Download VTU (Visvesvaraya Technological University) MBA 2nd Semester (Second Semester) 17MBA23-Research Methodology RM Module 6.1 Important Lecture Notes (MBA Study Material Notes)
Hypothesis
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Module 6
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
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Module 6
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.
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
PROCEDURE FOR HYPOTHESIS TESTING
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
PROCEDURE FOR HYPOTHESIS TESTING
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
PROCEDURE FOR HYPOTHESIS TESTING (i) State Ho and H1:
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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.
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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.
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
Non Parametric tests
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
Bivariate Analysis
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
Bivariate Analysis
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
Bivariate Analysis
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
Bivariate Analysis
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
Bivariate Analysis
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
Bivariate Analysis
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
Bivariate Analysis
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
Bivariate Analysis
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
Bivariate Analysis
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
Bivariate Analysis
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
Bivariate Analysis
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
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).
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
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.
? Behaviours, 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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
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.
? Behaviours, 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
(behaviour, 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.
FirstRanker.com - FirstRanker's Choice
Module 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
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.
? Behaviours, 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
(behaviour, 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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
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.
? Behaviours, 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
(behaviour, 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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
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.
? Behaviours, 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
(behaviour, 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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
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.
? Behaviours, 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
(behaviour, 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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
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.
? Behaviours, 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
(behaviour, 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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
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.
? Behaviours, 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
(behaviour, 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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
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.
? Behaviours, 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
(behaviour, 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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
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.
? Behaviours, 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
(behaviour, 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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
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.
? Behaviours, 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
(behaviour, 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 6
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. But one
must remember that simplicity of hypothesis has nothing to do with its
significance.
(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.
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:
? Yet another step consists in 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.
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.
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.
K-W test
? is?a?non-parametric?method?for?testing?whether?samples?originate?from?the?
same?distribution.
? It? is? used? for? comparing? two? or? more? independent? samples? of? equal? or?
different?sample?sizes.?It?extends?the?Mann?Whitney?U?test?when?there?are?
more?than?two?groups.
? The?parametric?equivalent?of?the?Kruskal-Wallis?test?is?the?one-way?analysis?
of?variance?(ANOVA).
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.
? Behaviours, 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
(behaviour, 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
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This post was last modified on 18 February 2020