variables (IVs) and
DVs
? Can help control for Type I Error
Disadvantages
? Larger Ns are often required
? More difficult to interpret
? Less known about the robustness of assumptions
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis Multivariate analysis
? Multivariate analysis is essentially the statistical process of
simultaneously analysing multiple independent (or predictor)
variables with multiple dependent (outcome or criterion) variables
using matrix algebra (most multivariate analyses are correlational).
Purpose.
? Behaviors, emotions, cognitions, and attitudes can rarely be
described in terms of one or two variables.
? Furthermore, these traits cannot be measured directly, as say running
speed, but must be inferred from constructs which in turn are
measured by multiple factors or variables.
? Importance is usually based upon how much common or shared
variance can be extracted from the data.
? Variance is a numerical representation of the distribution of a trait
(behavior, emotion, cognition, etc.) in the population.
? We assume it represents how much of that trait is present in each
individual.
? If two variables are associated or correlated with one another, then
they share some common underlying trait/factor that causes some
equality in how they vary on the scores in the data set.
? That underlying trait is causing them to co-vary together.
Why the multivariate approach?
With univariate analyses we have just one dependent variable of interest
Although any analysis of data involving more than one variable could be seen
as ?multivariate?, we typically reserve the term for multiple dependent
variables
So MV analysis is an extension of UV ones, or conversely, many of the UV
analyses are special cases of MV ones
Multivariate Pros and Cons Summary
Advantages of using a multivariate statistic
? Richer realistic design
? Looks at phenomena in an overarching way (provides multiple levels of
analysis)
? Each method differs in amount or type of Independent Variables (IVs) and
DVs
? Can help control for Type I Error
Disadvantages
? Larger Ns are often required
? More difficult to interpret
? Less known about the robustness of assumptions
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis Multivariate analysis
? Multivariate analysis is essentially the statistical process of
simultaneously analysing multiple independent (or predictor)
variables with multiple dependent (outcome or criterion) variables
using matrix algebra (most multivariate analyses are correlational).
Purpose.
? Behaviors, emotions, cognitions, and attitudes can rarely be
described in terms of one or two variables.
? Furthermore, these traits cannot be measured directly, as say running
speed, but must be inferred from constructs which in turn are
measured by multiple factors or variables.
? Importance is usually based upon how much common or shared
variance can be extracted from the data.
? Variance is a numerical representation of the distribution of a trait
(behavior, emotion, cognition, etc.) in the population.
? We assume it represents how much of that trait is present in each
individual.
? If two variables are associated or correlated with one another, then
they share some common underlying trait/factor that causes some
equality in how they vary on the scores in the data set.
? That underlying trait is causing them to co-vary together.
Why the multivariate approach?
With univariate analyses we have just one dependent variable of interest
Although any analysis of data involving more than one variable could be seen
as ?multivariate?, we typically reserve the term for multiple dependent
variables
So MV analysis is an extension of UV ones, or conversely, many of the UV
analyses are special cases of MV ones
Multivariate Pros and Cons Summary
Advantages of using a multivariate statistic
? Richer realistic design
? Looks at phenomena in an overarching way (provides multiple levels of
analysis)
? Each method differs in amount or type of Independent Variables (IVs) and
DVs
? Can help control for Type I Error
Disadvantages
? Larger Ns are often required
? More difficult to interpret
? Less known about the robustness of assumptions
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis Multivariate analysis
? Multivariate analysis is essentially the statistical process of
simultaneously analysing multiple independent (or predictor)
variables with multiple dependent (outcome or criterion) variables
using matrix algebra (most multivariate analyses are correlational).
Purpose.
? Behaviors, emotions, cognitions, and attitudes can rarely be
described in terms of one or two variables.
? Furthermore, these traits cannot be measured directly, as say running
speed, but must be inferred from constructs which in turn are
measured by multiple factors or variables.
? Importance is usually based upon how much common or shared
variance can be extracted from the data.
? Variance is a numerical representation of the distribution of a trait
(behavior, emotion, cognition, etc.) in the population.
? We assume it represents how much of that trait is present in each
individual.
? If two variables are associated or correlated with one another, then
they share some common underlying trait/factor that causes some
equality in how they vary on the scores in the data set.
? That underlying trait is causing them to co-vary together.
Why the multivariate approach?
With univariate analyses we have just one dependent variable of interest
Although any analysis of data involving more than one variable could be seen
as ?multivariate?, we typically reserve the term for multiple dependent
variables
So MV analysis is an extension of UV ones, or conversely, many of the UV
analyses are special cases of MV ones
Multivariate Pros and Cons Summary
Advantages of using a multivariate statistic
? Richer realistic design
? Looks at phenomena in an overarching way (provides multiple levels of
analysis)
? Each method differs in amount or type of Independent Variables (IVs) and
DVs
? Can help control for Type I Error
Disadvantages
? Larger Ns are often required
? More difficult to interpret
? Less known about the robustness of assumptions
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis Multivariate analysis
? Multivariate analysis is essentially the statistical process of
simultaneously analysing multiple independent (or predictor)
variables with multiple dependent (outcome or criterion) variables
using matrix algebra (most multivariate analyses are correlational).
Purpose.
? Behaviors, emotions, cognitions, and attitudes can rarely be
described in terms of one or two variables.
? Furthermore, these traits cannot be measured directly, as say running
speed, but must be inferred from constructs which in turn are
measured by multiple factors or variables.
? Importance is usually based upon how much common or shared
variance can be extracted from the data.
? Variance is a numerical representation of the distribution of a trait
(behavior, emotion, cognition, etc.) in the population.
? We assume it represents how much of that trait is present in each
individual.
? If two variables are associated or correlated with one another, then
they share some common underlying trait/factor that causes some
equality in how they vary on the scores in the data set.
? That underlying trait is causing them to co-vary together.
Why the multivariate approach?
With univariate analyses we have just one dependent variable of interest
Although any analysis of data involving more than one variable could be seen
as ?multivariate?, we typically reserve the term for multiple dependent
variables
So MV analysis is an extension of UV ones, or conversely, many of the UV
analyses are special cases of MV ones
Multivariate Pros and Cons Summary
Advantages of using a multivariate statistic
? Richer realistic design
? Looks at phenomena in an overarching way (provides multiple levels of
analysis)
? Each method differs in amount or type of Independent Variables (IVs) and
DVs
? Can help control for Type I Error
Disadvantages
? Larger Ns are often required
? More difficult to interpret
? Less known about the robustness of assumptions
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis Multivariate analysis
? Multivariate analysis is essentially the statistical process of
simultaneously analysing multiple independent (or predictor)
variables with multiple dependent (outcome or criterion) variables
using matrix algebra (most multivariate analyses are correlational).
Purpose.
? Behaviors, emotions, cognitions, and attitudes can rarely be
described in terms of one or two variables.
? Furthermore, these traits cannot be measured directly, as say running
speed, but must be inferred from constructs which in turn are
measured by multiple factors or variables.
? Importance is usually based upon how much common or shared
variance can be extracted from the data.
? Variance is a numerical representation of the distribution of a trait
(behavior, emotion, cognition, etc.) in the population.
? We assume it represents how much of that trait is present in each
individual.
? If two variables are associated or correlated with one another, then
they share some common underlying trait/factor that causes some
equality in how they vary on the scores in the data set.
? That underlying trait is causing them to co-vary together.
Why the multivariate approach?
With univariate analyses we have just one dependent variable of interest
Although any analysis of data involving more than one variable could be seen
as ?multivariate?, we typically reserve the term for multiple dependent
variables
So MV analysis is an extension of UV ones, or conversely, many of the UV
analyses are special cases of MV ones
Multivariate Pros and Cons Summary
Advantages of using a multivariate statistic
? Richer realistic design
? Looks at phenomena in an overarching way (provides multiple levels of
analysis)
? Each method differs in amount or type of Independent Variables (IVs) and
DVs
? Can help control for Type I Error
Disadvantages
? Larger Ns are often required
? More difficult to interpret
? Less known about the robustness of assumptions
FirstRanker.com - FirstRanker's Choice
Module 5
Hypothesis
WHAT IS A HYPOTHESIS ?
? Hypothesis may be defined as a proposition or a set of proposition set
forth as an explanation for the occurrence of some specified group of
phenomena either asserted merely as a provisional conjecture to
guide some investigation or accepted as highly probable in the light of
established facts.
? Quite often a research hypothesis is a predictive statement, capable
of being tested by scientific methods, that relates an independent
variable to some dependent variable
Characteristics of hypothesis:
(i) Hypothesis should be clear and precise. If the hypothesis is not clear
and precise, the inferences drawn on its basis cannot be taken as
reliable.
(ii) Hypothesis should be capable of being tested.
(iii) Hypothesis should state relationship between variables, if it
happens to be a relational hypothesis.
(iv) Hypothesis should be limited in scope and must be specific. A
researcher must remember that narrower hypotheses are generally
more testable and he should develop such hypotheses.
Characteristics of hypothesis:
(v) Hypothesis should be stated as far as possible in most simple terms
so that the same is easily understandable by all concerned.
(vi) Hypothesis should be consistent with most known facts i.e., it must
be consistent with a substantial body of established facts. In other
words, it should be one which judges accept as being the most likely.
Characteristics of hypothesis:
? (vii) Hypothesis should be amenable to testing within a reasonable
time.
One should not use even an excellent hypothesis, if the same cannot be
tested in reasonable time for one cannot spend a life-time collecting
data to test it.
Characteristics of hypothesis:
(viii) Hypothesis must explain the facts that gave rise to the need for
explanation.
? Thus hypothesis must actually explain what it claims to explain; it
should have empirical reference
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1: (ii) Selecting a Significance level:
? The hypotheses are tested on a pre-determined level of significance
and as such the same should be specified.
? Generally, in practice, either 5% level or 1% level is adopted for the
purpose.
? The 5 per cent level of significance means that researcher is willing to
take as much as a 5 per cent risk of rejecting the null hypothesis when
it (H ) happens to be true.
In a two-tailed test,
there are two rejection
regions
one on each tail of the
curve which can be
illustrated
(iii) Deciding the distribution to use:
After deciding the level of significance, the next step in hypothesis
testing is to determine the appropriate sampling distribution.
The choice generally remains between normal distribution and the
t-distribution.
(iv) Selecting a random sample and
computing an appropriate value:
? Another step is to select a random sample(s) and compute an
appropriate value from the sample data concerning the test statistic
utilizing the relevant distribution.
? In other words, draw a sample to furnish empirical data.
(v) Calculation of the probability:
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis
were in fact true.
(vi) Comparing the probability:
? Comparing the probability thus calculated with the specified value for ? ,
the significance level.
? If the calculated probability is equal to or smaller than the ? value in case
of one-tailed test, then reject the null hypothesis (i.e., accept the
alternative hypothesis),
? but if the calculated probability is greater, then accept the null hypothesis.
Errors in hypothesis
? Type 1 error
? Hypothesis is rejected when it is true
? Type 2 error
? Hypothesis is not rejected when it is false
Types of tests
? Parametric test
? Non-parametric test
z-test
? A?type?of?statistical?analysis?that?considers?the?difference?between?
The?mean?of?the?variable?in?a?sample?set?and?
The?mean?of?the?variable?in?a?larger?population.
Circumstance where the Z test is used
? A z-test is used for testing the mean of a population versus a standard, or
comparing the means of two populations, with large (n ? 30) samples whether
you know the population standard deviation or not.
? It is also used for testing the proportion of some characteristic versus a
standard proportion, or comparing the proportions of two populations.
Example: Comparing the average engineering salaries of men versus women.
Example: Comparing the fraction defectives from 2 production lines.
T - test
? A?T-test?is?a?statistical?examination?of?two?population?means.
? A?two-sample?t-test examines?
whether?two?samples?are?different?and?is?commonly?used?
-?when?the?variances?of?two?normal?distributions?are?unknown?and?
-?when?an?experiment?uses?a?small?sample?size
t-test ? when to use
? A t-test is used for testing the mean of one population against a standard or
comparing the means of two populations if you do not know the populations?
standard deviation and when you have a limited sample (n < 30).
? If you know the populations? standard deviation, you may use a z-test.
Example: Measuring the average diameter of shafts from a certain machine when
you have a small sample.
1.Testing difference between means of two
samples (independent Samples)
When is a one-tailed test appropriate?
? If?you?are?using?a?significance?level?of?0.05,?a?one-tailed?test?allots?all?
of?your?alpha?to?testing?the?statistical?significance?in?the?one?direction?
of?interest.??
? This?means?that?0.05?is?in?one?tail?of?the?distribution?of?your?test?
statistic.?
? When?using?a?one-tailed?test,?you?are?testing?for?the?possibility?of?the?
relationship?in?one direction and?completely?disregarding?the?
possibility?of?a?relationship?in?the?other?direction.
? For?example,?imagine?that?you?have?developed?a?new?drug.?
? It?is?cheaper?than?the?existing?drug?and,?you?believe,?not?less?effective.?
? In?testing?this?drug,?you?are?only?interested?in?testing?if?it?less?effective?
than?the?existing?drug.?
? You?do?not?care?if?it?is?significantly?more?effective.??
? You?only?wish?to?show?that?it?is?not?less?effective.?In?this?scenario,?a?one-
tailed?test?would?be?appropriate.?
? Our?null?hypothesis?is?that?the?mean?is?equal?to? x.
? ?A?one-tailed?test?will?test?either?if?the?mean?is?significantly?greater?
than? x?
Or
? ?if?the?mean?is?significantly?less?than?x,?but?not?both.
Two-tailed test
? If you are using a significance level of 0.05, a two-tailed test allots half
of your alpha to testing the statistical significance in one direction and
half of your alpha to testing statistical significance in the other
direction.
? This means that .025 is in each tail of the distribution of your test
statistic.
? Our?null?hypothesis?is?that?the?mean?is?equal?to?x.
? A?two-tailed?test?will?test?both?if?the?mean?is?significantly?greater?
than?x?and?if?the?mean?significantly?less?than?x.?
? The?mean?is?considered?significantly?different?from?x?if?the?test?
statistic?is?in?the?top?2.5%?or?bottom?2.5%?of?its?probability?
distribution,?resulting?in?a?p-value?less?than?0.05.
2. To test Significance of the mean of a
Random sample
Fiducial limits of Population Mean
? Assuming that the saple is a random sample from a normal
population of unknown mean the 95% fiducial limits of the population
mean (?) are
Example
The manufacturer of a certain make of electric bulbs claims that his
bulbs have a mean life of 25 months with a standard deviation of 5
months. A random sample of 6 such bulbs gave the following values.
Life of months : 24 26 30 20 20 18
Can you regard the producer?s claim to be valid at 1% level of
significance?
in this two samples are said to be dependent when the elements in one
sample are related to those in the other in any pairs of observations made
on the same subject.
Two samples may consists of pairs of observations made on same object, on
the same selected population.
Say, for effectiveness of coaching class or training programme
3.Testing difference between means of two
samples (Dependent Samples or matched pair
observations)
3.Testing difference between means of two samples
(Dependent Samples or matched pair observations)
It is defined by
?d =the mean of the differences
S = the standard deviation of the differences
If t > table value then Ho is rejected
If t < table value then Ho is accepted
? The value of the S is calculated by,
? It should be noted that t is based on n-1 degree of freedom
Example
? To verify whether a course in accounting improved performance, a
similar test was given to 12 participants both before and after the
course. The original marks recorded in alphabetical order of the
participants were- 44,40,61,52,32,44,70,41,67,72,53 and 72.
After the course, the marks were in the same order,
53,38,69,57,46,39,73,48,73,74,60 and 78. was the course useful?
Hypothesis :
there is no difference in the marks obtained before and after the
course, i.e. the course has not been useful
4.Testing the Significance of an observed
Correlation Coefficient
? Given random sample from bivariate population.
? If we are to test the hypothesis that the correlation coefficient of the
population is zero.
? i.e. the variable in the population are uncorrelated, we have to apply
the test.
? Here t is based on n-2 degree of freedom.
? We say that the value of r is significant at 5% level.
? If t < t
0.05
the data are considered with the hypothesis i.e
uncorrelated.
? r= correlation coefficient
Example 1
? A random sample of 27 pairs of observations from a
normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated?
? Solution
? Ho : there is no significant difference in the sample correlation
and correlation in the population
Example 2
? How many pairs of observations must be included in a sample in
order that an observed correlation coefficient of value 0.42 shall have
a calculated value of r greater than 2.72?
Solution-
Here, r value is 0.42, we have to find out n
Example 3
? The following table gives the ages in years of 10 husbands and their
wives at marriage. Compute the correlation coefficient and test for its
significance.
? Husband?s Age : 23 27 28 29 30 31 33 35
36 39
? Wife?s Age : 18 22 23 24 25 26 28 29 30
32
? Solution
? Ho : there is no correlation in the population
? Hear you need to find our
F-test
? Definition:??F-test?is?a?statistical?test?that?is?used?to?determine?whether?two?
populations? having? normal? distribution? have? the? same? variances? or?
standard? deviation.? This? is? an? important? part? of? Analysis? of? Variance?
(ANOVA).
WHEN?
? An?F-test?is?used?to?compare?2?populations??variances.?The?samples?can?be?
any?size.?It?is?the?basis?of?ANOVA.
Example:?Comparing?the?variability?of?bolt?diameters?from?two?machines.
? The F-Test is named in honor of the great statistician R.A Fisher.
? The object of the F-test is to find out whether the two independent
estimates of population variance differ significantly, or whether the
two samples may be regarded as drawn from the normal populations
having the same variance. For carrying out the test of significance, we
calculate the ratio F.
S
1
2
> S
2
2 means S1 is always
larger estimate of variance.
V1 = Degree of Freedom for samples having larger variance
V2 = Degree of Freedom for samples having Smaller variance
V1 =
n
1
-1
V2
= n
2
-1
If F > table value then F ratio is considered as significant and Ho is rejected &
H1 is accept
If F < table value accept Ho, means both the samples
have come from the
population having same
variance.
H
0
= Two Populations Have Same Variance V1 = 10
V2 = 8
F
0.05 = 3.36
CALCULATED VALUE OF F = 0.707
The Calculated F Value Is Less Than Table Value HENCE THE HYPOTHESIS IS ACCEPTED.
Non Parametric tests
? U?test?
? (also?called?the?Mann?Whitney?Wilcoxon?(MWW),?Wilcoxon rank-
sum test,?or?Wilcoxon?Mann?Whitney test)
? is?a?nonparametric?test?of?the?null?hypothesis?that?two?samples?come?
from? the? same? population? against? an?alternative? hypothesis,?
especially? that? a? particular? population? tends? to? have? larger? values?
than?the?other.
Mann?Whitney-U test
? This test is to determine whether two independent samples have
been drawn from the same population.
? This test applies in very general conditions and requires only
populations sampled are continuous.
Mann?Whitney-U test Steps to solve
? We first of all rank the data jointly, taking them as belonging to a
single sample in either an increasing or decreasing order of
magnitude.
? We usually adopt low to high ranking process which means we assign
rank 1 to an item with lowest value, rank 2 to the next higher item ad
so on.
? In case there are ties, then we would assign each of the tied
observation the mean of the ranks which they jointly occupy.
? (for 11 11 11 rank may be (6+7+8)3=7 )
Mann?Whitney-U test Steps to solve
? After this we find the sum of the ranks assigned to the values of the
first sample(R1) and also the sum of the ranks assigned to the values
of the second sample(R2)
? Then we work out the test staststic i. e.U which is a measurement of
the difference between the ranked observations o the two samples as
under
? U = n1 x n2 + n1 (n1+1) ? R1
2
n1 and n2 are the sample sizes and R1 is the sum of ranks assigned to
the values of the first Sample
Mann?Whitney-U test Steps to solve
? ? ? If n1 and n2 are sufficiently large ( i.e. both greater than 8), the
sampling distribution of U can be approximately closely with normal
distribution and the limits of the acceptance region can be
determined in the usual way at a given level of significance.
Example 1
? The value in one sample are 53 38 69 57 46 39 73 48 73 74 60 & 78.
In another sample they are 44 40 61 52 32 44 70 41 67 72 53 & 72.
test at the 10% level the hypothesis that they come from population
with the same mean. Apply U-test
Solution
size of sample item in
ascending order
Rank Related sample A or B
32 1 B
38 2 A
39 3 A
40 4 B
41 5 B
44 6.5 B
44 6.5 B
? R1 = 2+3+8+9+11.5+13+14+17+21.5+21.5+23+24 = 167.5
? R2= 1+4+5+6.5+6.5+10+11.5+15+16+18+19.5+19.5= 132.5
? N1 = 12
? N2 = 12
? U = 54.5
? Since in the given proble n1 and n2 both are greater than 8, so the
sampling distribution of U approximately closely with normal curve.
? Keeping this in view, we work out the mean and SD taking the null
hypothesis that the two samples come from identical populations as
under
? = 72
? = 17.32
? As the alternative hapothesis is that the means of the two
populations are not equal, a two-tailed test is appropriate. According
the limits of acceptance region, keeping in view 10% level of
significance as given, can be worked out as under,
? As Z value for 0.45 of the are under the normal curve is 1.64 we have
following limits of acceptance region
? Upper limit = ? +1.64 ?U = 100.40
? Lower limit = ? -1.64 ?U = 43.60
The Kruskal-Wallis test (or H test ):
? This test is conducted in a way similar to the U test described above.
? This test is used to test the null hypothesis that ?k? independent
random samples come from identical universes against the
alternative hypothesis that the means of these universes are not
equal.
? This test is analogous to the one-way analysis of variance, but unlike
the latter it does not require the assumption that the samples come
from approximately normal populations or the universes having the
same standard deviation.
The Kruskal-Wallis test (or H test ):
? In this test, like the U test, the data are ranked jointly from low to
high or high to low as if they constituted a single sample.
? The test statistic is H for this test which is worked out as under:
where n = n1 + n2 + ... + nk and Ri being the sum of the ranks assigned to n i
observations in the ith sample.
The Kruskal-Wallis test (or H test ):
? If the null hypothesis is true that there is no difference between the
sample means and each sample has at least five items*, then the
sampling distribution of H can be approximated with a chi square
distribution with (k ? 1) degrees of freedom.
? As such we can reject the null hypothesis at a given level of significance
if H value calculated, as stated above, exceeds the concerned table
value of chi-square. * If any of the given samples has less than five items then chi-square
distribution approximation can not be used and the exact tests may be
based on table
The Kruskal-Wallis test (or H test ):
Illustration
? Use the Kruskal-Wallis test at 5% level of significance to test the null
hypothesis that a professional bowler performs equally well with the
four bowling balls, given the following results:
? As the four samples have five items* each, the sampling distribution
of H approximates closely with chi-square distribution.
? Now taking the null hypothesis that the bowler performs equally well
with the four balls, we have the value of chi-square = 7.815 for (k ? 1)
or 4 ? 1 = 3 degrees of freedom at 5% level of significance.
? Since the calculated value of H is only 4.51 and does not exceed the
c2 value of 7.815, so we accept the null hypothesis and conclude that
bowler performs equally well with the four bowling balls.
Illustration: Bivariate Analysis Multivariate analysis
? Multivariate analysis is essentially the statistical process of
simultaneously analysing multiple independent (or predictor)
variables with multiple dependent (outcome or criterion) variables
using matrix algebra (most multivariate analyses are correlational).
Purpose.
? Behaviors, emotions, cognitions, and attitudes can rarely be
described in terms of one or two variables.
? Furthermore, these traits cannot be measured directly, as say running
speed, but must be inferred from constructs which in turn are
measured by multiple factors or variables.
? Importance is usually based upon how much common or shared
variance can be extracted from the data.
? Variance is a numerical representation of the distribution of a trait
(behavior, emotion, cognition, etc.) in the population.
? We assume it represents how much of that trait is present in each
individual.
? If two variables are associated or correlated with one another, then
they share some common underlying trait/factor that causes some
equality in how they vary on the scores in the data set.
? That underlying trait is causing them to co-vary together.
Why the multivariate approach?
With univariate analyses we have just one dependent variable of interest
Although any analysis of data involving more than one variable could be seen
as ?multivariate?, we typically reserve the term for multiple dependent
variables
So MV analysis is an extension of UV ones, or conversely, many of the UV
analyses are special cases of MV ones
Multivariate Pros and Cons Summary
Advantages of using a multivariate statistic
? Richer realistic design
? Looks at phenomena in an overarching way (provides multiple levels of
analysis)
? Each method differs in amount or type of Independent Variables (IVs) and
DVs
? Can help control for Type I Error
Disadvantages
? Larger Ns are often required
? More difficult to interpret
? Less known about the robustness of assumptions
FirstRanker.com - FirstRanker's Choice
This post was last modified on 18 February 2020