Download MBA IB and Marketing 1st and 2nd Semester Reseach Methodology

Download MBA (Master of Business Administration) IB and Marketing 1st and 2nd Semester Reseach Methodology

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INTRODUCTION

Learning Objectives:

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After reading this lesson, you should be able to understand:

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Meaning, objectives and types of research

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Qualities of researcher

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Significance of research

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Research process

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Research problem

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Features, importance, characteristics, concepts and types of Research

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design

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Case study research

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Hypothesis and its testing

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Sample survey and sampling methods

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1.1 Meaning of Research:

Research in simple terms refers to search for knowledge. It is a scientific

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and systematic search for information on a particular topic or issue. It is also

known as the art of scientific investigation. Several social scientists have

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defined research in different ways.

In

the

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Encyclopedia of Social Sciences, D. Slesinger and M. Stephension

(1930) defined research as "the manipulation of things, concepts or symbols for

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the purpose of generalizing to extend, correct or verify knowledge, whether that

knowledge aids in the construction of theory or in the practice of an art".

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According to Redman and Mory (1923), research is a "systematized

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effort to gain new knowledge". It is an academic activity and therefore the term

should be used in a technical sense. According to Clifford Woody (Kothari,

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1988), research comprises "defining and redefining problems, formulating

hypotheses or suggested solutions; collecting, organizing and evaluating data;

making deductions and reaching conclusions; and finally, carefully testing the

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conclusions to determine whether they fit the formulated hypotheses".

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Thus, research is an original addition to the available knowledge, which

contributes to its further advancement. It is an attempt to pursue truth through

the methods of study, observation, comparison and experiment. In sum,

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research is the search for knowledge, using objective and systematic methods to

find solution to a problem.

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1.1.1 Objectives of Research:

The objective of research is to find answers to the questions by applying

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scientific procedures. In other words, the main aim of research is to find out the

truth which is hidden and has not yet been discovered. Although every research

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study has its own specific objectives, the research objectives may be broadly

grouped as follows:

1. to gain familiarity with new insights into a phenomenon (i.e., formulative

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research studies);

2. to accurately portray the characteristics of a particular individual, group, or a

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situation (i.e., descriptive research studies);

3. to analyse the frequency with which something occurs (i.e., diagnostic research

studies); and

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4. to examine the hypothesis of a causal relationship between two variables (i.e.,

hypothesis-testing research studies).

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1.1.2 Research Methods versus Methodology:

Research methods include all those techniques/methods that are adopted

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for conducting research. Thus, research techniques or methods are the methods

that the researchers adopt for conducting the research studies.

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On the other hand, research methodology is the way in which research

problems are solved systematically. It is a science of studying how research is

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conducted scientifically. Under it, the researcher acquaints himself/herself with

the various steps generally adopted to study a research problem, along with the

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underlying logic behind them. Hence, it is not only important for the researcher

to know the research techniques/methods, but also the scientific approach called

methodology.

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1.1.3 Research Approaches:

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There are two main approaches to research, namely quantitative

approach and qualitative approach. The quantitative approach involves the

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collection of quantitative data, which are put to rigorous quantitative analysis in

a formal and rigid manner. This approach further includes experimental,

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inferential, and simulation approaches to research. Meanwhile, the qualitative

approach uses the method of subjective assessment of opinions, behaviour and

attitudes. Research in such a situation is a function of the researcher's

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impressions and insights. The results generated by this type of research are

either in non-quantitative form or in the form which cannot be put to rigorous

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quantitative analysis. Usually, this approach uses techniques like indepth

interviews, focus group interviews, and projective techniques.

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1.1.4 Types of Research:

There are different types of research. The basic ones are as follows:

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1) Descriptive versus Analytical:

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Descriptive research consists of surveys and fact-finding enquiries of

different types. The main objective of descriptive research is describing the

state of affairs as it prevails at the time of study. The term `ex post facto

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research' is quite often used for descriptive research studies in social sciences

and business research. The most distinguishing feature of this method is that the

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researcher has no control over the variables here. He/she has to only report what

is happening or what has happened. Majority of the ex post facto research

projects are used for descriptive studies in which the researcher attempts to

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examine phenomena, such as the consumers' preferences, frequency of

purchases, shopping, etc. Despite the inability of the researchers to control the

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variables, ex post facto studies may also comprise attempts by them to discover

the causes of the selected problem. The methods of research adopted in

conducting descriptive research are survey methods of all kinds, including

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correlational and comparative methods.

Meanwhile in the Analytical research, the researcher has to use the

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already available facts or information, and analyse them to make a critical

evaluation of the subject.

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2)

Applied versus Fundamental:

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Research can also be applied or fundamental in nature. An attempt to

find a solution to an immediate problem encountered by a firm, an industry, a

business organisation, or the society is known as Applied Research. Researchers

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engaged in such researches aim at drawing certain conclusions confronting a

concrete social or business problem.

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On the other hand, Fundamental Research mainly concerns

generalizations and formulation of a theory. In other words, "Gathering

knowledge for knowledge's sake is termed `pure' or `basic' research" (Young in

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Kothari, 1988). Researches relating to pure mathematics or concerning some

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natural phenomenon are instances of Fundamental Research. Likewise, studies

focusing on human behaviour also fall under the category of fundamental

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research.

Thus, while the principal objective of applied research is to find a

solution to some pressing practical problem, the objective of basic research is to

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find information with a broad base of application and add to the already existing

organized body of scientific knowledge.

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3)

Quantitative versus Qualitative:

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Quantitative research relates to aspects that can be quantified or can be

expressed in terms of quantity. It involves the measurement of quantity or

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amount. The various available statistical and econometric methods are adopted

for analysis in such research. Some such includes correlation, regressions and

time series analysis.

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On the other hand, Qualitative research is concerned with qualitative

phenomena, or more specifically, the aspects related to or involving quality or

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kind. For example, an important type of qualitative research is `Motivation

Research', which investigates into the reasons for human behaviour. The main

aim of this type of research is discovering the underlying motives and desires of

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human beings by using in-depth interviews. The other techniques employed in

such research are story completion tests, sentence completion tests, word

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association tests, and other similar projective methods. Qualitative research is

particularly significant in the context of behavioural sciences, which aim at

discovering the underlying motives of human behaviour. Such research helps to

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analyse the various factors that motivate human beings to behave in a certain

manner, besides contributing to an understanding of what makes individuals like

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or dislike a particular thing. However, it is worth noting that conducting

qualitative research in practice is considerably a difficult task. Hence, while

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undertaking such research, seeking guidance from experienced expert

researchers is important.

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Conceptual versus Empirical:

The research related to some abstract idea or theory is known as

Conceptual Research. Generally, philosophers and thinkers use it for

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developing new concepts or for reinterpreting the existing ones. Empirical

Research, on the other hand, exclusively relies on the observation or experience

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with hardly any regard for theory and system. Such research is data based,

which often comes up with conclusions that can be verified through experiments

or observation. Empirical research is also known as experimental type of

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research, in which it is important to first collect the facts and their sources, and

actively take steps to stimulate the production of desired information. In this

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type of research, the researcher first formulates a working hypothesis, and then

gathers sufficient facts to prove or disprove the stated hypothesis. He/she

formulates the experimental design, which according to him/her would

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manipulate the variables, so as to obtain the desired information. This type of

research is thus characterized by the researcher's control over the variables

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under study. Empirical research is most appropriate when an attempt is made to

prove that certain variables influence the other variables in some way.

Therefore, the results obtained by using the experimental or empirical studies

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are considered to be the most powerful evidences for a given hypothesis.

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5)

Other Types of Research:

The remaining types of research are variations of one or more of the

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afore-mentioned methods. They vary in terms of the purpose of research, or the

time required to complete it, or may be based on some other similar factor. On

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the basis of time, research may either be in the nature of one-time or

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longitudinal research. While the research is restricted to a single time-period in

the former case, it is conducted over several time-periods in the latter case.

Depending upon the environment in which the research is to be conducted, it can

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also be laboratory research or field-setting research, or simulation research,

besides being diagnostic or clinical in nature. Under such research, in-depth

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approaches or case study method may be employed to analyse the basic causal

relations. These studies usually undertake a detailed in-depth analysis of the

causes of certain events of interest, and use very small samples and sharp data

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collecting methods. The research may also be explanatory in nature.

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Formalized research studies consist of substantial structure and specific

hypotheses to be verified. As regards historical research, sources like historical

documents, remains, etc. are utilized to study past events or ideas. It also

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includes philosophy of persons and groups of the past or any remote point of

time.

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Research has also been classified into decision-oriented and conclusion-

oriented categories. The Decision-oriented research is always carried out as per

the need of a decision maker and hence, the researcher has no freedom to

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conduct the research according to his/her own desires. On the other hand, in the

case of Conclusion-oriented research, the researcher is free to choose the

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problem, redesign the enquiry as it progresses and even change

conceptualization as he/she wishes to. Further, Operations research is a kind of

decision-oriented research, because it is a scientific method of providing the

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departments, a quantitative basis for decision-making with respect to the

activities under their purview.

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1.1.5 Importance of Knowing How to Conduct Research:

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The importance of knowing how to conduct research is listed below:

(i) the knowledge of research methodology provides training to new

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researchers and enables them to do research properly. It helps them to

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develop disciplined thinking or a `bent of mind' to objectively observe

the field;

(ii) the knowledge of doing research inculcates the ability to evaluate and

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utilise the research findings with confidence;

(iii) the knowledge of research methodology equips the researcher with the

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tools that help him/her to make the observations objectively; and

(iv) the knowledge of methodology helps the research consumer to evaluate

research and make rational decisions.

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1.1.6 Qualities of a Researcher:

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It is important for a researcher to possess certain qualities to conduct

research. First and foremost, he being a scientist should be firmly committed to

the `articles of faith' of the scientific methods of research. This implies that a

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researcher should be a social science person in the truest sense. Sir Michael

Foster (Wilkinson and Bhandarkar, 1979) identified a few distinct qualities of a

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scientist. According to him, a true research scientist should possess the

following qualities:

(1) First of all, the nature of a researcher must be of the temperament that

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vibrates in unison with the theme which he is searching. Hence, the seeker of

knowledge must be truthful with truthfulness of nature, which is much more

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important, much more exacting than what is sometimes known as truthfulness.

The truthfulness relates to the desire for accuracy of observation and precision

of statement. Ensuring facts is the principle rule of science, which is not an easy

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matter. The difficulty may arise due to untrained eye, which fails to see

anything beyond what it has the power of seeing and sometimes even less than

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that. This may also be due to the lack of discipline in the method of science. An

unscientific individual often remains satisfied with the expressions like

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approximately, almost, or nearly, which is never what nature is. It cannot see

two things which differ, however minutely, as the same.

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(2) A researcher must possess an alert mind. Nature is constantly

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changing and revealing itself through various ways. A scientific researcher must

be keen and watchful to notice such changes, no matter how small or

insignificant they may appear. Such receptivity has to be cultivated slowly and

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patiently over time by the researcher through practice. An individual who is

ignorant or not alert and receptive during his research will not make a good

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researcher. He will fail as a good researcher if he has no keen eyes or mind to

observe the unusual behind the routine. Research demands a systematic

immersion into the subject matter for the researcher to be able to grasp even the

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slightest hint that may culminate into significant research problems. In this

context, Cohen and Negal (Selltiz et al, 1965; Wilkinson and Bhandarkar, 1979)

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state that "the ability to perceive in some brute experience the occasion of a

problem is not a common talent among men... It is a mark of scientific genius to

be sensitive to difficulties where less gifted people pass by untroubled by

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doubt".

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(3) Scientific enquiry is pre-eminently an intellectual effort. It requires

the moral quality of courage, which reflects the courage of a steadfast

endurance. The science of conducting research is not an easy task. There are

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occasions when a research scientist might feel defeated or completely lost. This

is the stage when a researcher would need immense courage and the sense of

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conviction. The researcher must learn the art of enduring intellectual hardships.

In the words of Darwin, "It's dogged that does it".

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In order to cultivate the afore-mentioned three qualities of a researcher, a

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fourth one may be added. This is the quality of making statements cautiously.

According to Huxley, the assertion that outstrips the evidence is not only a

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blunder but a crime (Thompson, 1975). A researcher should cultivate the habit

of reserving judgment when the required data are insufficient.

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1.1.7 Significance of Research:

According to a famous Hudson Maxim, "All progress is born of inquiry.

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Doubt is often better than overconfidence, for it leads to inquiry, and inquiry

leads to invention". It brings out the significance of research, increased amounts

of which make the progress possible. Research encourages scientific and

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inductive thinking, besides promoting the development of logical habits of

thinking and organisation. The role of research in applied economics in the

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context of an economy or business is greatly increasing in modern times. The

increasingly complex nature of government and business has raised the use of

research in solving operational problems. Research assumes significant role in

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the formulation of economic policy for both, the government and business. It

provides the basis for almost all government policies of an economic system.

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Government budget formulation, for example, depends particularly on the

analysis of needs and desires of people, and the availability of revenues, which

requires research. Research helps to formulate alternative policies, in addition

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to examining the consequences of these alternatives. Thus, research also

facilitates the decision-making of policy-makers, although in itself it is not a part

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of research. In the process, research also helps in the proper allocation of a

country's scarce resources.

Research is also necessary for collecting information on the social and

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economic structure of an economy to understand the process of change

occurring in the country. Collection of statistical information, though not a

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routine task, involves various research problems. Therefore, large staff of

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research technicians or experts is engaged by the government these days to

undertake this work. Thus, research as a tool of government economic policy

formulation involves three distinct stages of operation: (i) investigation of

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economic structure through continual compilation of facts; (ii) diagnosis of

events that are taking place and analysis of the forces underlying them; and (iii)

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the prognosis i.e., the prediction of future developments (Wilkinson and

Bhandarkar, 1979).

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Research also assumes a significant role in solving various operational

and planning problems associated with business and industry. In several ways,

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operations research, market research and motivational research are vital and

their results assist in taking business decisions. Market research refers to the

investigation of the structure and development of a market for the formulation of

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efficient policies relating to purchases, production and sales. Operational

research relates to the application of logical, mathematical, and analytical

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techniques to find solution to business problems, such as cost minimization or

profit maximization, or the optimization problems. Motivational research helps

to determine why people behave in the manner they do with respect to market

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characteristics. More specifically, it is concerned with the analysis of the

motivations underlying consumer behaviour. All these researches are very

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useful for business and industry, and are responsible for business decision-

making.

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Research is equally important to social scientists for analyzing the social

relationships and seeking explanations to various social problems. It gives

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intellectual satisfaction of knowing things for the sake of knowledge. It also

possesses the practical utility for the social scientist to gain knowledge so as to

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be able to do something better or in a more efficient manner. The research in

social sciences is concerned with both knowledge for its own sake, and

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knowledge for what it can contribute to solve practical problems.

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1.2 Research Process:

Research process consists of a series of steps or actions required for

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effectively conducting research. The following are the steps that provide useful

procedural guidelines regarding the conduct of research:

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(1) formulating the research problem;

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(2) extensive literature survey;

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(3) developing hypothesis;

(4) preparing the research design;

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(5)

determining

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sample

design;

(6)

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collecting

data;

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(7) execution of the project;

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(8) analysis of data;

(9)

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hypothesis

testing;

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(10) generalization and interpretation, and

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(11) preparation of the report or presentation of the results. In other

words, it involves the formal write-up of conclusions.

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1.3 Research Problem:

The first and foremost stage in the research process is to select and

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properly define the research problem. A researcher should first identify a

problem and formulate it, so as to make it amenable or susceptible to research.

In general, a research problem refers to an unanswered question that a researcher

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might encounter in the context of either a theoretical or practical situation,

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which he/she would like to answer or find a solution to. A research problem is

generally said to exist if the following conditions emerge (Kothari, 1988):

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(i) there should be an individual or an organisation, say X, to whom the

problem can be attributed. The individual or the organization is situated

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in an environment Y, which is governed by certain uncontrolled variables

Z;

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(ii)

there should be atleast two courses of action to be pursued, say A1 and

A2. These courses of action are defined by one or more values of the

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controlled variables. For example, the number of items purchased at a

specified time is said to be one course of action.

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(iii)

there should be atleast two alternative possible outcomes of the said

courses of action, say B1 and B2. Of them, one alternative should be

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preferable to the other. That is, atleast one outcome should be what the

researcher wants, which becomes an objective.

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(iv)

the courses of possible action available must offer a chance to the

researcher to achieve the objective, but not the equal chance. Therefore,

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if P(Bj / X, A, Y) represents the probability of the occurrence of an

outcome Bj when X selects Aj in Y, then P(B1 / X, A1,Y) P (B1 / X, A2,

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Y). Putting it in simple words, it means that the choices must not have

equal efficiencies for the desired outcome.

Above all these conditions, the individual or organisation may be said to have

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arrived at the research problem only if X does not know what course of action to

be taken is the best. In other words, X should have a doubt about the solution.

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Thus, an individual or a group of persons can be said to have a problem if they

have more than one desired outcome. They should have two or more alternative

courses of action, which have some but not equal efficiency. This is required for

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probing the desired objectives, such that they have doubts about the best course

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of action to be taken. Thus, the components of a research problem may be

summarised as:

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(i)

there should be an individual or a group who have some difficulty or

problem.

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(ii)

there should be some objective(s) to be pursued. A person or an

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organization who wants nothing cannot have a problem.

(iii)

there should be alternative ways of pursuing the objective the researcher

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wants to pursue. This implies that there should be more than one

alternative means available to the researcher. This is because if the

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researcher has no choice of alternative means, he/she would not have a

problem.

(iv)

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there should be some doubt in the mind of the researcher about the

choice of alternative means. This implies that research should answer

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the question relating to the relative efficiency or suitability of the

possible alternatives.

(v)

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there should be a context to which the difficulty relates.

Thus, identification of a research problem is the pre-condition to conducting

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research. A research problem is said to be the one which requires a researcher to

find the best available solution to the given problem. That is, the researcher

needs to find out the best course of action through which the research objective

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may be achieved optimally in the context of a given situation. Several factors

may contribute to making the problem complicated. For example, the

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environment may alter, thus affecting the efficiencies of the alternative courses

of action taken or the quality of the outcomes. The number of alternative courses

of action might be very large and the individual not involved in making the

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decision may be affected by the change in environment and may react to it

favorably or unfavorably. Other similar factors are also likely to cause such

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changes in the context of research, all of which may be considered from the

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point of view of a research problem.

1.4 Research Design:

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The most important step after defining the research problem is preparing the

design of the research project, which is popularly known as the `research

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design'. A research design helps to decide upon issues like what, when, where,

how much, by what means etc. with regard to an enquiry or a research study.

A research design is the arrangement of conditions for collection and analysis of

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data in a manner that aims to combine relevance to the research purpose with

economy in procedure. Infact, research design is the conceptual structure within

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which research is conducted; it constitutes the blueprint for the collection,

measurement and analysis of data (Selltiz et al, 1962). Thus, research design

provides an outline of what the researcher is going to do in terms of framing the

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hypothesis, its operational implications and the final data analysis. Specifically,

the research design highlights decisions which include:

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(i)

the nature of the study

(ii)

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the purpose of the study

(iii)

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the location where the study would be conducted

(iv)

the nature of data required

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(v)

from where the required data can be collected

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(vi)

what time period the study would cover

(vii) the type of sample design that would be used

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(viii) the techniques of data collection that would be used

(ix)

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the methods of data analysis that would be adopted and

(x)

the manner in which the report would be prepared

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In view of the stated research design decisions, the overall research

design may be divided into the following (Kothari 1988):

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(a)

the sampling design that deals with the method of selecting items to be

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observed for the selected study;

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(b)

the observational design that relates to the conditions under which the

observations are to be made;

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(c)

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the statistical design that concerns with the question of how many items are

to be observed, and how the information and data gathered are to be

analysed; and

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(d)

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the operational design that deals with the techniques by which the

procedures specified in the sampling, statistical and observational designs

can be carried out.

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1.4.1 Features of Research Design:

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The important features of research design may be outlined as follows:

(i) it constitutes a plan that identifies the types and sources of information

required for the research problem;

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(ii) it constitutes a strategy that specifies the methods of data collection and

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analysis which would be adopted; and

(iii) it also specifies the time period of research and monetary budget involved

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in conducting the study, which comprise the two major constraints of

undertaking any research.

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1.4.2

Concepts Relating to Research Design:

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Some of the important concepts relating to Research Design are

discussed below:

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1. Dependent and Independent Variables:

A magnitude that varies is known as a variable. The concept may

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assume different quantitative values like height, weight, income etc. Qualitative

variables are not quantifiable in the strictest sense of the term. However, the

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qualitative phenomena may also be quantified in terms of the presence or

absence of the attribute(s) considered. The phenomena that assume different

values quantitatively even in decimal points are known as `continuous

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variables'. But all variables need not be continuous. Values that can be

expressed only in integer values are called `non-continuous variables'. In

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statistical terms, they are also known as `discrete variables'. For example, age

is a continuous variable, whereas the number of children is a non-continuous

variable. When changes in one variable depend upon the changes in other

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variable or variables, it is known as a dependent or endogenous variable, and the

variables that cause the changes in the dependent variable are known as the

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independent or explanatory or exogenous variables. For example, if demand

depends upon price, then demand is a dependent variable, while price is the

independent variable. And, if more variables determine demand, like income

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and price of the substitute commodity, then demand also depends upon them in

addition to the price of original commodity. In other words, demand is a

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dependent variable which is determined by the independent variables like price

of the original commodity, income and price of substitutes.

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2 Extraneous Variable:

The independent variables which are not directly related to the purpose

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of the study but affect the dependent variable are known as extraneous variables.

For instance, assume that a researcher wants to test the hypothesis that there is a

relationship between children's school performance and their self-concepts, in

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which case the latter is an independent variable and the former, a dependent

variable. In this context, intelligence may also influence the school

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performance. However, since it is not directly related to the purpose of the

study undertaken by the researcher, it would be known as an extraneous

variable. The influence caused by the extraneous variable(s) on the dependent

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variable is technically called the `experimental error'. Therefore, a research

study should always be framed in such a manner that the influence of extraneous

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variables on the dependent variable/s is completely controlled, and the influence

of independent variable/s is clearly evident.

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3. Control:

One of the most important features of a good research design is to

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minimize the effect of extraneous variable(s). Technically, the term `control' is

used when a researcher designs the study in such a manner that it minimizes the

effects of extraneous variables. The term `control' is used in experimental

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research to reflect the restrain in experimental conditions.

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4. Confounded Relationship:

The relationship between the dependent and independent variables is

said to be confounded by an extraneous variable, when the dependent variable is

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not free from its effects.

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5. Research Hypothesis:

When a prediction or a hypothesized relationship is tested by adopting

scientific methods, it is known as research hypothesis. The research hypothesis

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is a predictive statement which relates to a dependent variable and an

independent variable. Generally, a research hypothesis must consist of at least

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one dependent variable and one independent variable. Whereas, the

relationships that are assumed but not to be tested are predictive statements that

are not to be objectively verified, thus are not classified as research hypotheses.

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6. Experimental and Non-experimental Hypothesis Testing Research:

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When the objective of a research is to test a research hypothesis, it is known as

hypothesis-testing research. Such research may be in the nature of experimental

design or non-experimental design. The research in which the independent

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variable is manipulated is known as `experimental hypothesis-testing research',

whereas the research in which the independent variable is not manipulated is

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termed as `non-experimental hypothesis-testing research'. For example, assume

that a researcher wants to examine whether family income influences the school

attendance of a group of students, by calculating the coefficient of correlation

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between the two variables. Such an example is known as a non-experimental

hypothesis-testing research, because the independent variable - family income is

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not manipulated here. Again assume that the researcher randomly selects 150

students from a group of students who pay their school fees regularly and then

classifies them into two sub-groups by randomly including 75 in Group A,

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whose parents have regular earning, and 75 in group B, whose parents do not

have regular earning. Assume that at the end of the study, the researcher

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conducts a test on each group in order to examine the effects of regular earnings

of the parents on the school attendance of the student. Such a study is an

example of experimental hypothesis-testing research, because in this particular

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study the independent variable regular earnings of the parents have been

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manipulated.

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7. Experimental and Control Groups:

When a group is exposed to usual conditions in an experimental

hypothesis-testing research, it is known as `control group'. On the other hand,

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when the group is exposed to certain new or special condition, it is known as an

`experimental group'. In the afore-mentioned example, Group A can be called

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as control group and Group B as experimental group. If both the groups, A and

B are exposed to some special feature, then both the groups may be called as

`experimental groups'. A research design may include only the experimental

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group or both the experimental and control groups together.

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8. Treatments:

Treatments refer to the different conditions to which the experimental

and control groups are subject to. In the example considered, the two treatments

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are the parents with regular earnings and those with no regular earnings.

Likewise, if a research study attempts to examine through an experiment the

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comparative effect of three different types of fertilizers on the yield of rice crop,

then the three types of fertilizers would be treated as the three treatments.

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9. Experiment:

Experiment refers to the process of verifying the truth of a statistical

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hypothesis relating to a given research problem. For instance, an experiment

may be conducted to examine the yield of a certain new variety of rice crop

developed. Further, Experiments may be categorized into two types, namely,

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`absolute experiment' and `comparative experiment'. If a researcher wishes to

determine the impact of a chemical fertilizer on the yield of a particular variety

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of rice crop, then it is known as absolute experiment. Meanwhile, if the

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researcher wishes to determine the impact of chemical fertilizer as compared to

the impact of bio-fertilizer, then the experiment is known as a comparative

experiment.

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10. Experimental Unit(s):

Experimental Units refer to the pre-determined plots, characteristics or

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the blocks, to which different treatments are applied. It is worth mentioning

here that such experimental units must be selected with great caution.

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1.4.3 Types of Research Design:

There are different types of research designs. They may be broadly categorized

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as:

(1) Exploratory Research Design;

(2) Descriptive and Diagnostic Research Design; and

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(3) Hypothesis-Testing Research Design.

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1. Exploratory Research Design:

The Exploratory Research Design is known as formulative research design.

The main objective of using such a research design is to formulate a research

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problem for an in-depth or more precise investigation, or for developing a

working hypothesis from an operational aspect. The major purpose of such

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studies is the discovery of ideas and insights. Therefore, such a research design

suitable for such a study should be flexible enough to provide opportunity for

considering different dimensions of the problem under study. The in-built

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flexibility in research design is required as the initial research problem would be

transformed into a more precise one in the exploratory study, which in turn may

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necessitate changes in the research procedure for collecting relevant data.

Usually, the following three methods are considered in the context of a research

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design for such studies. They are (a) a survey of related literature; (b)

experience survey; and (c) analysis of `insight-stimulating' instances.

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Descriptive and Diagnostic Research Design:

A Descriptive Research Design is concerned with describing the

characteristics of a particular individual or a group. Meanwhile, a diagnostic

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research design determines the frequency with which a variable occurs or its

relationship with another variable. In other words, the study analyzing whether

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a certain variable is associated with another comprises a diagnostic research

study. On the other hand, a study that is concerned with specific predictions or

with the narration of facts and characteristics related to an individual, group or

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situation, are instances of descriptive research studies. Generally, most of the

social research design falls under this category. As a research design, both the

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descriptive and diagnostic studies share common requirements, hence they are

grouped together. However, the procedure to be used and the research design

must be planned carefully. The research design must also make appropriate

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provision for protection against bias and thus maximize reliability, with due

regard to the completion of the research study in an economical manner. The

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research design in such studies should be rigid and not flexible. Besides, it must

also focus attention on the following:

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(a) formulation of the objectives of the study,

(b) proper designing of the methods of data collection ,

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(f) reporting the findings.

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3. Hypothesis-testing Research Design:

Hypothesis-testing Research Designs are those in which the researcher tests

the hypothesis of causal relationship between two or more variables. These

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studies require procedures that would not only decrease bias and enhance

reliability, but also facilitate deriving inferences about the causality. Generally,

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experiments satisfy such requirements. Hence, when research design is

discussed in such studies, it often refers to the design of experiments.

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1.4.4 Importance of Research Design:

The need for a research design arises out of the fact that it facilitates the

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smooth conduct of the various stages of research. It contributes to making

research as efficient as possible, thus yielding the maximum information with

minimum effort, time and expenditure. A research design helps to plan in

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advance, the methods to be employed for collecting the relevant data and the

techniques to be adopted for their analysis. This would help in pursuing the

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objectives of the research in the best possible manner, provided the available

staff, time and money are given. Hence, the research design should be prepared

with utmost care, so as to avoid any error that may disturb the entire project.

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Thus, research design plays a crucial role in attaining the reliability of the results

obtained, which forms the strong foundation of the entire process of the research

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work.

Despite its significance, the purpose of a well-planned design is not

realized at times. This is because it is not given the importance that it deserves.

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As a consequence, many researchers are not able to achieve the purpose for

which the research designs are formulated, due to which they end up arriving at

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misleading conclusions. Therefore, faulty designing of the research project

tends to render the research exercise meaningless. This makes it imperative that

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an efficient and suitable research design must be planned before commencing

the process of research. The research design helps the researcher to organize

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his/her ideas in a proper form, which in turn facilitates him/her to identify the

inadequacies and faults in them. The research design is also discussed with

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other experts for their comments and critical evaluation, without which it would

be difficult for any critic to provide a comprehensive review and comments on

the proposed study.

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1.4.5 Characteristics of a Good Research Design:

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A good research design often possesses the qualities of being flexible,

suitable, efficient, economical and so on. Generally, a research design which

minimizes bias and maximizes the reliability of the data collected and analysed

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is considered a good design (Kothari 1988). A research design which does not

allow even the smallest experimental error is said to be the best design for

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investigation. Further, a research design that yields maximum information and

provides an opportunity of viewing the various dimensions of a research

problem is considered to be the most appropriate and efficient design. Thus, the

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question of a good design relates to the purpose or objective and nature of the

research problem studied. While a research design may be good, it may not be

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equally suitable to all studies. In other words, it may be lacking in one aspect or

the other in the case of some other research problems. Therefore, no single

research design can be applied to all types of research problems.

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A research design suitable for a specific research problem would usually

involve the following considerations:

(i) the methods of gathering the information;

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(ii) the skills and availability of the researcher and his/her staff, if any;

(iii) the objectives of the research problem being studied;

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(iv) the nature of the research problem being studied; and

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(v) the available monetary support and duration of time for the research

work.

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1.5 Case Study Research:

The method of exploring and analyzing the life or functioning of a social

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or economic unit, such as a person, a family, a community, an institution, a firm

or an industry is called case study method. The objective of case study method

is to examine the factors that cause the behavioural patterns of a given unit and

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its relationship with the environment. The data for a study are always gathered

with the purpose of tracing the natural history of a social or economic unit, and

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its relationship with the social or economic factors, besides the forces involved

in its environment. Thus, a researcher conducting a study using the case study

method attempts to understand the complexity of factors that are operative

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within a social or economic unit as an integrated totality. Burgess (Kothari,

1988) described the special significance of the case study in understanding the

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complex behaviour and situations in specific detail. In the context of social

research, he called such data as social microscope.

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1.5.1 Criteria for Evaluating Adequacy of Case Study:

John Dollard (Dollard, 1935) specified seven criteria for evaluating the

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adequacy of a case or life history in the context of social research. They are:

(i)

The subject being studied must be viewed as a specimen in a cultural set

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up. That is, the case selected from its total context for the purpose of study

should be considered a member of the particular cultural group or community.

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The scrutiny of the life history of the individual must be carried out with a view

to identify the community values, standards and shared ways of life.

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(ii) The organic motors of action should be socially relevant. This is to say

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that the action of the individual cases should be viewed as a series of

reactions to social stimuli or situations. To Put in simple words, the social

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meaning of behaviour should be taken into consideration.

(iii) The crucial role of the family-group in transmitting the culture should be

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recognized. This means, as an individual is the member of a family, the

role of the family in shaping his/her behaviour should never be ignored.

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(iv) The specific method of conversion of organic material into social

behaviour should be clearly demonstrated. For instance, case-histories that

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discuss in detail how basically a biological organism, that is man,

gradually transforms into a social person are particularly important.

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(v) The constant transformation of character of experience from childhood to

adulthood should be emphasized. That is, the life-history should portray

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the inter-relationship between the individual's various experiences during

his/her life span. Such a study provides a comprehensive understanding of

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an individual's life as a continuum.

(vi) The `social situation' that contributed to the individual's gradual

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transformation should carefully and continuously be specified as a factor.

One of the crucial criteria for life-history is that an individual's life should

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be depicted as evolving itself in the context of a specific social situation

and partially caused by it.

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(vii) The life-history details themselves should be organized according to some

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conceptual framework, which in turn would facilitate their generalizations

at higher levels.

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These criteria discussed by Dollard emphasize the specific link of co-

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ordinated, related, continuous and configured experience in a cultural pattern

that motivated the social and personal behaviour. Although, the criteria

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indicated by Dollard are principally perfect, some of them are difficult to put to

practice.

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Dollard (1935) attempted to express the diverse events depicted in the

life-histories of persons during the course of repeated interviews by utilizing

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psycho-analytical techniques in a given situational context. His criteria of life-

history originated directly from this experience. While the life-histories possess

independent significance as research documents, the interviews recorded by the

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investigators can afford, as Dollard observed, "rich insights into the nature of the

social situations experienced by them".

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It is a well-known fact that an individual's life is very complex. Till date

there is hardly any technique that can establish some kind of uniformity, and as

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a result ensure the cumulative of case-history materials by isolating the complex

totality of a human life. Nevertheless, although case history data are difficult to

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put to rigorous analysis, a skilful handling and interpretation of such data could

help in developing insights into cultural conflicts and problems arising out of

cultural-change.

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Gordon Allport (Kothari 1988) has recommended the following aspects

so as to broaden the perspective of case-study data:

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(i) if the life-history is written in first person, it should be as comprehensive

and coherent as possible.

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(ii) Life-histories must be written for knowledgeable persons. That is, if the

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enquiry of study is sociological in nature, the researcher should write it on

the assumption that it would be read largely by sociologists only.

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(iii) It would be advisable to supplement case study data by observational,

statistical and historical data, as they provide standards for assessing the

reliability and consistency of the case study materials. Further, such data

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offer a basis for generalizations.

(iv) Efforts must be made to verify the reliability of life-history data by

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examining the internal consistency of the collected material, and by

repeating the interviews with the concerned person. Besides this, personal

interviews with the persons who are well-acquainted with him/her,

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belonging to his/her own group should be conducted.

(v) A judicious combination of different techniques for data-collection is

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crucial for collecting data that are culturally meaningful and scientifically

significant.

(vi) Life-histories or case-histories may be considered as an adequate basis for

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generalization to the extent that they are typical or representative of a

certain group.

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(vii) The researcher engaged in the collection of case study data should never

ignore the unique or typical cases. He/she should include them as

exceptional cases.

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Case histories are filled with valuable information of a personal or

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private nature. Such information not only helps the researcher to portray the

personality of the individual, but also the social background that contributed to

it. Besides, it also helps in the formulation of relevant hypotheses. In general,

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although Blummer (in Wilkinson and Bhandarkar, 1979) was critical of

documentary material, he gave due credit to case histories by acknowledging the

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fact that the personal documents offer an opportunity to the researcher to

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develop his/her spirit of enquiry. The analysis of a particular subject would be

more effective if the researcher acquires close acquaintance with it through

personal documents. However, Blummer also acknowledges the limitations of

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the personal documents. According to him, such documents do not entirely

fulfill the criteria of adequacy, reliability, and representativeness. Despite these

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shortcomings, avoiding their use in any scientific study of personal life would be

wrong, as these documents become necessary and significant for both theory-

building and practice.

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In spite of these formidable limitations, case study data are used by

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anthropologists, sociologists, economists and industrial psychiatrists. Gordon

Allport (Kothari, 1988) strongly recommends the use of case study data for in-

depth analysis of a subject. For, it is one's acquaintance with an individual that

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instills a desire to know his/her nature and understand them. The first stage

involves understanding the individual and all the complexity of his/her nature.

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Any haste in analyzing and classifying the individual would create the risk of

reducing his/her emotional world into artificial bits. As a consequence, the

important emotional organizations, anchorages and natural identifications

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characterizing the personal life of the individual might not yield adequate

representation. Hence, the researcher should understand the life of the subject.

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Therefore, the totality of life-processes reflected in the well-ordered life-history

documents become invaluable source of stimulating insights. Such life-history

documents provide the basis for comparisons that contribute to statistical

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generalizations and help to draw inferences regarding the uniformities in human

behaviour, which are of great value. Even if some personal documents do not

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provide ordered data about personal lives of people, which is the basis of

psychological science, they should not be ignored. This is because the final aim

of science is to understand, control and make predictions about human life. Once

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they are satisfied, the theoretical and practical importance of personal

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documents must be recognized as significant. Thus, a case study may be

considered as the beginning and the final destination of abstract knowledge.

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1.6 Hypothesis:

"Hypothesis may be defined as a proposition or a set of propositions set

forth as an explanation for the occurrence of some specified group of

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phenomena either asserted merely as a provisional conjecture to guide some

investigation in the light of established facts" (Kothari, 1988). A research

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hypothesis is quite often a predictive statement, which is capable of being tested

using scientific methods that involve an independent and some dependent

variables. For instance, the following statements may be considered:

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i) "students who take tuitions perform better than the others who do not receive

tuitions" or,

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ii) "the female students perform as well as the male students".

These two statements are hypotheses that can be objectively verified and tested.

Thus, they indicate that a hypothesis states what one is looking for. Besides, it

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is a proposition that can be put to test in order to examine its validity.

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1.6.1 Characteristics of Hypothesis:

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A hypothesis should have the following characteristic features:-

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(i) A hypothesis must be precise and clear. If it is not precise and clear, then

the inferences drawn on its basis would not be reliable.

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(ii) A hypothesis must be capable of being put to test. Quite often, the

research programmes fail owing to its incapability of being subject to

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testing for validity. Therefore, some prior study may be conducted by the

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researcher in order to make a hypothesis testable. A hypothesis "is tested

if other deductions can be made from it, which in turn can be confirmed or

disproved by observation" (Kothari, 1988).

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(iii) A hypothesis must state relationship between two variables, in the case of

relational hypotheses.

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(iv) A hypothesis must be specific and limited in scope. This is because a

simpler hypothesis generally would be easier to test for the researcher.

And therefore, he/she must formulate such hypotheses.

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(v) As far as possible, a hypothesis must be stated in the simplest language, so

as to make it understood by all concerned. However, it should be noted

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that simplicity of a hypothesis is not related to its significance.

(vi) A hypothesis must be consistent and derived from the most known facts.

In other words, it should be consistent with a substantial body of

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established facts. That is, it must be in the form of a statement which

Judges accept as being the most likely to occur.

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(vii) A hypothesis must be amenable to testing within a stipulated or reasonable

period of time. No matter how excellent a hypothesis, a researcher should

not use it if it cannot be tested within a given period of time, as no one can

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afford to spend a life-time on collecting data to test it.

(viii) A hypothesis should state the facts that give rise to the necessity of looking

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for an explanation. This is to say that by using the hypothesis, and other

known and accepted generalizations, a researcher must be able to derive

the original problem condition. Therefore, a hypothesis should explain

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what it actually wants to explain, and for this it should also have an

empirical reference.

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1.6.2

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Concepts Relating to Testing of Hypotheses:

Testing of hypotheses requires a researcher to be familiar with various

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concepts concerned with it such as:

1)

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Null Hypothesis and Alternative Hypothesis:

In the context of statistical analysis, hypothesis is of two types viz., null

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hypothesis and alternative hypothesis. When two methods A and B are

compared on their relative superiority, and it is assumed that both the methods

are equally good, then such a statement is called as the null hypothesis. On the

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other hand, if method A is considered relatively superior to method B, or vice-

versa, then such a statement is known as an alternative hypothesis. The null

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hypothesis is expressed as H0, while the alternative hypothesis is expressed as

Ha. For example, if a researcher wants to test the hypothesis that the population

mean (?) is equal to the hypothesized mean (H0) = 100, then the null hypothesis

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should be stated as the population mean is equal to the hypothesized mean 100.

Symbolically it may be written as:-

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H0: = ? = ? H0 = 100

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If sample results do not support this null hypothesis, then it should be

concluded that something else is true. The conclusion of rejecting the null

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hypothesis is called as alternative hypothesis. To put it in simple words, the set

of alternatives to the null hypothesis is termed as the alternative hypothesis. If

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H0 is accepted, then it implies that Ha is being rejected. On the other hand, if H0

is rejected, it means that Ha is being accepted. For H0: ? = ? H0 = 100, the

following three possible alternative hypotheses may be considered:

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Alternative hypothesis

to be read as follows

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the alternative hypothesis is that the

Ha: ? ? H0

population mean is not equal to 100,

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i.e., it could be greater than or less

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than 100

the alternative hypothesis is that the

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Ha : ? > ? H0

population mean is greater than 100

the alternative hypothesis is that the

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Ha : ? < ? H0

population mean is less than 100

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Before the sample is drawn, the researcher has to state the null

hypothesis and the alternative hypothesis. While formulating the null

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hypothesis, the following aspects need to be considered:

(a) Alternative hypothesis is usually the one which a researcher wishes to prove,

whereas the null hypothesis is the one which he/she wishes to disprove. Thus, a

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null hypothesis is usually the one which a researcher tries to reject, while an

alternative hypothesis is the one that represents all other possibilities.

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(b) The rejection of a hypothesis when it is actually true involves great risk, as it

indicates that it is a null hypothesis because then the probability of rejecting it

when it is true is (i.e., the level of significance) which is chosen very small.

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(c) Null hypothesis should always be specific hypothesis i.e., it should not state

about or approximately a certain value.

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(2) The Level of Significance:

In the context of hypothesis testing, the level of significance is a very

important concept. It is a certain percentage that should be chosen with great

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care, reason and thought. If for instance, the significance level is taken at 5 per

cent, then it means that H0 would be rejected when the sampling result has a less

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than 0.05 probability of occurrence when H0 is true. In other words, the five per

cent level of significance implies that the researcher is willing to take a risk of

five per cent of rejecting the null hypothesis, when (H0) is actually true. In sum,

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the significance level reflects the maximum value of the probability of rejecting

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H0 when it is actually true, and which is usually determined prior to testing the

hypothesis.

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(3) Test of Hypothesis or Decision Rule:

Suppose the given hypothesis is H0 and the alternative hypothesis Ha,

then the researcher has to make a rule known as the decision rule. According to

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the decision rule, the researcher accepts or rejects H0. For example, if the H0 is

that certain students are good against the Ha that all the students are good, then

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the researcher should decide the number of items to be tested and the criteria on

the basis of which to accept or reject the hypothesis.

(4) Type I and Type II Errors:

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As regards the testing of hypotheses, a researcher can make basically two

types of errors. He/she may reject H0 when it is true, or accept H0 when it is

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not true. The former is called as Type I error and the latter is known as Type II

error. In other words, Type I error implies the rejection of a hypothesis when it

must have been accepted, while Type II error implies the acceptance of a

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hypothesis which must have been rejected. Type I error is denoted by (alpha)

and is known as error, while Type II error is usually denoted by (beta) and is

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known as error.

(5) One-tailed and two-tailed Tests:

These two types of tests are very important in the context of hypothesis

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testing. A two-tailed test rejects the null hypothesis, when the sample mean is

significantly greater or lower than the hypothesized value of the mean of the

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population. Such a test is suitable when the null hypothesis is some specified

value, the alternative hypothesis is a value that is not equal to the specified value

of the null hypothesis.

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1.6.3 Procedure of Hypothesis Testing:

Testing a hypothesis refers to verifying whether the hypothesis is valid

or not. Hypothesis testing attempts to check whether to accept or not to accept

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the null hypothesis. The procedure of hypothesis testing includes all the steps

that a researcher undertakes for making a choice between the two alternative

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actions of rejecting or accepting a null hypothesis. The various steps involved in

hypothesis testing are as follows:

(i)

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Making a Formal Statement:

This step involves making a formal statement of the null hypothesis (H0)

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and the alternative hypothesis (Ha). This implies that the hypotheses should be

clearly stated within the purview of the research problem. For example, suppose

a school teacher wants to test the understanding capacity of the students which

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must be rated more than 90 per cent in terms of marks, the hypotheses may be

stated as follows:

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Null Hypothesis H0 : = 100

Alternative Hypothesis Ha : > 100

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(ii)

Selecting a Significance Level:

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The hypotheses should be tested on a pre-determined level of

significance, which should be specified. Usually, either 5% level or 1% level is

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considered for the purpose. The factors that determine the levels of significance

are: (a) the magnitude of difference between the sample means; (b) the sample

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size: (c) the variability of measurements within samples; and (d) whether the

hypothesis is directional or non-directional (Kothari, 1988). In sum, the level of

significance should be sufficient in the context of the nature and purpose of

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enquiry.

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(iii)

Deciding the Distribution to Use:

After making decision on the level of significance for hypothesis testing,

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the researcher has to next determine the appropriate sampling distribution. The

choice to be made generally relates to normal distribution and the t-distribution.

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The rules governing the selection of the correct distribution are similar to the

ones already discussed with respect to estimation.

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(iv)

Selection of a Random Sample and Computing an Appropriate

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Value:

Another step involved in hypothesis testing is the selection of a random

sample and then computing a suitable value from the sample data relating to test

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statistic by using the appropriate distribution. In other words, it involves

drawing a sample for furnishing empirical data.

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(v)

Calculation of the Probability:

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The next step for the researcher is to calculate the probability that the

sample result would diverge as far as it can from expectations, under the

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situation when the null hypothesis is actually true.

(vi) Comparing the Probability:

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Another step involved consists of making a comparison of the

probability calculated with the specified value for , the significance level. If

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the calculated probability works out to be equal to or smaller than the value in

case of one-tailed test, then the null hypothesis is to be rejected. On the other

hand, if the calculated probability is greater, then the null hypothesis is to be

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accepted. In case the null hypothesis H0 is rejected, the researcher runs the risk

of committing the Type I error. But, if the null hypothesis H0 is accepted, then it

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involves some risk (which cannot be specified in size as long as H0 is vague and

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not specific) of committing the Type II error.

1.7 Sample Survey:

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A sample design is a definite plan for obtaining a sample from a given

population (Kothari, 1988). Sample constitutes a certain portion of the

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population or universe. Sampling design refers to the technique or the

procedure the researcher adopts for selecting items for the sample from the

population or universe. A sample design helps to decide the number of items to

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be included in the sample, i.e., the size of the sample. The sample design should

be determined prior to data collection. There are different kinds of sample

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designs which a researcher can choose. Some of them are relatively more

precise and easier to adopt than the others. A researcher should prepare or select

a sample design, which must be reliable and suitable for the research study

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proposed to be undertaken.

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1.8.1 Steps in Sampling Design:

A researcher should take into consideration the following aspects while

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developing a sample design:

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(i) Type of universe:

The first step involved in developing sample design is to clearly define the

number of cases, technically known as the Universe, to be studied. A universe

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may be finite or infinite. In a finite universe the number of items is certain,

whereas in the case of an infinite universe the number of items is infinite (i.e.,

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there is no idea about the total number of items). For example, while the

population of a city or the number of workers in a factory comprise finite

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universes, the number of stars in the sky, or throwing of a dice represent infinite

universe.

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(ii) Sampling Unit:

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Prior to selecting a sample, decision has to be made about the sampling unit. A

sampling unit may be a geographical area like a state, district, village, etc., or a

social unit like a family, religious community, school, etc., or it may also be an

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individual. At times, the researcher would have to choose one or more of such

units for his/her study.

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(iii) Source List:

Source list is also known as the `sampling frame', from which the sample is to

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be selected. The source list consists of names of all the items of a universe. The

researcher has to prepare a source list when it is not available. The source list

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must be reliable, comprehensive, correct, and appropriate. It is important that

the source list should be as representative of the population as possible.

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(iv) Size of Sample:

Size of the sample refers to the number of items to be chosen from the universe

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to form a sample. For a researcher, this constitutes a major problem. The size of

sample must be optimum. An optimum sample may be defined as the one that

satisfies the requirements of representativeness, flexibility, efficiency, and

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reliability. While deciding the size of sample, a researcher should determine the

desired precision and the acceptable confidence level for the estimate. The size

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of the population variance should be considered, because in the case of a larger

variance generally a larger sample is required. The size of the population should

be considered, as it also limits the sample size. The parameters of interest in a

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research study should also be considered, while deciding the sample size.

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Besides, costs or budgetary constraint also plays a crucial role in deciding the

sample size.

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(a) Parameters of Interest:

The specific population parameters of interest should also be considered

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while determining the sample design. For example, the researcher may want to

make an estimate of the proportion of persons with certain characteristic in the

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population, or may be interested in knowing some average regarding the

population. The population may also consist of important sub-groups about

whom the researcher would like to make estimates. All such factors have strong

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impact on the sample design the researcher selects.

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(b) Budgetary Constraint:

From the practical point of view, cost considerations exercise a major

influence on the decisions related to not only the sample size, but also on the

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type of sample selected. Thus, budgetary constraint could also lead to the

adoption of a non-probability sample design.

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to be adopted for selecting the items for a sample. This technique or procedure

itself may represent the sample design. There are different sample designs from

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which a researcher should select one for his/her study. It is clear that the

researcher should select that design which, for a given sample size and budget

constraint, involves a smaller error.

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1.7.2 Criteria for Selecting a Sampling Procedure:

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Basically, two costs are involved in a sampling analysis, which govern

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the selection of a sampling procedure. They are:

(i) the cost of data collection, and

(ii) the cost of drawing incorrect inference from the selected data.

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There are two causes of incorrect inferences, namely systematic bias and

sampling error. Systematic bias arises out of errors in the sampling procedure.

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They cannot be reduced or eliminated by increasing the sample size. Utmost,

the causes of these errors can be identified and corrected. Generally, a

systematic bias arises out of one or more of the following factors:

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a. inappropriate sampling frame,

b. defective measuring device,

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c. non-respondents,

d. indeterminacy principle, and

e. natural bias in the reporting of data.

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Sampling error refers to the random variations in the sample estimates

around the true population parameters. Because they occur randomly and likely

to be equally in either direction, they are of compensatory type, the expected

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value of which errors tend to be equal to zero. Sampling error tends to decrease

with the increase in the size of the sample. It also becomes smaller in magnitude

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when the population is homogenous.

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Sampling error can be computed for a given sample size and design. The

measurement of sampling error is known as `precision of the sampling plan'.

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When the sample size is increased, the precision can be improved. However,

increasing the sample size has its own limitations. The large sized sample not

only increases the cost of data collection, but also increases the systematic bias.

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Thus, an effective way of increasing the precision is generally to choose a better

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sampling design, which has smaller sampling error for a given sample size at a

specified cost. In practice, however, researchers generally prefer a less precise

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design owing to the ease in adopting the same, in addition to the fact that

systematic bias can be controlled better way in such designs.

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In sum, while selecting the sample, a researcher should ensure that the

procedure adopted involves a relatively smaller sampling error and helps to

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control systematic bias.

1.7.3 Characteristics of a Good Sample Design:

The following are the characteristic features of a good sample design:

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(a)

the sample design should yield a truly representative sample;

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(b)

the sample design should be such that it results in small sampling error;

(c)

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the sample design should be viable in the context of budgetary

constraints of the research study;

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(d)

the sample design should be such that the systematic bias can be

controlled; and

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(e)

the sample must be such that the results of the sample study would be

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applicable, in general, to the universe at a reasonable level of confidence.

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1.7.4 Different Types of Sample Designs:

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Sample designs may be classified into different categories based on two

factors, namely, the representation basis and the element selection technique.

Under the representation basis, the sample may be classified as:

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I.

non-probability sampling

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II.

probability sampling

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While probability sampling is based on random selection, the non-

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probability sampling is based on `non-random' sampling.

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I. Non-Probability Sampling:

Non-probability sampling is the sampling procedure that does not afford any

basis for estimating the probability that each item in the population would have

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an equal chance of being included in the sample. Non-probability sampling is

also known as deliberate sampling, judgment sampling and purposive sampling.

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Under this type of sampling, the items for the sample are deliberately chosen by

the researcher; and his/her choice concerning the choice of items remains

supreme. In other words, under non-probability sampling the researchers select

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a particular unit of the universe for forming a sample on the basis that the small

number that is thus selected out of a huge one would be typical or representative

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of the whole population. For example, to study the economic conditions of

people living in a state, a few towns or village may be purposively selected for

an intensive study based on the principle that they are representative of the

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entire state. In such a case, the judgment of the researcher of the study assumes

prime importance in this sampling design.

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Quota Sampling:

Quota sampling is also an example of non-probability sampling. Under

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this sampling, the researchers simply assume quotas to be filled from different

strata, with certain restrictions imposed on how they should be selected. This

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type of sampling is very convenient and is relatively less expensive. However,

the samples selected using this method certainly do not satisfy the characteristics

of random samples. They are essentially judgment samples and inferences

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drawn based on that would not be amenable to statistical treatment in a formal

way.

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II.

Probability Sampling:

Probability sampling is also known as `choice sampling' or `random sampling'.

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Under this sampling design, every item of the universe has an equal chance of

being included in the sample. In a way, it is a lottery method under which

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individual units are selected from the whole group, not deliberately, but by using

some mechanical process. Therefore, only chance would determine whether an

item or the other would be included in the sample or not. The results obtained

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from probability or random sampling would be assured in terms of probability.

That is, the researcher can measure the errors of estimation or the significance of

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results obtained from the random sample. This is the superiority of random

sampling design over the deliberate sampling design. Random sampling

satisfies the law of Statistical Regularity, according to which if on an average

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the sample chosen is random, then it would have the same composition and

characteristics of the universe. This is the reason why the random sampling

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method is considered the best technique of choosing a representative sample.

The following are the implications of the random sampling:

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(i) it provides each element in the population an equal probability chance of

being chosen in the sample, with all choices being independent of one another

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and

(ii) it offers each possible sample combination an equal probability

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opportunity of being selected.

1.7.5 Method of Selecting a Random Sample:

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The process of selecting a random sample involves writing the name of

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each element of a finite population on a slip of paper and putting them into a box

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or a bag. Then they have to be thoroughly mixed and then the required number

of slips for the sample should be picked one after the other without replacement.

While doing this, it has to be ensured that in successive drawings each of the

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remaining elements of the population has an equal chance of being chosen. This

method results in the same probability for each possible sample.

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1.7.6 Complex random sampling designs:

Under restricted sampling technique, the probability sampling may result in

complex random sampling designs. Such designs are known as mixed sampling

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designs. Many of such designs may represent a combination of non-probability

and probability sampling procedures in choosing a sample.

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Some of the prominent complex random sampling designs are as follows:

(i) Systematic sampling: In some cases, the best way of sampling is to select

every first item on a list. Sampling of this kind is called as systematic sampling.

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An element of randomness is introduced in this type of sampling by using

random numbers to select the unit with which to start. For example, if a 10 per

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cent sample is required, the first item would be selected randomly from the first

and thereafter every 10th item. In this kind of sampling, only the first unit is

selected randomly, while rest of the units of the sample is chosen at fixed

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intervals.

(ii) Stratified Sampling: When a population from which a sample is to be

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selected does not comprise a homogeneous group, stratified sampling technique

is generally employed for obtaining a representative sample. Under stratified

sampling, the population is divided into many sub-populations in such a manner

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that they are individually more homogeneous than the rest of the total

population. Then, items are selected from each stratum to form a sample. As

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each stratum is more homogeneous than the remaining total population, the

researcher is able to obtain a more precise estimate for each stratum and by

estimating each of the component parts more accurately, he/she is able to obtain

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a better estimate of the whole. In sum, stratified sampling method yields more

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reliable and detailed information.

(iii) Cluster Sampling: When the total area of research interest is large, a

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convenient way in which a sample can be selected is to divide the area into a

number of smaller non-overlapping areas and then randomly selecting a number

of such smaller areas. In the process, the ultimate sample would consist of all

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the units in these small areas or clusters. Thus in cluster sampling, the total

population is sub-divided into numerous relatively smaller subdivisions, which

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in themselves constitute clusters of still smaller units. And then, some of such

clusters are randomly chosen for inclusion in the overall sample.

(iv) Area Sampling: When clusters are in the form of some geographic

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subdivisions, then cluster sampling is termed as area sampling. That is, when

the primary sampling unit represents a cluster of units based on geographic area,

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the cluster designs are distinguished as area sampling. The merits and demerits

of cluster sampling are equally applicable to area sampling.

(v) Multi-stage Sampling: A further development of the principle of cluster

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sampling is multi-stage sampling. When the researcher desires to investigate the

working efficiency of nationalized banks in India and a sample of few banks is

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required for this purpose, the first stage would be to select large primary

sampling unit like the states in the country. Next, certain districts may be

selected and all banks interviewed in the chosen districts. This represents a two-

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stage sampling design, with the ultimate sampling units being clusters of

districts.

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On the other hand, if instead of taking census of all banks within the

selected districts, the researcher chooses certain towns and interviews all banks

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in it, this would represent three-stage sampling design. Again, if instead of

taking a census of all banks within the selected towns, the researcher randomly

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selects sample banks from each selected town, then it represents a case of using

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a four-stage sampling plan. Thus, if the researcher selects randomly at all

stages, then it is called as multi-stage random sampling design.

(vi) Sampling with Probability Proportional to Size: When the case of cluster

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sampling units does not have exactly or approximately the same number of

elements, it is better for the researcher to adopt a random selection process,

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where the probability of inclusion of each cluster in the sample tends to be

proportional to the size of the cluster. For this, the number of elements in each

cluster has to be listed, irrespective of the method used for ordering it. Then the

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researcher should systematically pick the required number of elements from the

cumulative totals. The actual numbers thus chosen would not however reflect

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the individual elements, but would indicate as to which cluster and how many

from them are to be chosen by using simple random sampling or systematic

sampling. The outcome of such sampling is equivalent to that of simple random

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sample. The method is also less cumbersome and is also relatively less

expensive.

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Thus, a researcher has to pass through various stages of conducting

research once the problem of interest has been selected. Research methodology

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familiarizes a researcher with the complex scientific methods of conducting

research, which yield reliable results that are useful to policy-makers,

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government, industries etc. in decision-making.

References:

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Claire Sellitiz and others, Research Methods in Social Sciences, 1962, p.50

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Dollard,J., Criteria for the Life-history, Yale University Press, New York,1935,

pp.8-31.

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C.R. Kothari, Research Methodology, Methods and Techniques, Wiley Eastern

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Limited, New Delhi, 1988.

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Marie Jahoda, Morton Deutsch and Staurt W. Cook, Research Methods in

Social Relations, p.4.

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Pauline V. Young, Scientific Social Surveys and Research, p.30

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L.V. Redman and A.V.H. Mory, The Romance of Research, 1923.

The Encylopaedia of Social Sciences, Vol. IX, MacMillan, 1930.

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T.S. Wilkinson and P.L. Bhandarkar, Methodology and Techniques of Social

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Research, Himalaya Publishing House, Bombay, 1979.

Questions:

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1. Define research.

2. What are the objectives of research?

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3. State the significance of research.

4. What is the importance of knowing how to do research?

5. Briefly outline research process.

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6. Highlight the different research approaches.

7. Discuss the qualities of a researcher.

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8. Explain the different types of research.

9. What is a research problem?

10. Outline the features of research design.

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11. Discuss the features of a good research design.

12. Describe the different types of research design.

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13. Explain the significance of research design.

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14. What is a case study?

15. Discuss the criteria for evaluating case study.

16. Define hypothesis.

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17. What are the characteristic features of a hypothesis?

18. Distinguish between null and alternative hypothesis.

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19. Differentiate Type I error and Type II error.

20. How is a hypothesis tested?

21. Define the concept of sampling design.

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22. Describe the steps involved in sampling design.

23. Discuss the criteria for selecting a sampling procedure.

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24. Distinguish between probability and non-probability sampling.

25. How is a random sample selected?

26. Explain complex random sampling designs.

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***

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UNIT--II

DATA COLLECTION

1. SOURCES OF DATA

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Lesson Outline:

Primary data

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investigation

Indirect oral Methods of collecting primary data

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Direct personal interviews

Information received through local

agencies

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Mailed questionnaire method

Schedules sent through enumerators

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Learning Objectives:

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After reading this lesson, you should be able to

? Understand the meaning of primary data
? Preliminaries of data collection

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? Method of data collection
? Methods of collecting primary data
? Usefulness of primary data
? Merits and demerits of different methods of primary data collection
? Precautions while collecting primary data.

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Introduction:

It is important for a researcher to know the sources of data which he

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requires for different purposes. Data are nothing but the information. There are

two sources of information or data - Primary data and Secondary data. Primary

data refers to the data collected for the first time, whereas secondary data refers

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to the data that have already been collected and used earlier by somebody or

some agency. For example, the statistics collected by the Government of India

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relating to the population is primary data for the Government of India since it

has been collected for the first time. Later when the same data are used by a

researcher for his study of a particular problem, then the same data become the

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secondary data for the researcher. Both the sources of information have their

merits and demerits. The selection of a particular source depends upon the (a)

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purpose and scope of enquiry, (b) availability of time, (c) availability of finance,

(d) accuracy required, (e) statistical tools to be used, (f) sources of information

(data), and (g) method of data collection.

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(a)

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Purpose and Scope of Enquiry: The purpose and scope of data

collection or survey should be clearly set out at the very beginning. It requires

the clear statement of the problem indicating the type of information which is

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needed and the use for which it is needed. If for example, the researcher is

interested in knowing the nature of price change over a period of time, it would

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be necessary to collect data of commodity prices. It must be decided whether it

would be helpful to study wholesale or retail prices and the possible uses to

which such information could be put. The objective of an enquiry may be either

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to collect specific information relating to a problem or adequate data to test a

hypothesis. Failure to set out clearly the purpose of enquiry is bound to lead to

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confusion and waste of resources.

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After the purpose of enquiry has been clearly defined, the next step is to

decide about the scope of the enquiry. Scope of the enquiry means the coverage

with regard to the type of information, the subject-matter and geographical area.

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For instance, an enquiry may relate to India as a whole or a state or an industrial

town wherein a particular problem related to a particular industry can be studied.

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(b)Availability of Time: - The investigation should be carried out within a

reasonable period of time, failing which the information collected may become

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outdated, and would have no meaning at all. For instance, if a producer wants to

know the expected demand for a product newly launched by him and the result

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of the enquiry that the demand would be meager takes two years to reach him,

then the whole purpose of enquiry would become useless because by that time

he would have already incurred a huge loss. Thus, in this respect the information

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is quickly required and hence the researcher has to choose the type of enquiry

accordingly.

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(c) Availability of Resources: The investigation will greatly depend on the

resources available like number of skilled personnel, the financial position etc. If

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the number of skilled personnel who will carry out the enquiry is quite sufficient

and the availability of funds is not a problem, then enquiry can be conducted

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over a big area covering a good number of samples, otherwise a small sample

size will do.

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(d)The Degree of Accuracy Desired: Deciding the degree of accuracy required

is a must for the investigator, because absolute accuracy in statistical work is

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seldom achieved. This is so because (i) statistics are based on estimates, (ii)

tools of measurement are not always perfect and (iii) there may be unintentional

bias on the part of the investigator, enumerator or informant. Therefore, a desire

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of 100% accuracy is bound to remain unfulfilled. Degree of accuracy desired

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primarily depends upon the object of enquiry. For example, when we buy gold,

even a difference of 1/10th gram in its weight is significant, whereas the same

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will not be the case when we buy rice or wheat. However, the researcher must

aim at attaining a higher degree of accuracy, otherwise the whole purpose of

research would become meaningless.

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(e) Statistical Tools to be used: A well defined and identifiable object or a

group of objects with which the measurements or counts in any statistical

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investigation are associated is called a statistical unit. For example, in socio-

economic survey the unit may be an individual, a family, a household or a block

of locality. A very important step before the collection of data begins is to define

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clearly the statistical units on which the data are to be collected. In number of

situations the units are conventionally fixed like the physical units of

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measurement, such as meters, kilometers, quintals, hours, days, weeks etc.,

which are well defined and do not need any elaboration or explanation.

However, in many statistical investigations, particularly relating to socio-

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economic studies, arbitrary units are used which must be clearly defined. This is

a must because in the absence of a clear cut and precise definition of the

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statistical units, serious errors in the data collection may be committed in the

sense that we may collect irrelevant data on the items, which should have, in

fact, been excluded and omit data on certain items which should have been

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included. This will ultimately lead to fallacious conclusions.

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(f) Sources of Information (data): After deciding about the unit, a researcher

has to decide about the source from which the information can be obtained or

collected. For any statistical inquiry, the investigator may collect the data first

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hand or he may use the data from other published sources, such as publications

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of the government/semi-government organizations or journals and magazines

etc.

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(g) Method of Data Collection: - There is no problem if secondary data are

used for research. However, if primary data are to be collected, a decision has to

be taken whether (i) census method or (ii) sample technique is to be used for

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data collection. In census method, we go for total enumeration i.e., all the units

of a universe have to be investigated. But in sample technique, we inspect or

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study only a selected representative and adequate fraction of the population and

after analyzing the results of the sample data we draw conclusions about the

characteristics of the population. Selection of a particular technique becomes

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difficult because where population or census method is more scientific and

100% accuracy can be attained through this method, choosing this becomes

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difficult because it is time taking, it requires more labor and it is very expensive.

Therefore, for a single researcher or for a small institution it proves to be

unsuitable. On the other hand, sample method is less time taking, less laborious

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and less expensive but a 100% accuracy cannot be attained through this method

because of sampling and non-sampling errors attached to this method. Hence, a

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researcher has to be very cautious and careful while choosing a particular

method.

Methods of Collecting Primary Data:

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Primary data may be obtained by applying any of the following methods:

1. Direct Personal Interviews.

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2. Indirect oral interviews.

3. Information from correspondents.

4. Mailed questionnaire methods.

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5. Schedule sent through enumerators.

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1. Direct personal interviews: A face to face contact is made with the

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informants (persons from whom the information is to be obtained) under this

method of collecting data. The interviewer asks them questions pertaining to the

survey and collects the desired information. Thus, if a person wants to collect

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data about the working conditions of the workers of the Tata Iron and Steel

Company, Jamshedpur, he would go to the factory, contact the workers and

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obtain the desired information. The information collected in this manner is first

hand and also original in character. There are many merits and demerits of this

method, which are discussed as under:

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Merits:

1.

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Most often respondents are happy to pass on the information required

from them when contacted personally and thus response is encouraging.

2.

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The information collected through this method is normally more accurate

because interviewer can clear doubts of the informants about certain

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questions and thus obtain correct information. In case the interviewer

apprehends that the informant is not giving accurate information, he may

cross-examine him and thereby try to obtain the information.

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3.

This method also provides the scope for getting supplementary

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information from the informant, because while interviewing it is possible

to ask some supplementary questions which may be of greater use later.

4.

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There might be some questions which the interviewer would find

difficult to ask directly, but with some tactfulness, he can mingle such

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questions with others and get the desired information. He can twist the

questions keeping in mind the informant's reaction. Precisely, a delicate

situation can usually he handled more effectively by a personal interview

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than by other survey techniques.

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5.

The interviewer can adjust the language according to the status and

educational level of the person interviewed, and thereby can avoid

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inconvenience and misinterpretation on the part of the informant.

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Demerits:

1. This method can prove to be expensive if the number of informants is large

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and the area is widely spread.

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2. There is a greater chance of personal bias and prejudice under this method as

compared to other methods.

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3. The interviewers have to be thoroughly trained and experienced; otherwise

they may not be able to obtain the desired information. Untrained or poorly

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trained interviewers may spoil the entire work.

4. This method is more time taking as compared to others. This is because

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interviews can be held only at the convenience of the informants. Thus, if

information is to be obtained from the working members of households,

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interviews will have to be held in the evening or on week end. Even during

evening only an hour or two can be used for interviews and hence, the work

may have to be continued for a long time, or a large number of people may

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have to be employed which may involve huge expenses.

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Conclusion:

Though there are some demerits in this method of data collection still we cannot

say that it is not useful. The matter of fact is that this method is suitable for

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intensive rather than extensive field surveys. Hence, it should be used only in

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those cases where intensive study of a limited field is desired.

In the present time of extreme advancement in the communication system,

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the investigator instead of going personally and conducting a face to face

interview may also obtain information over telephone. A good number of

surveys are being conducted every day by newspapers and television channels

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by sending the reply either by e-mail or SMS. This method has become very

popular nowadays as it is less expensive and the response is extremely quick.

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But this method suffers from some serious defects, such as (a) very few people

own a phone or a television and hence a limited number of people can be

approached by this method, (b) only few questions can be asked over phone or

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through television, (c) the respondents may give a vague and reckless answers

because answers on phone or through SMS would have to be very short.

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2. Indirect Oral Interviews: Under this method of data collection, the

investigator contacts third parties generally called `witnesses' who are capable

of supplying necessary information. This method is generally adopted when the

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information to be obtained is of a complex nature and informants are not

inclined to respond if approached directly. For example, when the researcher is

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trying to obtain data on drug addiction or the habit of taking liquor, there is high

probability that the addicted person will not provide the desired data and hence

will disturb the whole research process. In this situation taking the help of such

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persons or agencies or the neighbours who know them well becomes necessary.

Since these people know the person well, they can provide the desired data.

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Enquiry Committees and Commissions appointed by the Government generally

adopt this method to get people's views and all possible details of the facts

related to the enquiry.

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Though this method is very popular, its correctness depends upon a number of

factors which are discussed below:

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1. The person or persons or agency whose help is solicited must be of proven

integrity; otherwise any bias or prejudice on their part will not bring the correct

information and the whole process of research will become useless.

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2. The ability of the interviewers to draw information from witnesses by means

of appropriate questions and cross-examination.

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3. It might happen that because of bribery, nepotism or certain other reasons

those who are collecting the information give it such a twist that correct

conclusions are not arrived at.

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Therefore, for the success of this method it is necessary that the evidence of

one person alone is not relied upon. Views from other persons and related

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agencies should also be ascertained to find the real position .Utmost care must

be exercised in the selection of these persons because it is on their views that the

final conclusions are reached.

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3. Information from Correspondents: The investigator appoints local agents

or correspondents in different places to collect information under this method.

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These correspondents collect and transmit the information to the central office

where data are processed. This method is generally adopted by news paper

agencies. Correspondents who are posted at different places supply information

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relating to such events as accidents, riots, strikes, etc., to the head office. The

correspondents are generally paid staff or sometimes they may be honorary

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correspondents also. This method is also adopted generally by the government

departments in such cases where regular information is to be collected from a

wide area. For example, in the construction of a wholesale price index numbers

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regular information is obtained from correspondents appointed in different areas.

The biggest advantage of this method is that it is cheap and appropriate for

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extensive investigation. But a word of caution is that it may not always ensure

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accurate results because of the personal prejudice and bias of the

correspondents. As stated earlier, this method is suitable and adopted in those

cases where the information is to be obtained at regular intervals from a wide

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area.

4. Mailed Questionnaire Method: Under this method, a list of questions

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pertaining to the survey which is known as `Questionnaire' is prepared and

sent to the various informants by post. Sometimes the researcher himself too

contacts the respondents and gets the responses related to various

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questions in the questionnaire. The questionnaire contains questions and

provides space for answers. A request is made to the informants through a

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covering letter to fill up the questionnaire and send it back within a specified

time. The questionnaire studies can be classified on the basis of:

i.

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The degree to which the questionnaire is formalized or structured.

ii.

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The disguise or lack of disguise of the questionnaire and

iii. The communication method used.

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When no formal questionnaire is used, interviewers adapt their questioning

to each interview as it progresses. They might even try to elicit responses by

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indirect methods, such as showing pictures on which the respondent comments.

When a researcher follows a prescribed sequence of questions, it is referred to as

structured study. On the other hand, when no prescribed sequence of questions

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exists, the study is non-structured.

When questionnaires are constructed in such a way that the objective is clear

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to the respondents then these questionnaires are known as non- disguised; on the

other hand, when the objective is not clear, the questionnaire is a disguised one.

On the basis of these two classifications, four types of studies can he

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distinguished:

i.

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Non-disguised structured,

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ii.

Non-disguised non-structured,

iii. Disguised structured and

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iv. Disguised non-structured.

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There are certain merits and demerits or limitations of this method of data

collection which are discussed below:

Merits:

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1. Questionnaire method of data collection can be easily adopted where the

field of investigation is very vast and the informants are spread over a

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wide geographical area.

2. This method is relatively cheap and expeditious provided the informants

respond in time.

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3. This method has proved to be superior when compared to other methods like

personal interviews or telephone method. This is because when questions

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pertaining to personal nature or the ones requiring reaction by the family are

put forth to the informants, there is a chance for them to be embarrassed in

answering them.

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Demerits:

1. This method can be adopted only where the informants are literate

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people so that they can understand written questions and lend the

answers in writing.

2. It involves some uncertainty about the response. Co-operation on the part of

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informants may be difficult to presume.

3. The information provided by the informants may not be correct and it may

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be difficult to verify the accuracy.

However, by following the guidelines given below, this method can be made

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more effective:

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The questionnaires should be made in such a manner that they do not

become an undue burden on the respondents; otherwise the respondents may not

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return them back.

i.

Prepaid postage stamp should be affixed

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ii.

The sample should be large

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iii.

It should be adopted in such enquiries where it is expected that the

respondents would return the questionnaire because of their own interest

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in the enquiry.

iv.

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It should be preferred in such enquiries where there could be a legal

compulsion to provide the information.

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5. Schedules sent through Enumerators: Another method of data collection is

sending schedules through the enumerators or interviewers. The enumerators

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contact the informants, get replies to the questions contained in a schedule and

fill them in their own handwriting in the questionnaire form. There is difference

between questionnaire and schedule. Questionnaire refers to a device for

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securing answers to questions by using a form which the respondent fills in him

self, whereas Schedule is the name usually applied to a set of questions which

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are asked in a face-to face situation with another person. This method is free

from most of the limitations of the mailed questionnaire method.

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Merits:

The main merits or advantages of this method are listed below:

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i.

It can be adopted in those cases where informants are illiterate.

ii.

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There is very little scope of non-response as the enumerators go personally

to obtain the information.

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iii. The information received is more reliable as the accuracy of statements

can be checked by supplementary questions wherever necessary.

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This method too like others is not free from defects or limitations. The

main limitations are listed below:

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Demerits:

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i.

In comparison to other methods of collecting primary data, this method is

quite costly as enumerators are generally paid persons.

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ii.

The success of the method depends largely upon the training imparted to

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the enumerators.

iii. Interviewing is a very skilled work and it requires experience and training.

Many statisticians have the tendency to neglect this extremely important

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part of the data collecting process and this result in bad interviews.

Without good interviewing most of the information collected is of doubtful

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value.

iv. Interviewing is not only a skilled work but it also requires a great degree of

politeness and thus the way the enumerators conduct the interview would

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affect the data collected. When questions are asked by a number of

different interviewers, it is possible that variations in the personalities of

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the interviewers will cause variation in the answers obtained. This

variation will not be obvious. Hence, every effort must be made to remove

as much of variation as possible due to different interviewers.

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Secondary Data: As stated earlier, secondary data are those data which have

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already been collected and analyzed by some earlier agency for its own use, and

later the same data are used by a different agency. According to

W.A.Neiswanger, "A primary source is a publication in which the data are

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published by the same authority which gathered and analyzed them. A

secondary source is a publication, reporting the data which was gathered by

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other authorities and for which others are responsible."

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Sources of secondary data:-The various sources of secondary data can be

divided into two broad categories:

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1.

Published sources, and

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2.

Unpublished sources.

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1. Published Sources: The governmental, international and local agencies

publish statistical data, and chief among them are explained below:

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(a) International Publications: There are some international institutions and

bodies like I.M.F, I.B.R.D, I.C.A.F.E and U.N.O who publish regular and

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occasional reports on economic and statistical matters.

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(b) Official publications of Central and State Governments: Several

departments of the Central and State Governments regularly publish reports on a

number of subjects. They gather additional information. Some of the important

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publications are: The Reserve Bank of India Bulletin, Census of India, Statistical

Abstracts of States, Agricultural Statistics of India, Indian Trade Journal, etc.

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(c) Semi-official publications: Semi-Government institutions like Municipal

Corporations, District Boards, Panchayats, etc. publish reports relating to

different matters of public concern.

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(d) Publications of Research Institutions: Indian Statistical Institute (I.S.I),

Indian Council of Agricultural Research (I.C.A.R), Indian Agricultural Statistics

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Research Institute (I.A.S.R.I), etc. publish the findings of their research

programmes.

(e) Publications of various Commercial and Financial Institutions

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(f) Reports of various Committees and Commissions appointed by the

Government as the Raj Committee's Report on Agricultural Taxation, Wanchoo

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Committee's Report on Taxation and Black Money, etc. are also important

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sources of secondary data.

(g) Journals and News Papers: Journals and News Papers are very important

and powerful source of secondary data. Current and important materials on

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statistics and socio-economic problems can be obtained from journals and

newspapers like Economic Times, Commerce, Capital, Indian Finance, Monthly

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Statistics of trade etc.

2. Unpublished Sources: Unpublished data can be obtained from many

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unpublished sources like records maintained by various government and private

offices, the theses of the numerous research scholars in the universities or

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institutions etc.

Precautions in the Use of Secondary Data: Since secondary data have already

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been obtained, it is highly desirable that a proper scrutiny of such data is made

before they are used by the investigator. In fact the user has to be extra-cautious

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while using secondary data. In this context Prof. Bowley rightly points out that

"Secondary data should not be accepted at their face value." The reason being

that data may be erroneous in many respects due to bias, inadequate size of the

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sample, substitution, errors of definition, arithmetical errors etc. Even if there is

no error such data may not be suitable and adequate for the purpose of the

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enquiry. Prof. Simon Kuznet's view in this regard is also of great importance.

According to him, "The degree of reliability of secondary source is to be

assessed from the source, the compiler and his capacity to produce correct

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statistics and the users also, for the most part, tend to accept a series particularly

one issued by a government agency at its face value without enquiring its

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reliability".

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Therefore, before using the secondary data the investigators should

consider the following factors:

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4.

The suitability of data: The investigator must satisfy himself that the data

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available are suitable for the purpose of enquiry. It can be judged by the

nature and scope of the present enquiry with the original enquiry. For

example, if the object of the present enquiry is to study the trend in retail

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prices, and if the data provide only wholesale prices, such data are

unsuitable.

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(a) Adequacy of data: If the data are suitable for the purpose of investigation

then we must consider whether the data are useful or adequate for the

present analysis. It can be studied by the geographical area covered by the

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original enquiry. The time for which data are available is very important

element. In the above example, if our object is to study the retail price

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trend of India, and if the available data cover only the retail price trend in

the State of Bihar, then it would not serve the purpose.

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(b) Reliability of data: The reliability of data is must. Without which there is

no meaning in research. The reliability of data can be tested by finding out

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the agency that collected such data. If the agency has used proper methods

in collection of data, statistics may be relied upon.

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It is not enough to have baskets of data in hand. In fact, data in a raw form are

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nothing but a handful of raw material waiting for proper processing so that they

can become useful. Once data have been obtained from primary or secondary

source, the next step in a statistical investigation is to edit the data i.e. to

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scrutinize the same. The chief objective of editing is to detect possible errors and

irregularities. The task of editing is a highly specialized one and requires great

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care and attention. Negligence in this respect may render useless the findings of

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an otherwise valuable study. Editing data collected from internal records and

published sources is relatively simple but the data collected from a survey need

excessive editing.

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While editing primary data, the following considerations should be borne in

mind:

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1. The data should be complete in every respect

2. The data should be accurate

3. The data should be consistent, and

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4. The data should be homogeneous.

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Data to posses the above mentioned characteristics have to undergo the

same type of editing which is discussed below:

5.

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Editing for completeness: While editing, the editor should see that each

schedule and questionnaire is complete in all respects. He should see to it that

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the answers to each and every question have been furnished. If some questions

are not answered and if they are of vital importance, the informants should be

contacted again either personally or through correspondence. Even after all the

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efforts it may happen that a few questions remain unanswered. In such

questions, the editor should mark `No answer' in the space provided for answers

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and if the questions are of vital importance then the schedule or questionnaire

should be dropped.

1.

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Editing for Consistency: At the time of editing the data for consistency,

the editor should see that the answers to questions are not contradictory in

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nature. If they are mutually contradictory answers, he should try to obtain the

correct answers either by referring back the questionnaire or by contacting,

wherever possible, the informant in person. For example, if amongst others, two

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questions in questionnaire are (a) Are you a student? (b) Which class do you

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study and the reply to the first question is `no' and to the latter `tenth' then there

is contradiction and it should be clarified.

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2.

Editing for Accuracy: The reliability of conclusions depends basically

on the correctness of information. If the information supplied is wrong,

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conclusions can never be valid. It is, therefore, necessary for the editor to see

that the information is accurate in all respects. If the inaccuracy is due to

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arithmetical errors, it can be easily detected and corrected. But if the cause of

inaccuracy is faulty information supplied, it may be difficult to verify it and an

example of this kind is information relating to income, age etc.

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3.

Editing for Homogeneity: Homogeneity means the condition in which

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all the questions have been understood in the same sense. The editor must check

all the questions for uniform interpretation. For example, as to the question of

income, if some informants have given monthly income, others annual income

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and still others weekly income or even daily income, no comparison can be

made. Therefore, it becomes an essential duty of the editor to check up that the

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information supplied by the various people is homogeneous and uniform.

Choice between Primary and Secondary Data: As we have already seen,

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there are a lot of differences in the methods of collecting Primary and Secondary

data. Primary data which is to be collected originally involves an entire scheme

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of plan starting with the definitions of various terms used, units to be employed,

type of enquiry to be conducted, extent of accuracy aimed at etc. For the

collection of secondary data, a mere compilation of the existing data would be

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sufficient. A proper choice between the type of data needed for any particular

statistical investigation is to be made after taking into consideration the nature,

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objective and scope of the enquiry; the time and the finances at the disposal of

the agency; the degree of precision aimed at and the status of the agency

(whether government- state or central-or private institution of an individual).

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In using the secondary data, it is best to obtain the data from the primary source

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as far as possible. By doing so, we would at least save ourselves from the errors

of transcription which might have inadvertently crept in the secondary source.

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Moreover, the primary source will also provide us with detailed discussion about

the terminology used, statistical units employed, size of the sample and the

technique of sampling (if sampling method was used), methods of data

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collection and analysis of results and we can ascertain ourselves if these would

suit our purpose.

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Now-a-days in a large number of statistical enquiries, secondary data are

generally used because fairly reliable published data on a large number of

diverse fields are now available in the publications of governments, private

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organizations and research institutions, agencies, periodicals and magazines etc.

In fact, primary data are collected only if there do not exist any secondary data

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suited to the investigation under study. In some of the investigations both

primary as well as secondary data may be used.

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SUMMARY:

There are two types of data, primary and secondary. Data which are collected

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first hand are called Primary data and data which have already been collected

and used by somebody are called Secondary data. There are two methods of

collecting data: (a) Survey method or total enumeration method and (b) Sample

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method. When a researcher goes for investigating all the units of the subject, it is

called as survey method. On the other hand if he/she resorts to investigating only

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a few units of the subject and gives the result on the basis of that, it is known as

sample survey method. There are different sources of collecting Primary and

Secondary data. Some of the important sources of Primary data are--Direct

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Personal Interviews, Indirect Oral Interviews, Information from correspondents,

Mailed questionnaire method, Schedules sent through enumerators and so on.

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Though all these sources or methods of Primary data have their relative merits

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and demerits, a researcher should use a particular method with lot of care. There

are basically two sources of collecting secondary data- (a) Published sources and

(b) Unpublished sources. Published sources are like publications of different

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government and semi-government departments, research institutions and

agencies etc. whereas unpublished sources are like records maintained by

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different government departments and unpublished theses of different

universities etc. Editing of secondary data is necessary for different purposes as

? editing for completeness, editing for consistency, editing for accuracy and

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editing for homogeneity.

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It is always a tough task for the researcher to choose between primary

and secondary data. Though primary data are more authentic and accurate, time,

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money and labor involved in obtaining these more often prompt the researcher

to go for the secondary data. There are certain amount of doubt about its

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authenticity and suitability, but after the arrival of many government and semi

government agencies and some private institutions in the field of data collection,

most of the apprehensions in the mind of the researcher have been removed.

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SELF ASSESMENT QUESTIONS (SAQs):

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1. Explain primary and secondary data and distinguish between them.

(Refer the introduction part of this lesson.)

2. Explain the different methods of collecting primary data.

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(Explain direct personal, indirect oral interview, information received

through agencies etc.)

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3. Explain the merits and demerits of different methods of collecting primary

data.

(Refer the methods of collecting primary data)

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4. Explain the different sources of secondary data and the precautions in using

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secondary data.

5. What is editing of secondary data? Why is it required?

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6. What are the different types of editing of secondary data?

GLOSSARY OF TERMS:

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Primary Source: It is one that itself collects the data.

Secondary Source: It is one that makes available data collected by some other

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agency.

Collection of Statistics: Collection means the assembling for the purpose of

particular investigation of entirely new data presumably not already available in

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published sources.

Questionnaire: A list of questions properly selected and arranged pertaining to

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the investigation.

Investigator: Investigator is a person who collects the information.

Respondent: A person who fills the questionnaire or provides the required

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information.

***

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UNIT II

QUESTIONNAIRE AND SAMPLING

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Lesson Outline

Meaning of questionnaire.

Drafting of questionnaire.

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Size of questions

Clarity of questions

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Logical sequence of questions

Simple meaning questions

Other requirements of a good questionnaire

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Meaning and essentials of sampling.

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Learning Objectives

After reading this lesson you should be able to

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Understand the meaning of questionnaire

Different requirements and characteristics of a good questionnaire

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Meaning of sampling

Essentials of sampling

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Introduction:

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Nowadays questionnaire is widely used for data collection in social research. It

is a reasonably fair tool for gathering data from large, diverse, varied and

scattered social groups. The questionnaire is the media of communication

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between the investigator and the respondents. According to Bogardus, a

questionnaire is a list of questions sent to a number of persons for their answers

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and which obtains standardized results that can be tabulated and treated

statistically. The Dictionary of Statistical Terms defines it as a "group of or

sequence of questions designed to elicit information upon a subject or sequence

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of subjects from information." A questionnaire should be designed or drafted

with utmost care and caution so that all the relevant and essential information

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for the enquiry may be collected without any difficulty, ambiguity and

vagueness. Drafting of a good questionnaire is a highly specialized job and

requires great care skill, wisdom, efficiency and experience. No hard and fast

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rule can be laid down for designing or framing a questionnaire. However, in this

connection, the following general points may be borne in mind:

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1. Size of the Questionnaire Should be Small: A researcher should try his

best to keep the number of questions as small as possible, keeping in view the

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nature, objectives and scope of the enquiry. Respondent's time should not be

wasted by asking irrelevant and unimportant questions. A large number of

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questions would involve more work for the investigator and thus result in delay

on his part in collecting and submitting the information. A large number of

unnecessary questions may annoy the respondent and he may refuse to

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cooperate. A reasonable questionnaire should contain from 15 to 25 questions at

large. If a still larger number of questions are a must in any enquiry, then the

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questionnaire should be divided into various sections or parts.

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2. The Questions Should be Clear: The questions should be easy, brief,

unambiguous, non-offending, courteous in tone, corroborative in nature and to

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the point, so that much scope of guessing is left on the part of the respondents.

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3. The Questions Should be Arranged in a Logical Sequence: Logical

arrangement of questions reduces lot of unnecessary work on the part of the

researcher because it not only facilitates the tabulation work but also does not

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leave any chance for omissions or commissions. For example, to find if a person

owns a television, the logical order of questions would be: Do you own a

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television? When did you buy it? What is its make? How much did it cost you?

Is its performance satisfactory? Have you ever got it serviced?

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4. Questions Should be Simple to Understand: The vague words like good,

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bad, efficient, sufficient, prosperity, rarely, frequently, reasonable, poor, rich

etc., should not be used since these may be interpreted differently by different

persons and as such might give unreliable and misleading information. Similarly

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the use of words having double meaning like price, assets, capital income etc.,

should also be avoided.

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5. Questions Should be Comprehensive and Easily Answerable: Questions

should be designed in such a way that they are readily comprehensible and easy

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to answer for the respondents. They should not be tedious nor should they tax

the respondents' memory. At the same time questions involving mathematical

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calculations like percentages, ratios etc., should not be asked.

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6. Questions of Personal and Sensitive Nature Should Not be Asked: There

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are some questions which disturb the respondents and he/she may be shy or

irritated by hearing such questions. Therefore, every effort should be made to

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avoid such questions. For example, `do you cook yourself or your wife cooks?'

`Or do you drink?' Such questions will certainly irk the respondents and thus be

avoided at any cost. If unavoidable then highest amount of politeness should be

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used.

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7. Types of Questions: Under this head, the questions in the questionnaire may

be classified as follows:

(a) Shut Questions: Shut questions are those where possible answers are

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suggested by the framers of the questionnaire and the respondent is required to

tick one of them. Shut questions can further be subdivided into the following

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forms:

(i) Simple Alternate Questions: In this type of questions the respondent has to

choose from the two clear cut alternatives like `Yes' or `No', `Right or Wrong'

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etc. Such questions are also called as dichotomous questions. This technique can

be applied with elegance to situations where two clear cut alternatives exist.

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(ii) Multiple Choice Questions: Many a times it becomes difficult to define a

clear cut alternative and accordingly in such a situation additional answers

between Yes and No, like Do not know, No opinion, Occasionally, Casually,

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Seldom etc. are added. For example, in order to find if a person smokes or

drinks, the following multiple choice answers may be used:

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Do you smoke?

(a)Yes regularly [ ] (b) No never [ ]

(c) Occasionally [ ] (d) Seldom [ ]

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Multiple choice questions are very easy and convenient for the respondents to

answer. Such questions save time and also facilitate tabulation. This method

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should be used if only a selected few alternative answers exist to a particular

question.

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8. Leading Questions Should be Avoided: Questions like `Why do you use a

particular type of car, say Maruti car' should preferably be framed into two

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questions-

(i) Which car do you use?

(ii) Why do you prefer it?

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It gives smooth ride [ ]

It gives more mileage [ ]

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It is cheaper [ ]

It is maintenance free [ ]

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9 Cross Checks: The questionnaire should be so designed as to provide

internal checks on the accuracy of the information supplied by the respondents

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by including some connected questions at least with respect to matters which are

fundamental to the enquiry.

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10 Pre testing the Questionnaire: It would be practical in every sense to try

out the questionnaire on a small scale before using it for the given enquiry on a

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large scale. This has been found extremely useful in practice. The given

questionnaire can be improved or modified in the light of the drawbacks,

shortcomings and problems faced by the investigator in the pre test.

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11 A Covering Letter: A covering letter from the organizers of the enquiry

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should be enclosed along with the questionnaire for the purposes regarding

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definitions, units, concepts used in the questionnaire, for taking the respondent's

confidence, self addressed envelop in case of mailed questionnaire, mention

about award or incentives for the quick response, a promise to send a copy of the

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survey report etc.

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SAMPLING

Though sampling is not new, the sampling theory has been developed

recently. People knew or not but they have been using the sampling technique in

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their day to day life. For example a house wife tests a small quantity of rice to

see whether it has been well-cooked and gives the generalized result about the

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whole rice boiling in the vessel. The result arrived at is most of the times 100%

correct. In another example, when a doctor wants to examine the blood for any

deficiency, takes only a few drops of blood of the patient and examines. The

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result arrived at is most of the times correct and represent the whole amount of

blood available in the body of the patient. In all these cases, by inspecting a few,

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they simply believe that the samples give a correct idea about the population.

Most of our decision are based on the examination of a few items only i.e.

sample studies. In the words of Croxton and Cowdon, "It may be too expensive

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or too time consuming to attempt either a complete or a nearly complete

coverage in a statistical study. Further to arrive at valid conclusions, it may not

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be necessary to enumerate all or nearly all of a population. We may study a

sample drawn from the large population and if that sample is adequately

representative of the population, we should be able to arrive at valid

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conclusions."

According to Rosander, "The sample has many advantages over a census

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or complete enumeration. If carefully designed, the sample is not only

considerably cheaper but may give results which are just accurate and

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sometimes more accurate than those of a census. Hence a carefully designed

sample may actually be better than a poorly planned and executed census."

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Merits:

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1.

It saves time: Sampling method of data collection saves time because

fewer items are collected and processed. When the results are urgently required,

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this method is very helpful.

2.

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It reduces cost: Since only a few and selected items are studied in

sampling, there is reduction in cost of money and reduction in terms of man

hours.

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3.

More reliable results can be obtained: Through sampling, more

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reliable results can be obtained because (a) there are fewer chances of sampling

statistical errors. If there is sampling error, it is possible to estimate and control

the results.(b) Highly experienced and trained persons can be employed for

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scientific processing and analyzing of relatively limited data and they can use

their high technical knowledge and get more accurate and reliable results.

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4.

It provides more detailed information: As it saves time, money and

labor, more detail information can be collected in a sample survey.

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5.

Sometimes only Sampling method to depend upon: Some times it so

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happens that one has to depend upon sampling method alone because if the

population under study is finite, sampling method is the only method to be used.

For example, if someone's blood has to be examined, it will become fatal to take

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all the blood out from the body and study depending upon the total enumeration

method.

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6.

Administrative convenience: The organization and administration of

sample survey are easy for the reasons which have been discussed earlier.

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More scientific: Since the methods used to collect data are based on

scientific theory and results obtained can be tested, sampling is a more scientific

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method of collecting data.

It is not that sampling is free from demerits or shortcomings. There are certain

shortcomings of this method which are discussed below:

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1.

Illusory conclusion: If a sample enquiry is not carefully planned and

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executed, the conclusions may be inaccurate and misleading.

2.

Sample not representative: To make the sample representative is a

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difficult task. If a representative sample is taken from the universe, the result is

applicable to the whole population. If the sample is not representative of the

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universe the result may be false and misleading.

3.

Lack of experts: As there are lack of experts to plan and conduct a

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sample survey, its execution and analysis, and its results would be

unsatisfactory and not trustworthy.

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4.

Sometimes more difficult than census method: Sometimes the

sampling plan may be complicated and requires more money, labor and time

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than a census method.

5.

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Personal bias: There may be personal biases and prejudices with regard

to the choice of technique and drawing of sampling units.

6.

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Choice of sample size: If the size of the sample is not appropriate then it

may lead to untrue characteristics of the population.

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7.

Conditions of complete coverage: If the information is required for

each and every item of the universe, then a complete enumeration survey is

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better.

Essentials of sampling: In order to reach a clear conclusion, the sampling

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should possess the following essentials:

1.

It must be representative: The sample selected should possess the

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similar characteristics of the original universe from which it has been drawn.

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2.

Homogeneity: Selected samples from the universe should have similar

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nature and should mot have any difference when compared with the universe.

3.

Adequate samples: In order to have a more reliable and representative

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result, a good number of items are to be included in the sample.

4.

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Optimization: All efforts should be made to get maximum results both

in terms of cost as well as efficiency. If the size of the sample is larger, there is

better efficiency and at the same time the cost is more. A proper size of sample

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is maintained in order to have optimized results in terms of cost and efficiency.

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STATISTICAL LAWS

One of the basic reasons for undertaking a sample survey is to predict

and generalize the results for the population as a whole. The logical process of

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drawing general conclusions from a study of representative items is called

induction. In statistics, induction is a generalization of facts on the assumption

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that the results provided by an adequate sample may be taken as applicable to

the whole. The fact that the characteristics of the sample provide a fairly good

idea about the population characteristics is borne out by the theory of

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probability. Sampling is based on two fundamental principles of statistics theory

viz, (i) the Law of Statistical Regularity and (ii) the Law of Inertia of Large

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Numbers.

THE LAW OF STATISTICAL REGULARITY

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The Law of Statistical Regularity is derived from the mathematical theory of

probability. According to W.I.King, "The Law of Statistical Regularity

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formulated in the mathematical theory of probability lays down that a

moderately large number of items chosen at random from a very large group are

almost sure to have the characteristics of the large group." For example, if we

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want to find out the average income of 10,000 people, we take a sample of 100

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people and find the average. Suppose another person takes another sample of

100 people from the same population and finds the average, the average income

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found out by both the persons will have the least difference. On the other hand if

the average income of the same 10,000 people is found out by the census

method, the result will be more or less the same.

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Characteristics

1. The item selected will represent the universe and the result is generalized

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to universe as a whole.

2. Since sample size is large, it is representative of the universe.

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3. There is a very remote chance of bias.

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LAW OF INERTIA OF LARGE NUMBERS

The Law of inertia of Large Numbers is an immediate deduction from

the Principle of Statistical Regularity. Law of Inertia of Large Numbers states,

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"Other things being equal, as the sample size increases, the results tend to be

more reliable and accurate." This is based on the fact that the behavior or a

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phenomenon en masse. i.e., on a large scale is generally stable. It implies that

the total change is likely to be very small, when a large number or items are

taken in a sample. The law will be true on an average. If sufficient large samples

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are taken from the patent population, the reverse movements of different parts in

the same will offset by the corresponding movements of some other parts.

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Sampling Errors: In a sample survey, since only a small portion of the

population is studied its results are bound to differ from the census results and

thus, have a certain amount of error. In statistics the word error is used to denote

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the difference between the true value and the estimated or approximated value.

This error would always be there no matter that the sample is drawn at random

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and that it is highly representative. This error is attributed to fluctuations of

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sampling and is called sampling error. Sampling error is due to the fact that only

a sub set of the population has been used to estimate the population parameters

and draw inferences about the population. Thus, sampling error is present only

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in a sample survey and is completely absent in census method.

Sampling errors occur primarily due to the following reasons:

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1.

Faulty selection of the sample: Some of the bias is introduced by the

use of defective sampling technique for the selection of a sample e.g. purposive

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or judgment sampling in which the investigator deliberately selects a

representative sample to obtain certain results. This bias can be easily overcome

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by adopting the technique of simple random sampling.

2.

Substitution: When difficulties arise in enumerating a particular

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sampling unit included in the random sample, the investigators usually substitute

a convenient member of the population. This obviously leads to some bias since

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the characteristics possessed by the substituted unit will usually be different

from those possessed by the unit originally included in the sample.

3.

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Faulty demarcation of sampling units: Bias due to defective

demarcation of sampling units is particularly significant in area surveys such as

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agricultural experiments in the field of crop cutting surveys etc. In such surveys,

while dealing with border line cases, it depends more or less on the discretion of

the investigator whether to include them in the sample or not.

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4.

Error due to bias in the estimation method: Sampling method consists

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in estimating the parameters of the population by appropriate statistics computed

from the sample. Improper choice of the estimation techniques might introduce

the error.

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5.

Variability of the population: Sampling error also depends on the

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variability or heterogeneity of the population to be sampled.

Sampling errors are of two types: Biased Errors and Unbiased Errors

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Biased Errors: The errors that occur due to a bias of prejudice on the part of

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the informant or enumerator in selecting, estimating measuring instruments are

called biased errors. Suppose for example, the enumerator used the deliberate

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sampling method in the place of simple random sampling method, then it is

called biased errors. These errors are cumulative in nature and increase when the

sample size also increases. These errors arise due to defect in the methods of

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collection of data, defect in the method of organization of data and defect in the

method of analysis of data.

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Unbiased errors: Errors which occur in the normal course of investigation or

enumeration on account of chance are called unbiased errors. They may arise

accidentally without any bias or prejudice. These errors occur due to faulty

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planning of statistical investigation.

To avoid these errors, the statistician must take proper precaution and care in

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using the correct measuring instrument. He must see that the enumerators are

also not biased. Unbiased errors can be removed with the proper planning of

statistical investigations. Both these errors should be avoided by the statisticians.

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Reducing Sampling Errors:

Errors in sampling can be reduced if the size of sample is increased. This is

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shown in the following diagram.

From the above diagram it is clear that when the size of the sample

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increases, sampling error decreases. And by this process samples can be made

more representatives to the population.

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Testing of Hypothesis:

As a part of investigation, samples are drawn from the population and

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results are derived to help in taking the decisions. But such decisions involve an

element of uncertainty causing wrong decisions. Hypothesis is an assumption

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which may or may not be true about a population parameter. For example, if we

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toss a coin 200 times, we may get 110 heads and 90 tails. At this instance, we

are interested in testing whether the coin is unbiased or not.

Therefore, we may conduct a test to judge the significance of the difference of

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sampling or otherwise. To carry out a test of significance, the following

procedure has to be followed:

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1. Framing the Hypothesis: To verify the assumption, which is based on

sample study, we collect data and find out the difference between the sample

value and the population value. If there is no difference found or the difference

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is very small then the hypothetical value is correct. Generally two hypotheses

are constructed, and if one is found correct, the other is rejected.

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(a) Null Hypothesis: The random selection of the samples from the given

population makes the tests of significance valid for us. For applying any test of

significance we first set up a hypothesis- a definite statement about the

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population parameter/s. Such a statistical hypothesis, which is under test, is

usually a hypothesis of no difference and hence is called Null hypothesis. It is

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usually denoted by Ho. In the words of Prof. R.A.Fisher "Null hypothesis is the

hypothesis which is tested for possible rejection under the assumption that

it is true."

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(b)

Alternative Hypothesis. Any hypothesis which is complementary to the

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null hypothesis is called an alternative hypothesis. It is usually denoted by H1. It

is very important to explicitly state the alternative hypothesis in respect of any

null hypothesis H0 because the acceptance or rejection of Ho is meaningful only

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if it is being tested against a rival hypothesis. For example, if we want to test the

null hypothesis that the population has a specified mean ?0(say), i.e.,

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H0:?=? then the alternative hypothesis could be:

(i) H1:?0 (i.e., ?>?0 or ?<?0)

(ii) H1: ?>?0

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(iii) H1: ?<?0

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The alternative hypothesis (i) is known as a two-tailed alternative and the

alternatives in (ii) and (iii) are known as right-tailed and left-tailed alternatives.

Accordingly, the corresponding tests of significance are called two-tailed, right-

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tailed and left-tailed tests respectively.

The null hypothesis consists of only a single parameter value and is

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usually simple while alternative hypothesis is usually composite.

Types of Errors in Testing of Hypothesis: As stated earlier, the inductive

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inference consists in arriving at a decision to accept or reject a null hypothesis

(Ho) after inspecting only a sample from it. As such an element of risk ? the risk

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of taking wrong decision is involved. In any test procedure, the four possible

mutually disjoint and exhaustive decisions are:

(i) Reject Ho when actually it is not true i.e., when Ho is false.

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(ii) Accept Ho when it is true.

(iii) Reject Ho when it is true.

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(iv)

Accept Ho when it is false.

The decisions in (i) and (ii) are correct decisions while the decisions in

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(iii) and (iv) are wrong decisions. These decisions may be expressed in the

following dichotomous table:

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Decision from sample

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Reject Ho

Accept Ho

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True State

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Ho True

Wrong

Correct

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Type I Error

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Ho False

Correct Wrong

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(H1True)

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Type II Error.

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Thus, in testing of hypothesis we are likely to commit two types of

errors. The error of rejecting Ho when Ho is true is known as Type I error and

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the error of accepting Ho when Ho is false is known as Type II Error.

For example, in the Industrial Quality Control, while inspecting the quality of a

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manufactured lot, the Inspector commits Type I Error when he rejects a good lot

and he commits Type II Error when he accepts a bad lot.

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SUMMARY

Nowadays questionnaire method of data collection has become very popular. It

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is a very powerful tool to collect required data in shortest period of time and

with little expense. It is scientific too. But drafting of questionnaire is a very

skilled and careful work. Therefore, there are certain requirements and essentials

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which should be followed at the time of framing the questionnaire. They include

the following viz., (i) the size of the questionnaire should be small, (ii) questions

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should be very clear in understanding, (iii) questions should be put in a logical

order, (iv) questions should have simple meaning etc. Apart from this, multiple

choice questions should be asked. Questionnaire should be pre tested before

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going for final data collection. Information supplied should be cross checked for

any false or insufficient information. After all these formalities have been

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completed, a covering note should accompany the questionnaire explaining

various purposes, designs, units and incentives.

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There are two ways of survey- Census survey and Sample survey through

which data can be collected. Census survey means total enumeration i.e.,

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collecting data from each and every unit of the universe, whereas sample survey

concentrates on collecting data from a few units of the universe selected

scientifically for the purpose. Since census method is more time taking,

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expensive and labor intensive, it becomes impractical to depend on it. Therefore,

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sample survey is preferred which is scientific, less expensive, less time taking

and less labor intensive too.

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But there are merits and demerits of this method which are detailed below:

Merits - It reduces cost, saves time and is more reliable. It provides

more detailed information and is sometimes the only method to depend upon for

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administrative convenience and scientifically.

Demerits - Sometimes samples may not be representative and may give

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illusory conclusions. There are lack of experts and sometimes it is more difficult

than the census method, since there might arise personal bias and the

determination of the size of the sample might be very difficult.

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Apart from these, there are some essentials of sampling which must be

followed. They are: Samples must be representative, samples must be

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homogeneous and the number of samples must be adequate. When a researcher

resorts to sampling, he intends to collect some data which would help him to

draw results and finally take a decision. When he takes a decision it's on the

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basis of hypothesis which is precisely assumption and is prone to two types of

errors-Type I Error and Type II Error. When a researcher rejects a correct

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hypothesis, he commits type I error and when he accepts a wrong hypothesis he

commits type II error. The researcher should try to avoid both types of errors but

committing type II error is more harmful than type I error.

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SELF ASSESMENT QUESTIONS (SAQs)

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1.

Explain questionnaire and examine its main characteristics.

(Refer to the introduction part of the questionnaire section)

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Explain main requirements of a good questionnaire.

(Refer to the sub points from 1 to 11)

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3. What is sampling? Explain its main merits and demerits.

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(Refer to the introduction and the following part of the lesson)

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4

What are null and alternative hypothesis? Explain.

(Refer the point Framing the Hypothesis)

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What are Type I error and Type II error? (Refer to types of error in

hypothesis)

***

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UNIT II

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3. EXPERIMENTS

Lesson Outline

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Procedures adopted in experiments

Meaning of Experiments

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Research design in case of hypothesis testing

research studies

Basic principles in experimental designs

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Prominent experimental designs

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Learning Objectives

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After reading this lesson you should be able to understand the

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Nature and meaning of Experiments

Kinds of experiments

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Introduction

The meaning of experiment lies in the process of examining the truth of

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a statistical hypothesis related to some research problem. For example, a

researcher can conduct an experiment to examine the newly developed

medicine. Experiment is of two types: absolute experiment and comparative

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experiment. When a researcher wants to determine the impact of a fertilizer on

the yield of a crop it is a case of absolute experiment. On the other hand, if he

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wants to determine the impact of one fertilizer as compared to the impact of

some other fertilizer, the experiment will then be called as a comparative

experiment. Normally, a researcher conducts a comparative experiment when he

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talks of designs of experiments.

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Research design can be of three types:

(a)

Research design in the case of descriptive and diagnostic research

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studies,

(b)

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Research design in the case of exploratory research studies, and

(c)

Research design in the case of hypothesis testing research studies.

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Here we are mainly concerned with the third one which is Research design

in the case of hypothesis testing research studies.

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Research design in the case of hypothesis testing research studies:

Hypothesis testing research studies are generally known as experimental studies.

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This is a study where a researcher tests the hypothesis of causal relationships

between variables. This type of study requires some procedures which will not

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only reduce bias and increase reliability, but will also permit drawing inferences

about causality. Most of the times, experiments meet these requirements. Prof.

Fisher is considered as the pioneer of this type of studies (experimental studies).

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He did pioneering work when he was working at Rothamsted Experimental

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Station in England which was a centre for Agricultural Research. While working

there, Prof. Fisher found that by dividing plots into different blocks and then by

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conducting experiments in each of these blocks whatever information is

collected and inferences drawn from them happened to be more reliable. This

was where he was inspired to develop certain experimental designs for testing

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hypotheses concerning scientific investigations. Nowadays, the experimental

design is used in researches relating to almost every discipline of knowledge.

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Prof. Fisher laid three principles of experimental designs:

(1) The Principle of Replication

(2) The Principle of Randomization and

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(3) The Principle of Local Control.

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(1) The Principle of Replication:

According to this principle, the experiment should be repeated more than

once. Thus, each treatment is applied in many experimental units instead of one.

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This way the statistical accuracy of the experiments is increased. For example,

suppose we are going to examine the effect of two varieties of wheat.

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Accordingly, we divide the field into two parts and grow one variety in one part

and the other variety in the other. Then we compare the yield of the two parts

and draw conclusion on that basis. But if we are to apply the principle of

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replication to this experiment, then we first divide the field into several parts,

grow one variety in half of these parts and the other variety in the remaining

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parts. Then we collect the data of yield of the two varieties and draw conclusion

by comparing the same. The result so obtained will be more reliable in

comparison to the conclusion we draw without applying the principle of

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replication. The entire experiment can be repeated several times for better

results.

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(2) The Principle of Randomization:

When we conduct an experiment, the principle of randomization

provides us a protection against the effects of extraneous factors by

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randomization. This means that this principle indicates that the researcher

should design or plan the experiment in such a way that the variations caused by

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extraneous factors can all be combined under the general heading of `chance'.

For example, when a researcher grows one variety of wheat , say , in the first

half of the parts of a field and the other variety he grows in the other half, then it

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is just possible that the soil fertility may be different in the first half in

comparison to the other half. If this is so the researcher's result is not realistic.

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In this situation, he may assign the variety of wheat to be grown in different

parts of the field on the basis of some random sampling technique i.e., he may

apply randomization principle and protect himself against the effects of the

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extraneous factors. Therefore, by using the principle of randomization, he can

draw a better estimate of the experimental error.

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(3). The Principle of Local Control:

This is another important principle of experimental designs. Under this

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principle, the extraneous factor which is the known source of variability is made

to vary deliberately over as wide a range as necessary. This needs to be done in

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such a way that the variability it causes can be measured and hence eliminated

from the experimental error. The experiment should be planned in such a way

that the researcher can perform a two-way analysis of variance, in which the

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total variability of the data is divided into three components attributed to

treatments (varieties of wheat in this case), the extraneous factor (soil fertility in

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this case) and experimental error. In short, through the principle of local control

we can eliminate the variability due to extraneous factors from the experimental

error.

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Kinds of experimental Designs and Control

Experimental designs refer to the framework of structure of an

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experiment and as such there are several experimental designs. Generally,

experimental designs are classified into two broad categories: informal

experimental designs and formal experimental designs. Informal experimental

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designs are those designs that normally use a less sophisticated form of analysis

based on differences in the magnitudes, whereas formal experimental designs

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offer relatively more control and use precise statistical procedures for analysis.

Important experimental designs are discussed below:

(1) Informal experimental designs:

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(i) Before and after without control design

(ii) After only with control design

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(iii) Before and after with control design

(2) Formal experimental designs:

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(i) Completely randomized design (generally called C.R design)

(ii) Randomized block design (generally called R.B design)

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(iii) Latin square design (generally called L.S design)

(iv) Factorial designs.

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(1)Before and after without control design:

In this design, a single test group or area is selected and the dependent

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variable is measured before introduction of the treatment. Then the treatment is

introduced and the dependent variable is measured again after the treatment has

been introduced. The effect of the treatment would be equal to the level of the

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phenomenon after the treatment minus the level of the phenomenon before the

treatment. Thus, the design can be presented in the following manner:

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Test area Level of phenomenon Treatment Level of phenomenon

Before treatment(X) introduced after treatment(Y)

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Treatment Effect =(Y)-(X)

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The main difficulty of such a design is that with the passage of time

considerable extraneous variations may be there in its treatment effect.

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(2) After-only with control design:

Two groups or areas are selected in this design and the treatment is

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introduced into the test area only. Then the dependent variable is measured in

both the areas at the same time. Treatment impact is assessed by subtracting the

value of the dependent variable in the control area from its value in the test area.

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The design can be presented in the following manner:

Test area: Treatment introduced Level of phenomenon

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after Treatment (Y)

Control area: Level of phenomenon

Without treatment (Z)

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Treatment Effect = (Y)-(Z)

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The basic assumption in this type of design is that the two areas are identical

with respect to their behavior towards the phenomenon considered. If this

assumption is not true, there is the possibility of extraneous variation entering

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into the treatment effect.

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(3) Before and after with control design:

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In this design, two areas are selected and the dependent variable is

measured in both the areas for an identical time-period before the treatment.

Thereafter, the treatment is introduced into the test area only, and the dependent

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variable id measured in both for and identical time ?period after the introduction

of the treatment. The effect of the treatment is determined by subtracting the

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change in the dependent variable in the control area from the change in the

dependent variable in test area. This design can be shown in the following way:

Time Period I Time Period II

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Test area: Level of phenomenon Treatment Level of phenomenon

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Before treatment (X) introduced after treatment (Y)

Control area: Level of phenomenon Level of phenomenon

Without treatment without treatment

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(A) (Z)

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Treatment Effect = (Y-X)-(Z-A)

This design is superior to the previous two designs because it avoids extraneous

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variation resulting both from the passage of time and from non-comparability of

the rest and control areas. But at times, due to lack of historical data time or a

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comparable control area, we should prefer to select one of the first two informal

designs stated above.

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(2) Formal Experimental Design

(i) Completely randomized design: -

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This design involves only two principles i.e., the principle of replication

and the principle of randomization of experimental designs. Among all other

designs this is the simpler and easier because it's procedure and analysis are

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simple. The important characteristic of this design is that the subjects are

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randomly assigned to experimental treatments. For example, if the researcher

has 20 subjects and if he wishes to test 10 under treatment A and 10 under

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treatment B, the randomization process gives every possible group of 10

subjects selected from a set of 20 an equal opportunity of being assigned to

treatment A and treatment B. One way analysis of variance (one way ANOVA)

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is used to analyze such a design.

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(ii) Randomized block design:-

R. B. design is an improvement over the C.R. design. In the R .B. design,

the principle of local control can be applied along with the other two principles

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of experimental designs. In the R.B. design, subjects are first divided into

groups, known as blocks, such that within each group the subjects are relatively

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homogenous in respect to some selected variable. The number of subjects in a

given block would be randomly assigned to each treatment. Blocks are the levels

at which we hold the extraneous factor fixed, so that its contribution to the total

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variability of data can be measured. The main feature of the R.B. design is that,

in this, each treatment appears the same number of times in each block. This

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design is analyzed by the two-way analysis of variance (two-way ANOVA)

technique.

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(3) Latin squares design:-

The Latin squares design (L.S design) is an experimental design which is

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very frequently used in agricultural research. Since agriculture depends upon

nature to a large extent, the condition of research and investigation in agriculture

is different than the other studies. For example, an experiment has to be made

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through which the effects of fertilizers on the yield of a certain crop, say wheat,

is to be judged. In this situation, the varying fertility of the soil in different

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blocks in which the experiment has to be performed must be taken into

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consideration; otherwise the results obtained may not be very dependable

because the output happens to be the effects of not only of fertilizers, but also of

the effect of fertility of soil. Similarly there may be the impact of varying seeds

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of the yield. In order to overcome such difficulties, the L.S. design is used when

there are two major extraneous factors such as the varying soil fertility and

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varying seeds. The Latin square design is such that each fertilizer will appear

five times but will be used only once in each row and in each column of the

design. In other words, in this design, the treatment is so allocated among the

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plots that no treatment occurs more than once in any one row or any one

column. This experiment can be shown with the help of the following diagram:

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FERTILITY LEVEL

I II III IV V

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X1

A B

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C

D E

X2

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B

C

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D

E

A

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X3

C

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D

E

A

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B

X4

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D

E

A

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B

C

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X5

E

A

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B

C

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D

From the above diagram, it is clear that in L.S. design the field is divided into as

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many blocks as there are varieties of fertilizers. Then, each block is again

divided into as many parts as there are varieties of fertilizers in such a way that

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each of the fertilizer variety is used in each of the block only once. The analysis

of L.S. design is very similar to the two-way ANOVA technique.

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4. Factorial design:

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Factorial designs are used in experiments where the effects of varying

more than one factor are to be determined. These designs are used more in

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economic and social matters where usually a large number of factors affect a

particular problem. Factorial designs are usually of two types:

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(i) Simple factorial designs and (ii) complex factorial designs.

(i) Simple factorial design:

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In simple factorial design, the effects of varying two factors on the

dependent variable are considered but when an experiment is done with more

than two factors, complex factorial designs are used. Simple factorial design is

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also termed as a `two-factor-factorial design,' whereas complex factorial design

is known as `multi-factor-factorial design.

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(ii) Complex factorial designs:-

When the experiments with more than two factors at a time are

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conducted, it involves the use of complex factorial designs. A design which

considers three or more independent variables simultaneously is called a

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complex factorial design. In case of three factors with one experimental

variable, two treatments and two levels, complex factorial design will contain a

total of eight cells. This can be seen through the following diagram:

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2x2x2 COMPLEX FACTORIAL DESIGN

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Experimental Variable

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Treatment A

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Treatment B

Control

Control

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Control

Control

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Variable 2

Variable 2

Variable 2

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Variable 2

Level I

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Level II

Level I

Level II

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Level I

Cell 1

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Cell 3

Cell 5

Cell 7

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Control

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Cell 2

Cell 4

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Cell 6

Cell 8

Variable 2

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Level II

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A pictorial presentation is given of the design shown above in the following:

Experimental Variable

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Treatment

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Treatment

A B

Control Variable II

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Level II

Level I

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Level I

Level II

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Control Variable I

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The dotted line cell in this diagram corresponds to cell I of the above stated

2x2x2 design and is for treatment A, level I of the control variable 1, and level I

of the control variable 2. From this design, it is possible to determine the main

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effects for three variables i.e., one experimental and true control variables. The

researcher can also determine the interaction between each possible pair of

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variables (such interactions are called `first order interactions') and interaction

between variable taken in triplets (such interactions are called second order

interactions). In case of a 2x2x2 design, the further given first order interactions

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are possible:

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Experimental variable with control variable 1 (or EV x CV 1);

Experimental variable with control variable 2 (or EV x CV 2);

Control variable 1 with control variable 2 (or CV 1 x CV 2);

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There will be one second order interaction as well in the given design (it is

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between all the three variables i.e., EV x CV 1 x CV 2).

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To determine the main effect for the experimental variable, the

researcher must necessarily compare the combined mean of data in cells 1, 2, 3

and 4 for Treatment A with the combined mean of data in cells 5,6,7 and 8 for

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Treatment B. In this way the main effect experimental variable, independent of

control variable 1 and variable 2, is obtained. Similarly, the main effect for

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control variable 1, independent experimental variable and control variable 2, is

obtained if we compare the combined mean of data in cells 1, 3, 5 and 7 with the

combined mean of data in cells 2, 4, 6 and 8 of our 2x2x2 factorial design. On

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similar lines, one can determine the effect of control variable 2 independent of

experimental variable and control variable 1, if the combined mean of data in

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cells 1,2,5 and 6 are compared with the combined mean of data in cells 3,4,7 and

8.

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To obtain the first order interaction, say, for EV x CV 1 in the above

stated design, the researcher must necessarily ignore control variable 2 for which

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purpose he may develop 2x2 design from the 2x2x2 design by combining the

data of the relevant cells of the latter design as has been shown on next page:

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Experimental Variable

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Treatment A

Treatment B

Control

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Level I

Cells 1.3

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Cells 5,7

Variable 1

Level II

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Cells 2,4

Cells 6,8

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Similarly, the researcher can determine other first order interactions. The

analysis of the first order interaction in the manner described above is essentially

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a simple factorial analysis as only two variables are considered at a time and the

remaining ones are ignored. But the analysis of the second order interaction

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would not ignore one of the three independent variables in case of a 2x2x2

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design. The analysis would be termed as a complex factorial analysis.

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It may, however, be remembered that the complex factorial design need not

necessarily be of 2x2x2 type design, but can be generalized to any number and

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combinations of experimental and control independent variables. Of course, the

greater the number of independent variables included in a complex factorial

design, the higher the order of the interaction analysis possible. But the overall

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task goes on becoming more and more complicated with the inclusion of more

and more independent variables in our design.

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Factorial designs are used mainly because of the two advantages -

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(i) They provide equivalent accuracy (as happens in the case of experiments

with only one factor) with less labour and as such are source of economy. Using

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factorial designs, we can determine the effects of two (in simple factorial

design) or more (in case of complex factorial design) factors (or variables) in

one single experiment. (ii) They permit various other comparisons of interest.

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For example, they give information about such effects which cannot be obtained

by treating one single factor at a time. The determination of interaction effects is

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possible in case of factorial designs.

Conclusion

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There are several research designs and the researcher must decide in advance of

collection and analysis of data as to which design would prove to be more

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appropriate for his research project. He must give due weight to various points

such as type of universe and it's nature, the objective of the study, the source list

or the sampling frame, desired standard accuracy and the like when taking a

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decision in respect of the design for his research project.

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SUMMARY

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Experiment is the process of examining the truth of a statistical hypothesis

related to some research problem. There are two types of experiments - absolute

and comparative. There are three types of research designs - research design for

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descriptive and diagnostic research, research design for exploratory research

studies and research design for hypothesis testing. Prof. Fisher has laid three

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principles of experimental design. They are Principle of Replication, Principle of

Randomization and Principle of Local control. There are different kinds of

experimental designs. Some of them are Informal experimental design, After

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only with control design, Formal experimental design, Completely randomized

design, Randomized block design, Latin square design and Factorial design.

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SELF ASSESMENT QUESTIONS (SAQs)

1.

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Explain the meaning and types of experiment.

(Ref. introduction and types of research design next to introduction)

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2.

Explain informal designs.

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(Ref. i,ii,iii in informal experiment design portion.)

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3. Explain formal experimental design and control.

(Ref. i,ii,iii,iv in formal experiment design section)

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4. Explain complex factorial design.

***

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UNIT II

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4. OBSERVATION

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Lesson Outline

Steps in obMeaning and Characteristics of

observation

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Types of observation

Stages of observation

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servation

Problems and

Merits

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and

Demerits

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Lesson Objectives

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After reading this lesson you will be able to know

Meaning and types of observation

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Stages through which observation passes

Steps followed and the problems coming in observation

Merits and Demerits

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Introduction

Observation is a method that employs vision as its main means of data

collection. It implies the use of eyes rather than of ears and the voice. It is

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accurate watching and noting of phenomena as they occur with regard to the

cause and effect or mutual relations. It is watching other persons' behavior as it

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actually happens without controlling it. For example, watching bonded

labourer's life, or treatment of widows and their drudgery at home, provide

graphic description of their social life and sufferings. Observation is also defined

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as "a planned methodical watching that involves constraints to improve

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accuracy".

CHARACTERISTICS OF OBSERVATION

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Scientific observation differs from other methods of data collection

specifically in four ways: (i) observation is always direct while other methods

could be direct or indirect; (ii) field observation takes place in a natural setting;

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(iii) observation tends to be less structured; and (iv) it makes only the qualitative

(and not the quantitative) study which aims at discovering subjects' experiences

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and how subjects make sense of them (phenomenology) or how subjects

understand their life (interpretivism).

Lofland (1955:101-113) has said that this method is more appropriate for

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studying lifestyles or sub-cultures, practices, episodes, encounters, relationships,

groups, organizations, settlements and roles etc. Black and Champion

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(1976:330) have given the following characteristics of observation:

? Behavior is observed in natural surroundings.
? It enables understanding significant events affecting social relations of the

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participants.

? It determines reality from the perspective of observed person himself.
? It identifies regularities and recurrences in social life by comparing data in

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our study with that of those in other studies.

Besides, four other characteristics are:

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? Observation involves some control pertaining to the observation and to the

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means he uses to record data. However, such controls do not exist for the

setting or the subject population.

? It is focused on hypotheses-free inquiry.

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? It avoids manipulations in the independent variable i.e., one that is supposed

to cause other variable(s) and is not caused by them.

? Recording is not selective.

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Since at times, observation technique is indistinguishable from

experiment technique, it is necessary to distinguish the two.

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(i) Observation involves few controls than the experiment technique.

(ii) The behaviour observed in observation is natural, whereas in

experiment it is not always so.

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(iii) The behavior observed in experiment is more molecular (of a

smaller unit), while one in observation is molar.

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(iv) In observation, fewer subjects are watched for long periods of time

in more varied circumstances than in experiment.

(v) Training required in observation study is directed more towards

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sensitizing the observer to the flow of events, whereas training in experiments

serves to sharpen the judgment of the subject.

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(vi) In observational study, the behavior observed is more diffused.

Observational methods differ from one another along several variables or

dimensions.

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***

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UNIT ? III

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STATISTICAL ANALYSIS

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CONTENTS

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1. Probability

2. Probability distribution

2.1 Binomial distribution

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2.2 Poisson distribution

2.3 Normal distribution

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3. Testing of Hypothesis

3.1 Small sample

3.2 Large sample test

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4. 2 test

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5. Index Number

6. Analysis of Time Series

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OBJECTIVES:

The objectives of the present chapter are:

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i)

To examine the utility of various statistical tools in decision making.

ii)

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To inquire about the testing of a hypothesis

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1. PROBABILITY

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If an experiment is repeated under essentially homogeneous and similar

conditions, we will arrive at two types of conclusions. They are: the results are

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unique and the outcome can be predictable and result is not unique but may be

one of the several possible outcomes. In this context, it is better to understand

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various terms pertaining to probability before examining the probability theory.

The main terms are explained as follows:

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(i)

Random experiment

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An experiment which can be repeated under the same conditions and the

outcome cannot be predicted under any circumstances is known as random

experiment. For example: An unbiased coin is tossed. Here we are not in a

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position to predict whether head or tail is going to occur. Hence, this type of

experiment is known as random experiment.

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(ii)

Sample Space

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A set of possible outcomes of a random experiment is known as sample

space. For example, in the case of tossing of an unbiased coin twice, the

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possible outcomes are HH, HT, TH and TT. This can be represented in a sample

space as S= (HH, HT, TH, TT).

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(iii)

An event

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Any possible outcomes of an experiment are known as an event. In the

case of tossing of an unbiased coin twice, HH is an event. An event can be

classified into two. They are: (a) Simple events, and (ii) compound events.

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Simple event is an event which has only one sample point in the sample space.

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Compound event is an event which has more than one sample point in the

sample space. In the case of tossing of an unbiased coin twice HH is a simple

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event and TH and TT are the compound events.

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(iv)

Complementary event

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A and A' are the complementary event if A' consists of all those sample

point which is not included in A. For instance, an unbiased dice is thrown once.

The probability of an odd number turns up are complementary to an even

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number turns up. Here, it is worth mentioning that the probability of sample

space is always is equal to one. Hence, the P (A') = 1 - P (A).

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(v)

Mutually exclusive events

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A and B are the two mutually exclusive events if the occurrence of A

precludes the occurrence of B. For example, in the case of tossing of an

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unbiased coin once, the occurrence of head precludes the occurrence of tail.

Hence, head and tail are the mutually exclusive event in the case of tossing of an

unbiased coin once. If A and B are mutually exclusive events, then the

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probability of occurrence of A or B is equal to sum of their individual

probabilities. Symbolically, it can be presented as:

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P (A U B) = P (A) + P (B)

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If A and B is joint sets, then the addition theorem of probability can be

stated as:

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P (A U B ) = P(A) + P(B) - P(AB)

(vi)

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Independent event

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A and B are the two independent event if the occurrence of A does not

influence the occurrence of B. In the case of tossing of an unbiased coin twice,

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the occurrence of head in the first toss does not influence the occurrence of head

or tail in the toss. Hence, these two events are called independent events. In the

case of independent event, the multiplication theorem can be stated as the

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probability of A and B is the product of their individual probabilities.

Symbolically, it can be presented as:-

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P (A B) = P (A) * P (B)

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Addition theorem of Probability

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Let A and B be the two mutually exclusive events, then the probability of

A or B is equal to the sum of their individual probabilities. (For detail refer

mutually exclusive events)

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Multiplication theorem of Probability

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Let A and B be the two independent events, then the probability of A and

B is equal to the product of their individual probabilities. (For details refer

independent events)

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Example: The odds that person X speaks the truth are 4:1 and the odds that Y

speaks the truth are 3:1. Find the probability that:-

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(i)

both of them speak the truth,

(ii)

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any one of them speak the truth and

(iii)

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truth may not be told.

Solution: The probability of X speaks the truth = 1/5

The probability that X speaks lie = 4/5

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The probability that Y speaks the truth = 1/4

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The probability that Y speaks lie = ?

(i) Both of them speak truth = P(X) * P(Y) = 1/5 * 1/4 = 1/20

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(independent event)

(ii) any one of them speak truth = P(X) + P(Y) - P(X*Y)

= 1/5 + 1/4 - 1/5*1/4 = 8/20 = 2/5 (not mutually exclusive events)

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(iii)

Truth may not be told

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= 1 ? P(any one of them speak truth)( complementary event)

= 1 ? 2/5 = 3/5.

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2. PROBABILITY

DISTRIBUTION

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If X is discrete random variable which takes the values of x1, x2,x3..... xn

and the corresponding probabilities are p1, p2, ..........pn, then, X follows the

probability distribution. The two main properties of probability distribution are:

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(i) P(Xi) is always greater than or equal to zero and less than or equal to one,

and (ii) the summation of probability distribution is always equal to one. For

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example, tossing of an unbiased coin twice.

Then the probability distribution is:

X (probability of obtaining head): 0 1 2

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P(Xi) : ? ? ?

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Expectation of probability

Let X be the discrete random variable which takes the value of x1, x2,...... xn

then the respective probability is p1, p2, ............ pn, Then the expectation of

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probability distribution is p1x1 + p2x2 + .............. + pnxn. In the above

example, the expectation of probability distribution is (0* ? +1*1/2+2*?) =1.

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2.1 BINOMIAL

DISTRIBUTION

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The binomial distribution also known as `Bernoulli Distribution' is

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associated with the name of a Swiss mathematician, James Bernoulli who is also

known as Jacques or Jakon (1654 ? 1705). Binomial distribution is a probability

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distribution expressing the probability of one set of dichotomous alternatives. It

can be explained as follows:

(i)

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If an experiment is repeated under the same conditions for a fixed

number of trials, say, n.

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(ii)

In each trial, there are only two possible outcomes of the experiment.

Let us define it as "success" or "failure". Then the sample space of possible

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outcomes of each experiment is:

S = [failure, success]

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(iii)

The probability of a success denoted by p remains constant from trial to

trial and the probability of a failure denoted by q which is equal to (1 ? p).

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(iv)

The trials are independent in nature i.e., the outcomes of any trial or

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sequence of trials do not affect the outcomes of subsequent trials. Hence, the

Multiplication theorem of probability can be applied for the occurrence of

success and failure. Thus, the probability of success or failure is p.q.

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(v) Let us assume that we conduct an experiment in n times. Out of which x

times be the success and failure is (n-x) times. The occurrence of success or

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failure in successive trials is mutually exclusive events. Hence, we can apply

addition theorem of probability.

(vi) Based on the above two theorems, the probability of success or failure is

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P(X) = nCxpxqn-x

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n !

--------------- . px qn-x

x ! (n ? x) !

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Where P = Probability of success in a single trail, q = 1 ? p, n = Number of trials

and x = no. of successes in n trials.

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Thus, for an event A with probability of occurrence p and non-

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occurrence q, if n trials are made,

probability distribution of the number of

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occurrences of A will be as set. If we want to obtain the probable frequencies of

the various outcomes in N sets of n trials, the following expression shall be

used: N(p + q)n

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N(p + q)n = Npn + nC1pn-1q + nC2pn-2q2 + ......+ nCrpn-rqr + ......qn.

The frequencies obtained by the above expansion are known as expected

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or theoretical frequencies. On the other hand, the frequencies actually obtained

by making experiments are called actual or observed frequencies. Generally,

there is some difference between the observed and expected frequencies but the

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difference becomes smaller and smaller as N increases.

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Obtaining Coefficient of the Binomial Distribution:

The following rules may be considered for obtaining coefficients from

the binomial expansion:

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(i)

The first term is qn.,

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(ii)

The second term is nC1qn-1p,

(iii)

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In each succeeding term the power of q is reduced by 1 and the power of

p is increased by 1.

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(iv)

The coefficient of any term is found by multiplying the coefficient of the

preceding term by the power of q in that preceding term, and dividing the

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products so obtained by one more than the power of p in that proceeding

term.

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Thus, when we expand (q + p)n, we will obtain the following:-

(p + q)n = pn + nC1pn-1q + nC2pn-2q2 + ......+ nCrpn-rqr + ......qn.

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Where, 1, nC1, nC2 ....... are called the binomial coefficient. Thus in the

expansion of (p + q)4 we will have (p + q)4 = p4 +4p3q +6p2q2 + 4p1q3 + q4 and

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the coefficients will be 1, 4, 6, 4, 1.

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From the above binomial expansion, the following general relationships

should be noted:

(i)

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The number of terms in a binomial expansion is always n + 1,

(ii)

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The exponents of p and q, for any single term, when added together,

always sum to n.

(iii)

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The exponents of p are n, (n ? 1), (n ? 2),.......1, 0, respectively and the

exponents of q are 0, 1, 2, ......(n ? 1), n, respectively.

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(iv)

The coefficients for the n + 1 terms of the distribution are always

symmetrical in nature.

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Properties of Binomial Distribution

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The main properties of Binomial Distribution are:-

(i)

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The shape and location of binomial distribution changes as p changes for

a given n or as n changes for a given p. As p increases for a fixed n, the

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binomial distribution shifts to the right.

(ii) The mode of the binomial distribution is equal to the value of x which

has the largest probability. The mean and mode are equal if np is an integer.

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(iii) As n increases for a fixed p, the binomial distribution moves to the right,

flattens and spreads out.

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(iv) The mean of the binomial distribution is np and it increases as n

increases with p held constant. For larger n there are more possible outcomes of

a binomial experiment and the probability associated with any particular

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outcome becomes smaller.

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(v) If n is larger and if neither p nor q is too close to zero, the binomial

distribution can be closely approximated by a normal distribution with

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standardized variable given by z = (X ? np) / npq.

(vi) The various constants of binomial distribution are:

Mean

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=

np

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Standard Deviation

=

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npq

?1

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= 0

?2

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npq

?3

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npq(q ? p)

?4

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=

3n2p2q2 + npq(1 ? 6pq).

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(q ? p)2

Skewness

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=

---------

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npq

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1 ? 6pq

Kurtosis

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3 + ---------

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npq

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Illustrations: A coin is tossed four times. What is the probability of obtaining

two or more heads?

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Solution: When a coin is tossed the probabilities of head and tail in case of an

unbiased coin are equal, i.e., p = q = ?

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The various possibilities for all the events are the terms of the expansion (q+p)4

(p ? q)4 = p4 + 4p3q + 6p2q2 + 4p1q3 + q4

Therefore, the probability of obtaining 2 heads is

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6p2q2 = 6 x (?)2(?)2 = 3/8

The probability of obtaining 3 heads is 6p3q1 = 4 x (?)3(?)1 = 1/4

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The probability of obtaining 4 heads is (q)4 = (?)4 = 1/16

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Therefore, the probability of obtaining 2 or more heads is

3 1 1 11

--- + --- + --- = -----

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8 4 16 16

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Illustration: Assuming that half the population is vegetarian so that the chance

of an individual being a vegetarian is ? and assuming that 100 investigations

can take sample of 10 individuals to verify whether they are vegetarians, how

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many investigation would you expect to report that three people or less were

vegetarians?

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Solution:

n = 10, p, i.e., probability of an individual being vegetarian = ?.q =1 ? p= ?

Using binomial distribution, we have P(r) = ncr qn-rpr

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Putting the various values, we have

1

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10c r(?)r(?)10 ? r = 10cr = (?)10 = --------10cr

1024

The probability that in a sample of 10, three or less people are vegetarian shall

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be given by: P(0) + P(1) + P(2) + P(3)

1

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= --------- [10c0 + 10c1 + 10c2 + 10c3]

1024

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176 11

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= --------- [ 1 + 10 + 45 + 120] = -------- = -----

1024

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1024 64

Hence out of 1000 investigators, the number of investigators who will

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report 3 or less vegetarians in a sample of 10 is 1000 x --- = 172.

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2.2 POISSON

DISTRIBUTION

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Poisson distribution was derived in 1837 by a French Mathematician Simeon

D Poisson (1731 ? 1840). In binomial distribution, the values of p and q and n

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are given. There is a certainty of the total number of events. But there are cases

where p is very small and n is very large and such case is normally related to

Poisson distribution. For example, persons killed in road accidents, the number

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of defective articles produced by a quality machine. Poisson distribution may be

obtained as a limiting case of binomial probability distribution, under the

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following condition.

(i) p, successes, approach zero (p 0)

(ii) np = m is finite.

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The Poisson distribution of the probabilities of occurrence of various rare

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events (successes) 0,1,2,.... are Given below:

Number of success (X)

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Probabilities p(X)

0

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e-m

1

me-m

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m2e-m

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2

--------

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r

mre-m

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--------

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mne-m

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--------

n!

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Where, e = 2.718, and m = average number of occurrence of given distribution.

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The Poisson distribution is a discrete distribution with a parameter m.

The various constants are:

(i)

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Mean

=

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m

=

p

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(ii) Standard

Deviation

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= m

(iii) Skewness

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1

= 1/m

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(iv) Kurtosis,

2

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=

3 + 1/m

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(v) Variance

= m

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Illustration: A book contains 100 misprints distributed randomly throughout its

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100 pages. What is the probability that a page observed at random contains at

least two misprints? Assume Poisson Distribution.

Solution:

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Total Number of misprints 100

m = ------------------------------- = ----- = 1

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Total number of pages 100

Probability that a page contains at least two misprints:

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p(r2) = 1 ? [p(0) + p (1)]

mre-m

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p(r) = --------

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r!

10e-1 1 1

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p(0) = ------ = e-1 = ---- = ---------

0! e 2.7183

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11e-1 1 1

p(1) = ------ = e-1 = ---- = ---------

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1! e 2.7183

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1 1

p(0) + p(1) = ----------- + ----------- = 0.736

2.718 2.718

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P(r2) = 1 ? [p(0) + p (1)] = 1-0.736 = 0.264

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Illustration: If the mean of a Poisson distribution is 16, find (1) S.D.(2) B1

(3) B2 (4) ?3 (5) ?4

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Solution: m = 16

1. S.D.

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=

m = 16 = 4

2.

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1 = 1/m = 1/16 = 0.625

3.

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2 = 3 + 1/m = 3 + 0.625 = 3.0625

4.

?3 = m = 16

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5.

?4 = m + 3m2 = 16 + 3(16)2 = 784

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2.3 NORMAL

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DISTRIBUTION

The normal distribution was first described by Abraham Demoivre (1667-1754)

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as the limiting form of binomial model in 1733. Normal distribution was

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rediscovered by Gauss in 1809 and by Laplace in 1812. Both Gauss and

Laplace were led to the distribution by their work on the theory of errors of

observations arising in physical measuring processes particularly in astronomy.

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The probability function of a Normal Distribution is defined as:

1 -(x - ?)2 / 22

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P(X) = ------------ e

2

Where, X = Values of the continuous random variable, ? = Mean of the normal

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random variable, e = 2.7183, = 3.1416

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Relation between Binomial, Poisson and Normal Distributions

Binomial, Poisson and Normal distribution are closely related to one other.

When N is large while the probability P of the occurrence of an event is close to

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zero so that q = (1-p) the binomial distribution is very closely approximated by

the Poisson distribution with m = np.

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The Poisson distribution approaches a normal distribution with standardized

variable (x ? m)/ m as m increases to infinity.

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Normal Distribution and its properties

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The important properties of the normal distribution are:-

1. The normal curve is "bell shaped" and symmetrical in nature. The distribution

of the frequencies on either side of the maximum ordinate of the curve is similar

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with each other.

2. The maximum ordinate of the normal curve is at x = ?. Hence the mean,

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median and mode of the normal distribution coincide.

3. It ranges between - to +

4. The value of the maximum ordinate is 1/ 2.

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5. The points where the curve change from convex to concave or vice versa is at X

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= ? ? .

6. The first and third quartiles are equidistant from median.

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7. The area under the normal curve distribution are:

a)

? ? 1 covers 68.27% area;

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b)

? ? 2 covers 95.45% area.

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c)

? ? 3 covers 99.73% area.

68.27%

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95.45%

99.73%

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? - 36 ? - 26

? - 16

? = 0

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? + 16

? + 26

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? + 36

- 3

- 2

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- 1

Z = 0

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+ 1

+ 2

+ 3

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8. When

? = 0 and = 1, then the normal distribution will be a standard

normal curve. The probability function of standard normal curve is

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1 -x2/2

P(X) = ------------ e

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2

The following table gives the area under the norm

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ability curve for

al prob

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some important value of Z.

Distance from the mean ordinate in Are

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a under the curve

Terms of ?

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Z

=

?

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0.6745

0.50

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?

1

Z

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=

.0

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0.6826

?

1

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Z

=

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.96

0.95

Z

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=

?

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2.00

0.9544

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Z

=

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?

2.58

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0.99

Z = ? 3.0

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0.9973

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All odd moments are equal to zero.

10.

Skewness = 0 and Kurtosis = 3 in normal distribution.

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Illustration: Find the probability that the standard normal value lies between 0

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and 1.5

0.4332 (43.32%)

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Z = 0

Z = 1.5

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As the mean, Z = 0.

To find the area between Z = 0 and Z = 1.5, look the area between 0 and 1.5,

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from the table. It is 0.4332 (shaded area)

Illustration: The results of a particular examination are given below in a

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summary form:

Result

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Percentage of candidates

Passed with distinction

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10

Passed

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60

Failed

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It is known that a candidate gets plucked if he obtains less than 40

marks, out of 100 while he must

in

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obta

at least 75 marks in order to pass with

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distinction. Determine the mean and standard deviation of the distribution of

marks assumi

a

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ng this to be norm l.

Solution:

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30% students get marks less tha 40.

n

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40 ?

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X

Z = ---------- = -0.52 (from the ta l

b e)

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30%

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20%

40%

10%

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40 ? X = -0.52 -----------

(i)

o

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10% students get m re than 75

40% area

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= 75 ? X = 1.28

------------ (ii)

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= 75 ? X = 1.28

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t

Subtrac (ii) from (i)

40 ? X = -0.52

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75 ? X = 1.28

--------------------

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-35 = -1.8

35 = 1.8

1.80 = 35

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35

= ------- = 19.4

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1.80

Mean

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40 ? X = -0.52 x (19.4)

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-X = -40 ? 10.09 = 50.09

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Illustration: The scores made by candidate in a certain test are normally

distributed with mean 1000 and standard deviation 200. what per cent of

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candidates receive scores (i) less than 800, (ii) between 800 and 1200?

(the area under the curve between Z = 0 and Z = 1 is 0.34134).

Solution:

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X = 1000; = 200

X ? X

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Z = ----------

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(i) For X = 800

800 ? 1000

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Z = ------------- = -1

200

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Area between Z = -1 and Z = 0 is 0.34134

Area for Z = -1 = 0.5 ? 0.34134 = 0.15866

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Therefore, the percentage = 0.15866 x 100 = 15.86%

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(ii)

en,

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Wh

X = 1200,

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1200 ? 1000

Z = -------------- = 1

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200

Area between Z = 0 and Z = 1 is 0.34134

Area between X = 400 to X = 600

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i.e., Z = -1 and Z = 1 is 0.34134 + 0.34134 = 0.6826 = 68.26%

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0.6826

0.1586

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800

1000

1200

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TESTING OF HYPOTHESIS

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3.1

Test of Significance for Large Samples

The test of significance for the large samples can be explained by the following

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assumptions:

(i)

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The random sampling distribution of statistics is approximately normal.

(ii)

Sampling values are sufficiently close to the population value and can be

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used for the calculation of standard error of estimate.

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1.

The standard error of mean.

In the case of large samples, when we are testing the significance of statistic, the

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concept of standard error is used. It measures only sampling errors. Sampling

errors are involved i

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n estimating a population parameter from a sample, instead

of including all the essential information in the population.

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(i)

when standard deviation of the population is known, the formula is

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p

S.E. X = ----

n

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Where,

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S.E.X = The standard error of the mean, p = Standard deviation of the

population, and n = Number of observations in the sample.

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(ii)

When standard deviation of population is not known, we have to use the

standard deviation of the sample in calculating standard error of mean. The

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formula is

(Sample)

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S.E. X = ------------

n

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Where,

= standard deviation of the sample, and n = sample size

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Illustration: A sample of 100 students from Pondicherry University was taken

and their average was found to be 116 lbs with a standard deviation of 20 lbs.

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Could the mean weight of students in the population be 125 pounds?

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Solution:

Let us take the hypothesis that there is no significant difference between the

sample mean and the hypothetical population mean.

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20 20

S.E. X = ---- = -------- = -------- = 2

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n 100 10

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Difference 125 ? 116 9

-------------- = ------------- = ------- = 4.5

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S.E.X 2 2

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Since, the difference is more than 2.58 S.E.( %

1

level) it could not have arisen

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due to fluctuations of sampling. Hence the mean weight of students in the

population could not be 125 lbs.

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3.2

Test of Significance for Small Samples

If the sample size is less than 30, then those samples may be regarded as

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small samples. As a rule, the methods and the theory of large samples are not

applicable to the small samples. The small samples are used in testing a given

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y

h pothesis, to find out the observed values, which could have arisen by

sampling fluctuations from some values given in advance. In a small sample,

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the investigator's estimate will vary widely from sample to sample. An

inference drawn from a smaller sample result is less precise than the inference

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drawn from a large sample result.

t-distribution will be employed, when the sample size is 30 or less and

the population standard deviation is unknown.

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The formula is

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( X - ?)

t = ------- x n

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Where, = (X ? X)

2/n ? 1

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Illustratio :

n The following results are obtained from a sample of 20 boxes of

--- Content provided by FirstRanker.com ---

a

m ngoes:

Mean weight of contents = 490gms,

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Standard deviation of the weight = 9 gms.

Could the sample come from a population having a mean of 500 gms?

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Solution:

Let us take the hypothesis that ? = 510 gms.

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( X - ?)

t = ------- x n

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X = 500; ? = 510; = 10; n = 20.

500 ? 510

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t = ------------- x 20

10

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Df = 20 ? 1 = 19 = (10/9) 20 = (10/9) x 4.47 = 44.7/9 = 4.96

Df = 19, t0.01 = 3.25

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The computed value is less than the table value. Hence, our null hypothesis is

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accepted.

4. CHI-SQUARE

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TEST

F, t and Z tests were based on the assumption that the samples were drawn from

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nor a

m lly distributed populations. The testing procedure requires assumption

about the type of population or parameters, and these tests are known as

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`parametric tests'.

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There are many situations in which it is not possible to make any rigid

assumption about the distribution of the population from which samples are

being drawn. This limitation has led to the development of a group of

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alternative techniques known as n

a

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on-p rametric tests. Chi-square test of

independence and goodness of fit is a prominent example of the use of non-

parametric tests.

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Though non-parametric theory developed as early as the middle of the

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nineteenth century, it was only after 1945 that non-parametric tests came to be

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used widely in sociological and psychological research. The main reasons for

the increasing use of non-parametric tests in business research are:-

(i)

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These statistical tests are distribution-free

(ii)

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They are usually computationally easier to handle and understand than

e

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param tric tests; and

(iii)

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They can be used with type of measurements that prohibit the use of

parametric tests.

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The 2 test is one of the simplest and most widely used non-parametric

tests in statistical work. It is defined as:

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(O ? E)2

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2 = ------------

E

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Where O = the observed frequencies, and E = the expected frequencies.

Steps: The steps required to determine the value of 2are:

(i)

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Calculate the expected frequencies. In general the expected frequency

for any cell can be calculated from

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:

the following equation

R X C

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E = ------------

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N

Where E = Expected frequency, R = row's total of the respective cell, C =

column's total of the respective cell and N = the total num

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bservations.

ber of o

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(ii)

Take the difference between observed and expected frequencies and

obtain the squares of these differences. Symbolically, it can be represented as

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(O ? E)2

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(iii)

Divide the values of (O ? E)2 obtained in step (ii) by the respective

expected frequency and obtain the total, which can be symbolically represented

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by [(O ? E)2/E]. This gives the value of 2 which can range from zero to

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infinity. If 2 is zero it means that the observed and expected frequencies

completely coincide. The greater the discrepancy between the observed and

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expected frequencies, the greater shall be the value of 2.

The computed value of 2 is compared with the table value of 2 for

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given degrees of freedom at a certain specified level of significance. If at the

stated level, the calculated value of 2 is less than the table value, the difference

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between theory and observation is not considered as significant.

The following observation may be made with regard to the 2

distribution:-

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(i)

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The sum of the observed and expected frequencies is always zero.

Symbolically,

) =

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(O ? E

O - E = N ? N = 0

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(ii)

The 2 test depends only on the set of observed and expected frequencies

and on degrees of freedom v. It is a non-parametric test.

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(iii)

2 distribution is a limiting approximation of the multinomial

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distribution.

(iv)

Even though 2 distribution is essentially a continuous distribution it can

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be applied to discrete random variables whose frequencies can be counted and

tabulated with or without grouping.

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The Chi-Square Distribution

For large sample sizes, the sampling distribution of 2 can be closely

approximated by a continuous curve known as the Chi-square distribution. The

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probability function of 2 distribution is:

F(2) = C (2)(v/2 ? 1)e ? x2/2

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Where e = 2.71828, v = number of degrees of freedom, C = a constant

depending only on v.

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The 2 distribution has only one parameter, v, the number of degrees of

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freedom. As in case of t-distribution there is a distribution for each different

number of degrees of freedom. For very small number of degrees of freedom,

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the Chi-square distribution is severely skewed to the right. As the number of

degrees of freedom increases, the curve rapi l

d y becomes more symmetrical. For

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large values of v the Chi-square distribution is closely approximated by the

normal curve.

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The following diagram gives 2 distribution for 1, 5 and 10 degrees of

freedom:

F(x2)

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v = 1

v = 5

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v = 10

0

2

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4

6

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8

10

12

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14

16 18

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20

22

2

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2 Distribution

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It is clear from the given diagram that as the degrees of freedom

increase, the curve becomes more and more symmetric. The Chi-square

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distribution is a probability distribution and the total area under the curve in

each chi-square distribution is unity.

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Properties of 2 distribution

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The main Properties of 2 distribution are:-

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(i)

the mean of the 2 distribution is equal to the number of degrees of freedom, i.e.,

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X = v

(ii)

the variance of the 2 distribution is twice the degrees of freedom, Variance =

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2v

(iii)

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?1 = 0,

(iv)

?2 = 2v,

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(v)

?3 = 8v,

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(vi)

?4 = 48v + 12v2.

2

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?3 64v2 8

(vii) 1 = --- = ----- = --

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2

?2 8v3 v

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?4 48v + 12v2

12

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)

(v

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1?3 = ---- = --------------- = 3 + ---

2

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?2

4v2

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The table values of 2

are available only up to 30 degrees of freedom.

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For degrees of freedom greater than 30, the distribution of 22 approximates

the normal distribution. For degrees of freedom greater than 30, the

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approximation is acceptable close. The mean of the distribution 22 is 2v ? 1,

and the standard deviation is equal to 1. Thus the application of the test is

simple, for deviation of 22 from 2v ? 1 may be interpreted as a normal

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deviate with units standard deviation. That is,

Z = 22 - 2v ? 1

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Alternative Method of Obtaining the Value of 2

In a 2x2 table where the cell frequencies and marginal totals are as below:

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(a+b)

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d

(c+d)

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(a+c)

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(b+d)

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N

N is the total frequency and ad the larger cross-product, the value of 2

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can easily be obtained by the following formula:

N (ad ? bc)2

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2 = --------------------------------- or

(a + c) (b + d) (c + d) (a + b)

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With Yate's corrections

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N (ab ? bc - ?N)2

2

= -----------------------------------

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(a + c) (b + d) (c + d) (a + b)

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Conditions for applying 2 test:

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The main conditions considered for employing the 2 test are:

(i) N must be to ensure the similarity between theoretically correct distribution and

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our sampling distribution of 2.

(ii) No theoretical cell frequency should be small when the expected frequencies are

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too small. If it is so, then the value of 2 will be overestimated and will result in

too many rejections of the null hypothesis. To avoid making incorrect

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inferences, a general rule is followed that expected frequency of less than 5 in

one cell of a contingency table is too small to use. When the table contains

more than one cell with an expected frequency of less than 5 then add with the

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preceding or succeeding frequency so that the resulting sum is 5 or more.

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However, in doing so, we reduce the number of categories of data and will gain

less information from contingency table.

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(iii) The constraints on the cell frequencies if any should be linear, i.e., they should

not involve square and higher powers of the frequencies such as O = E = N.

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Uses of 2 test:

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The main uses of 2 test are:

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(i)

2 test as a test of independence. With the help of 2 test, we can find

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out whether two or more attributes are associated or not. Let's assume that we

have N observations classified according to some attributes. We may ask

whether the attributes are related or independent. Thus, we can find out whether

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there is any association between skin colour of husband and wife. To examine

the attributes that are associated, we formulate the null hypothesis that there is

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no association against an alternative hypothesis and that there is an association

between the attributes under study. If the calculated value of 2 is less than the

table value at a certain level of significance, we say that the result of the

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experiment provides no evidence for doubting the hypothesis. On the other

hand, if the calculated value of 2 is greater than the table value at a certain level

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of significance, the results of the experiment do not support the hypothesis.

(ii)

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2 test as a test of goodness of fit. This is due to the f

le

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act that it enab s

us to ascertain how appropriately the theoretical distributions such as binomial,

Poisson, Normal, etc., fit empirical distributions. When an ideal frequency

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curve whether normal or some other type is fitted to the data, we are interested

in finding out how well this curve fits with the observed facts. A test of the

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concordance of the two can be made just by inspection, but such a test is

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obviously inadequate. Precision can be secured by applying the 2 test.

(iii)

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2 test as a test of Homogeneity. The 2 test of homogeneity is an

extension of the chi-square test of independence. Tests of homogeneity are

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designed to determine whether two or more independent random samples are

drawn from the same population or from different populations. Instead of one

sample as we use with independence pr

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em

obl

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we shall now have 2 or more

samples. For example, we may be interested in finding out whether or not

university students of various levels, i.e., mi

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e

ddle and richer poor incom groups

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are homogeneous in perform

ination.

ance in the exam

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Illustration: In an anti-diabetes campaign in a certain area, a particular

e

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m dicine, say x was administered to 812 persons out of a total population of

3248. The num er of

b

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diabetes cases is shown below:

Treatment

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Diabetes

No

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Total

Diabetes

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Medicine

x

20

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2

812

No

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Medicine

x 220

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2216

2436

Total

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240

08

3248

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Discuss the usefulness of medicine x in checking malaria.

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Solution: Let us take the hypothesis that quinine is not effective in checking

diabetes. Applying 2 test :

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(A) X (B) 240 x 812

Expectation of (AB) = ------------ = ------------ = 60

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N 3248

Or E1, i.e., expected frequency corresponding to fi

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nd first colum

rst row a

n is 60.

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the bale of expected frequencies shall be:

60 752 812

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180 2256 2436

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240 3008 3248

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(O ? E)2

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(O ? E)2/E

20

60

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1600

26.667

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220

180

1600

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8.889

792

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752

1600

2.218

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2216

2256

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1600

0.709

2

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[(O ? E) /E] = 38.593

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2

2

= [(O ? E) /E] = 38.593

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v = (r ? 1) (c ? 1) = (2 ? 1) (2 ? 1) = 1

2

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for v = 1, 0.05 = 3.84

The calculated value of 2 is greater than the table value. The hypothesis is

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rejected. Hence medicine x is useful in checking malaria.

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Illustration: In an experiment on immunization of cattle from tuberculosis the

following results were obtained:

Affected Not

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affected

Inoculated

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Not inoculated

15

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Calculate 2 and discuss the effect of vaccine in controlling susceptibility to

tuberculosis (5% value of 2 for one degree of freedom = 3.84).

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Solution: Let us take the hypothesis that the vaccine is not effective in

controlling susceptibility to

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is. Applying

tuberculos

2 test:

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N(ad ? bc)2 50 (11x5 ? 20x15)2

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2 = -------------------------- = ------------------------ = 8.3

(a+b) (c+d)(a+c)(b+d) 30x20x25x25

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Since the calculated value of 2 is greater than the table value the hypothesis is

not true. We, therefore, conclude the vaccine is effective in controlling

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susceptibility to tuberculosis.

5. INDEX

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NUMBERS

An Index Number is used to measure the level of a certain phenomenon

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as compared to the level of the same

enon at som

phenom

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e standard period. An

Index Number is a statistical device for comparing the general level of

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magnitude of a group of related variables in two or more situations. If we want

to compare the price level of 2004 with what it was in 2000, we may have to

look into a group of variables ? prices of rice, wheat, vegetables clothes, etc.

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Hence, we will have one figure to indicate the changes of different commodities

as a whole and it is cal

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b

led an Index Num er.

Utility of Index Number:

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The main uses of index numbers are:

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(i)

Index Numbers are particularly useful in measuring relative changes. Example

Changes in level of price, production, etc.

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(ii) Index numbers are economic barometers. Various index numbers computed for

different purposes, like employment, trade, agriculture are of immense value in

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dealing with different economic problems.

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(iii) Index numbers are useful to compute the standard of living. Index numbers may

measure the cost of living of different classes and comparison across groups

becomes easier.

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(iv) They help in form

policies.

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ulating

For instance increase or decrease in wages

require the study of the cost of living index numbers.

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Steps in construction of Index Numbers:

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The main steps involved in the construction of index numbers are:

)

(i

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Purpose. The researcher must clearly define the purpose for which the index

numbers are to be constructed. For example, cost of living index numbers of

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workers in an industrial area and those of the workers of an agricultural area are

different in respect of requirement. So, it is very essential to define the purpose

of the index numbers.

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(i

Selection of Base. The base period is important for the construction of index

numbers. When we select a base year, the year must be recent and normal. A

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normal year is one which is free from economic and natural, social and

economic disturbance. Besides, when we are selecting the base period one of

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the following criteria should be considered (a) Fixed base, (b) Average base, (b)

.

Chain Base

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(iii) Selection of commodities. We should in

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o

clude important c mmodities and they

are represen

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th

tative of

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e defined purpose. For the purpose of finding the cost of

living index number for lo

e

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w incom groups, the selected items should be mostly

consumed by that group.

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(iv) Sources of data. The price relating to

be

the thing to

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measured must be

ollected. If we want to study the ch

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c

anges in industrial production, we must

lect

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col

ting to the pr

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the prices rela

oduction of various goods of factories.

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(v) Weighting. All commodities are not equally important because different groups

of people will have different preferences on different commodities. For

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instance, when the price of rice is doubled than the price of ice-cream, then the

people suffer much, due

price of rice which is essential. Therefore, a

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to hike in

ould be given for each

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relative weight sh

commodity based on its importance.

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(vi) Choice of Formulae. The index number computed based on different formulas

roduce different results. Hence,

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usually p

the problem is perhaps of greater

theoretical than practical importance. In general, choice of the formula to be

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used depends upon the availabi

nature, purpose and scope of

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lity of data and the

the study.

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The various methods of construction of index number are:

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1.

Unweighted

(a)

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Simple Aggregate

(b)

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Simple average of price relative

2.

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Weighted

(a)

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Weighted Aggregate

(b)

Weighted average of price relative.

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Unweighted

(a)

Simple Aggregate method.

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The price of the different commodities of the current year is added to the total

and it is divided by the sum of the prices of the base year commodity and

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multiplied by 100: symbolically,

1

P x 100

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P01 = -------------

P0

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Where,

01

P = Price index number for the current year with reference to the base year.

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P1 = Aggregate of prices for the current year, and
P0 = Aggregate of prices for the base year.

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(b)

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Simple average of price relative method.

Under this method, the price relative of each item is calculated separately and

then averaged. A price relative is the price of the current year expressed as a

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percentage of the price of the base year:

P1 x 100

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------------

P0

P

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P01 = ------------------ = -----

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N

Where, N = Number of ite s

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m , P = P1 x 100 / P0

If we employ geometric ean in the pl

m

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ace of the arithmetic mean the

n the

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formula is

--- Content provided by‍ FirstRanker.com ---

P1 x 100

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log ------------

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P0 logP

P01 = antilog ------------------ = antilog -------

--- Content provided by FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

N

Illustration: Compute a price index for the following by (a) simple aggregate

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and (b) average of price relative method by using both arithmetic mean and

geometric mean:

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Co

mmodity

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A B C D E F

rice

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P

in

2000

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(Rs.)

20 30 10 25 40 50

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Price

in

2005

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(Rs.)

15 35 45 55

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25 30

Solution: Ca

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Price Index

lculation for

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Commodity Price

in

Price

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2000

in 2005 Price relative

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log P

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--- Content provided by‍ FirstRanker.com ---

P1 P= P1/P0 x 100

A

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20

25

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125

2.0969

B

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30

30

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100

2.0000

C

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10

15

--- Content provided by⁠ FirstRanker.com ---

150

2.1761

D

--- Content provided by FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

35

140

2.1461

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--- Content provided by‌ FirstRanker.com ---

40

45

112.5

--- Content provided by‌ FirstRanker.com ---

2.0511

F

--- Content provided by‌ FirstRanker.com ---

50

55

--- Content provided by⁠ FirstRanker.com ---

110

2.0414

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175 205

737.5 12.5116

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P1 x 100

--- Content provided by⁠ FirstRanker.com ---

(a) Simple Aggregative Index = -----------

P0

--- Content provided by FirstRanker.com ---

P0 = 175, P1 = 205

205

--- Content provided by FirstRanker.com ---

= ----- x 100 = 117.143

175

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(b) (i) Arithmetic mean of Price

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Relatives = P / N

P = 737.5, N = 6

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= 737.5 / 6 = 122.92

(ii) Geometric Mean of Price

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--- Content provided by‍ FirstRanker.com ---

logP

Relative Index = Antilog ----------

--- Content provided by FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

12.5116

= Antilog ----------- = Antilog 2.0853 = 121.7

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--- Content provided by FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

Weighted Index Numbers

ethod, prices them

Under this m

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selves are weighted by quantities, i.e., p*q. Thus

physical quantities are used

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ights. The diffe

as we

rent methods of assigning

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weights are:

(a)

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's

Laspeyre

method,

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(b)

Paasche's

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me

thod,

--- Content provided by FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

ethod,

Dorfish

m

--- Content provided by‌ FirstRanker.com ---

(d)

--- Content provided by‍ FirstRanker.com ---

Fisher's Ideal method,

(e)

--- Content provided by⁠ FirstRanker.com ---

Marshall Edgworth method,

(f)

--- Content provided by FirstRanker.com ---

Kelley's

Method

--- Content provided by‌ FirstRanker.com ---

)

(a

--- Content provided by FirstRanker.com ---

Laspeyre's method.

Under this me

uantitie

--- Content provided by FirstRanker.com ---

thod, the base year q

s are taken as weights: symbolically,

--- Content provided by⁠ FirstRanker.com ---

p1q 0

--- Content provided by‍ FirstRanker.com ---

P01(La) = --------- x 100

--- Content provided by‍ FirstRanker.com ---

p0q 0

--- Content provided by FirstRanker.com ---

(b)

Paasche's method.

The current year quantities are taken as weights under Paasche's method:

--- Content provided by FirstRanker.com ---

symbolically,

p1q 1

--- Content provided by‍ FirstRanker.com ---

P01(Pa) = -------- x 100

--- Content provided by‌ FirstRanker.com ---

p0q 1

--- Content provided by‌ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

methods; symbolically

p

--- Content provided by FirstRanker.com ---

1q 0 p1q 1

-------- + --------

--- Content provided by‍ FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

0q 0 p0q 1 L + P

P01(B) = ---------------------- x 100 = ------

--- Content provided by‍ FirstRanker.com ---

2 2

Where, L = Laspeyre's method & P = Paasche's method.

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(d)

Fisher's Ideal method.

--- Content provided by FirstRanker.com ---

Fisher's price index n m

u ber is given by the geometric mean of Laspeyre's and

--- Content provided by FirstRanker.com ---

Paasche's Index; symbolically,

p

--- Content provided by FirstRanker.com ---

1q 0 p1q 1

P01(F) = L x P = -------- x -------- x 100

--- Content provided by‌ FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

0q 0 p0q 1

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

)

(e

d

--- Content provided by‌ FirstRanker.com ---

Marshall E geworth Method

--- Content provided by FirstRanker.com ---

p1 (q 1 + q 0)

0

P M

--- Content provided by‌ FirstRanker.com ---

1( a) = -----------------

--- Content provided by FirstRanker.com ---

0

p (q 0 + q 1)

--- Content provided by‌ FirstRanker.com ---

By removal of brackets,

--- Content provided by‌ FirstRanker.com ---

p

1q 0 p1q 1

--- Content provided by‌ FirstRanker.com ---

P01(Ma) = -------- + -------- x 100

--- Content provided by FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

0q 0

p0

--- Content provided by⁠ FirstRanker.com ---

1

q

--- Content provided by FirstRanker.com ---

(f)

--- Content provided by FirstRanker.com ---

Kelley's method.

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p1q

--- Content provided by‌ FirstRanker.com ---

P01(K) = ------- x 100

p0q

--- Content provided by‍ FirstRanker.com ---

q = q 0 + q1 2

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--- Content provided by‍ FirstRanker.com ---

Illustrations:

Calculate various weighted index number from

--- Content provided by FirstRanker.com ---

lowing data:

the fol

--- Content provided by‍ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

Base ye

Current year

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Kilo

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Rate (Rs.)

Kilo

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Rate (Rs.)

Bread

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10

3

10

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--- Content provided by FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

Meat

20

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15

16

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Tea

2

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--- Content provided by‍ FirstRanker.com ---

3

30

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--- Content provided by‌ FirstRanker.com ---

Solution:

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Base year

Current year

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Kilo Rate Kilo Rate p1q 0

--- Content provided by‍ FirstRanker.com ---

p0q 0

p1q1

p0q1

--- Content provided by FirstRanker.com ---

(Rs.)

(Rs.)

--- Content provided by FirstRanker.com ---

Q 0

p0

Q1

--- Content provided by‍ FirstRanker.com ---

p1

Bread

--- Content provided by FirstRanker.com ---

10

3.00

10

--- Content provided by FirstRanker.com ---

4.00

40.00

--- Content provided by‌ FirstRanker.com ---

30.00

40.00

30.00

--- Content provided by FirstRanker.com ---

Meat

20

--- Content provided by FirstRanker.com ---

15.0

16

20.0

--- Content provided by FirstRanker.com ---

400.00

300.00

--- Content provided by‌ FirstRanker.com ---

320.00 240.00

Tea

2

--- Content provided by⁠ FirstRanker.com ---

0

3

--- Content provided by FirstRanker.com ---

0

60.00

40.00

--- Content provided by⁠ FirstRanker.com ---

90.00

60.00

--- Content provided by‌ FirstRanker.com ---

20.0

30.0

0

--- Content provided by⁠ FirstRanker.com ---

0

Total

--- Content provided by FirstRanker.com ---

500.00

370.00

450.00 330.00

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(a) Laspeyre's

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method

p1q 0 x 100 500 x 100

P01(La) = ----------------- = ---------------- = 135.1

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

p0q 0 370.00

(b)

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Paasche's method

p1q 1 x 100 450 x 100

--- Content provided by FirstRanker.com ---

P01(Pa) = ----------------- = -------------- = 136.4

--- Content provided by‍ FirstRanker.com ---

p0q 1 370

--- Content provided by FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

1q 0 p1q 1

-------- + --------

--- Content provided by FirstRanker.com ---

p0q 0 p0q 1 L + P

--- Content provided by‌ FirstRanker.com ---

P01( = ------------

B)

--- Content provided by‍ FirstRanker.com ---

---------- x 100 = ------

2 2

--- Content provided by FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

L + P 135.1 + 136.1

--- Content provided by‍ FirstRanker.com ---

= ------- = ------------------ = 135.8

2 2

--- Content provided by‌ FirstRanker.com ---

(d)

Fisher's ideal formula

--- Content provided by‍ FirstRanker.com ---

p

1q 0 p1q 1

--- Content provided by‌ FirstRanker.com ---

P01(F) = L x P = -------- x -------- x 100

--- Content provided by FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

p

0q 0 p0q 1

--- Content provided by FirstRanker.com ---

= L x P = (135.1 x 136.1) = 135.7

--- Content provided by FirstRanker.com ---

(e)

Marshall Edgeworth method

p

--- Content provided by‍ FirstRanker.com ---

1q 0 p1q 1 500 + 450

P01(Ma) = -------- + -------- x 100 = --------------- x 100

--- Content provided by FirstRanker.com ---

p

0q 0 p0q 1 500 + 330

--- Content provided by FirstRanker.com ---

950 x 100

--- Content provided by‌ FirstRanker.com ---

= --------------- = 1.14 x 100 = 114

830

--- Content provided by‍ FirstRanker.com ---

ANALYSIS OF TIME SERIES

An arrangement of statistical data in accordance with time of occurrence or in a

chronological order is called a time series. The numerical data which we get at

--- Content provided by FirstRanker.com ---

different points of time is known as time series. It plays an important role in

economics, statistics and commerce. For example, if we observe agricultural

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production, sales, national income etc., over a period of time, say the last 3 or 5

years, the set of observations is called time series. The analysis of time series is

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done ma

of for

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inly for the purpose

ecasts and for evaluating the past

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performances.

Utility of Time series.

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The main uses of time series are:

(i)

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It helps in understanding the past behaviour and estimating the future

behaviour.

(ii)

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It helps in planning and forecasting and is very essential for the business

and economics to prepare plans for the future.

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(iii)

Comparison between data of one period with that of another period is

possible.

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(iv)

We can evaluate the progress in any field of economic and business

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activity with the help of time series data.

(v)

Seasonal, cyclical, secular trend of data is useful not only to economists

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but also to the businessmen.

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Components of time series:

There are four basic types of variations, which are called the components or

elements of time series. They are:

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1.

Secular Trend,

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2.

Seasonal variation,

3.

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Cyclical fluctuations, and

4.

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irregular or random fluctuations.

1.

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Secular trend

The general tendency of the time series data to increase or decrease or stagnate

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during a long period of time is called the secular trend, also known as long-term

trend. This phenomenon is usually observed in most of the series relating to

--- Content provided by⁠ FirstRanker.com ---

Economics and Business. For instance, an upward tendency is usually observed

in time series relating to population, production, prices, income, money in

--- Content provided by‍ FirstRanker.com ---

l

circu ation etc. while a downward tendency is noticed in the time series relating

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to deaths, epidemics etc. due to an advancement in medical technology,

improved medical facilities, better sanitation, etc. In a long term trend, there are

two types of trend. They are:

--- Content provided by FirstRanker.com ---

(i)

Linear ? Straight Line Trend, and

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(ii)

Non-Linear or Curvilinear Trend.

--- Content provided by‍ FirstRanker.com ---

(i)

Linear or Straight Line Trend. When the values of time series are

--- Content provided by FirstRanker.com ---

plotted on a graph, then it is called the straight line trend or linear trend.

(ii)

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Non-linear or Curvilinear Trend. When we plot the time series values

on a graph and if it forms a curve or a non-linear one, then it is called Non-linear

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or Curvilinear Trend.

2.

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Seasonal Variation

A variation which occurs weekly, monthly or quarterly is known as Seasonal

--- Content provided by FirstRanker.com ---

may

Variation. The seasonal variation

occur due to the following reasons:

--- Content provided by‌ FirstRanker.com ---

)

(i

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Climate and natural forces:

The result of natural forces like climate is causing seasonal variation. For

example, umbrellas are sold more in rainy season (in winter season).

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(ii)

Customs and habits:

--- Content provided by FirstRanker.com ---

Man-made conventions are the customs, habits, fashion, etc. There is a custom

f wearing new clothes, preparing sweets fo

--- Content provided by⁠ FirstRanker.com ---

o

r Deepavali, Christmas etc. At that

--- Content provided by‌ FirstRanker.com ---

time there is more demand for clothes, sweets, etc.

3.

Cyclical Variation:

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According to Lincoln L. Chou, "Up and down movements are different from

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seasonal fluctuations, in that they extend over longer period of time-usually two

or more years". Most of economic and business time series are influenced by

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the wave-like changes of prosperity a

depression. There is periodic up and

nd

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down movement. This movement is known as cyclical variation. There are four

es

--- Content provided by‍ FirstRanker.com ---

phas

hey are

in a business cycle. T

--- Content provided by FirstRanker.com ---

a) Prosperity (boom), b) recession, c)

depression, and d) recovery.

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4.

Irregular variation:

Irregular variations arise owing to unforeseen and unpredictable forces at

--- Content provided by‌ FirstRanker.com ---

random and affect the data. These variations are not regular ones. These are

caused by war, flood, strike etc.

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In the classical time series model, the elements of trend, cyclical and

seasonal variations are viewed resulting from systematic influences. These

--- Content provided by FirstRanker.com ---

influences led to gradual growth, decline or recurrent movements and irregular

movements and are considered to be erratic movement. Therefore, the residual

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that remains after the elimination of systematic components is taken as

representing irregular fluctuations.

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Measurement of Secular Trend

The time series analysis is absolutely essential for planning. It guides the

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planners to achieve better results. The study of trend enables the planner to

project the plan in a better

he following

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direction. T

are the four methods which

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can be used for determining the trend.

(i)

Free-hand or Graphic Method,

--- Content provided by⁠ FirstRanker.com ---

(ii)

Semi-average Method,

--- Content provided by⁠ FirstRanker.com ---

(iii)

Moving Average Method, and

(iv)

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Method of Least Squares.

--- Content provided by⁠ FirstRanker.com ---

(i)

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Graphic or Free-hand Fitting Method:

This is the easiest, simplest and the most flexible method of estimating secular

trend. In this method we must plot the original data on the graph. Draw a

--- Content provided by FirstRanker.com ---

smooth curve carefully which will show the direction of the trend. Here time is

shown on the horizontal axis and the value of the variables is shown on the

--- Content provided by⁠ FirstRanker.com ---

vertical axis.

he fr

For fitting a trend line by t

--- Content provided by FirstRanker.com ---

ee-hand method, the following points

should be taken into consideration:

--- Content provided by‌ FirstRanker.com ---

a)

the curve should be smooth.

b)

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Approximately there must be equal number of points above and below

the curve.

--- Content provided by‍ FirstRanker.com ---

c)

The total deviations of the data above the trend line must be the same as

ine.

--- Content provided by⁠ FirstRanker.com ---

the vertical deviations below the l

d)

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The sum of the squares of

the trend should

the vertical deviations from

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be as small as possible.

i)

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(i

Semi-average Method:

In this me

--- Content provided by‌ FirstRanker.com ---

ual parts and averages are

thod, the original data is divided into two eq

--- Content provided by‌ FirstRanker.com ---

calculated for both the parts. These averages are called se i

m -averages. For

example, we can divide the 10 years, 1993 to 2002 into two equal parts; from

--- Content provided by‍ FirstRanker.com ---

1993 to 1997 and 1998 to 2002. If the period is odd number of years, the value

of the middle year is omitted.

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We can draw the line by a straight line by joining the two points of

average. By extending the line downward

--- Content provided by FirstRanker.com ---

we can predict the future

or upward,

--- Content provided by FirstRanker.com ---

values.

(iii)

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Moving Average Method:

In the moving average method, the averag

--- Content provided by‍ FirstRanker.com ---

for a number of periods is

e value

considered and placed at the centre of the time-span. It is calculated from

--- Content provided by FirstRanker.com ---

overlapping groups of successive time series data. It simplifies the analysis and

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removes periodic variations; and the influence of the fluctuations is also

reduced. The formula for calculating 3 yearly moving averages is:

--- Content provided by‍ FirstRanker.com ---

a + b + c b + c + d c + d + e

----------- , ---------- , ------------

3 3 3

--- Content provided by‌ FirstRanker.com ---

Steps for calculating odd number of years (3, 5, 7, 9)

--- Content provided by⁠ FirstRanker.com ---

If we want to calculate the three-yearly moving average, then:

(i)

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Compute the value of first three years (1, 2, 3) and place the three year

total against the middle year

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(ii)

Leave the first year's value and a

values of the next three years

--- Content provided by‌ FirstRanker.com ---

dd up the

and place the three-year total against the middle year.

--- Content provided by‍ FirstRanker.com ---

(iii)

this process must be continued until the last year's value is taken for

calculating moving average.

--- Content provided by FirstRanker.com ---

(iv)

the three-yearly total must be divided by 3 and placed in the next

--- Content provided by‌ FirstRanker.com ---

column. This is the trend value of moving average.

Even period of moving average:

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If the period of moving average is 4,6,8, it is an even number. The four-yearly

total cannot be placed against any year as the median 2.5 is between the second

--- Content provided by‍ FirstRanker.com ---

and the third year. So the total should be placed in between the 2nd and 3rd

years.

(iv)

--- Content provided by FirstRanker.com ---

Method of least square:

By the method of least square

--- Content provided by‌ FirstRanker.com ---

t

, a straigh

be

--- Content provided by FirstRanker.com ---

line trend can

tim

--- Content provided by‌ FirstRanker.com ---

fitted to the given

e

series data. With this method, economi

--- Content provided by FirstRanker.com ---

e series data can be

c and business tim

--- Content provided by⁠ FirstRanker.com ---

fitted and can derive the results for the forecasting and pr

tren

ediction. The

--- Content provided by‍ FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

line is called the line o b

f

--- Content provided by FirstRanker.com ---

est f

e

it. Th straight line

--- Content provided by⁠ FirstRanker.com ---

e

tr nd

--- Content provided by FirstRanker.com ---

gree

or the first de

parabola is represented by the mathematical equation.

--- Content provided by⁠ FirstRanker.com ---

Y = a + bX

Where, Y = required trend value, X = unit of time

--- Content provided by‌ FirstRanker.com ---

a and b are constants

the value of

or constants can be calculated by the following two

--- Content provided by‌ FirstRanker.com ---

the unknown

normal equation.

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Y = Na + bX

YX = a

X + bX2

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Where, N = the number of period

By solving the above two equation obtain the parameters of a and b.

--- Content provided by‍ FirstRanker.com ---

Illustration: Calculation of Trend Values by the Method of Least Square

Year Sales

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Deviation from 1988

X2

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Y X XY

2000 100

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-2

-200

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4

2001 110

-1

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-110

1

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2

200

130

--- Content provided by‌ FirstRanker.com ---

0

0

--- Content provided by FirstRanker.com ---

0

2003 140

--- Content provided by⁠ FirstRanker.com ---

+1

+140

--- Content provided by‌ FirstRanker.com ---

1

2004 150

+2

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+300

4

--- Content provided by‍ FirstRanker.com ---

N = 5 Y = 630 X = 0 xy = 130 X2=10

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Since

X = 0

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Y 400

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a = ----- = ------ = 80

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N 5

--- Content provided by‌ FirstRanker.com ---

XY 52

b = -------- = ----- = 5.2

--- Content provided by‌ FirstRanker.com ---

X2 10

--- Content provided by‍ FirstRanker.com ---

Hence, Y = 126 + 13X

The for

= 165

--- Content provided by‌ FirstRanker.com ---

ecasted value for 2005 is Y = 126 + 13(3)

--- Content provided by⁠ FirstRanker.com ---

Questions:

1.

Define probability and explain various concepts of probability.

--- Content provided by FirstRanker.com ---

2.

State and explain the addition and multiplication theorem of probability

--- Content provided by FirstRanker.com ---

with an example.

3.

Define Binomial distribution. Explain its properties.

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4.

What are the properties of Poisson distribution?

--- Content provided by FirstRanker.com ---

5.

What are the salient features of Normal distribution?

6.

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Explain the utility of normal distribution in statistical analysis.

7.

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Explain how Poisson, binomial and normal distribution are related.

8.

Distinguish between null and alternative hypothesis.

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9.

How will you conduct test pertaining to comparison between sample

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mean and population mean?

10.

What are the properties of 2 distribution?

--- Content provided by‌ FirstRanker.com ---

11.

What are the uses of 2 test?

--- Content provided by‌ FirstRanker.com ---

12.

Define Index Number. Explain its uses.

13.

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What are the steps involved in the construction of index number?

14.

--- Content provided by‍ FirstRanker.com ---

Explain any four weighted index number.

15.

What are the components of time series?

--- Content provided by FirstRanker.com ---

16.

What do you mean by time series? State its utility.

--- Content provided by‍ FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

17.

The probability of defective needle is 0.3 in a box, find (a) the mean and

standard deviation for the distribution of defective needles in a total of 1000

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box, and (b) the moment coefficient of skewness and kurtosis of the distribution.

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18.

The incidence of a certain disease is such that on the average 10% of

orkers suffer from it. If 10 workers ar

--- Content provided by‍ FirstRanker.com ---

w

e selected at random, find the probability

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that (i) Exactly 4 workers suffer from the disease, (ii) not more than 2 workers

suffer from the disease.

--- Content provided by⁠ FirstRanker.com ---

19.

Out of 1000 families with 4 children each, what percentage would be

--- Content provided by FirstRanker.com ---

expected to have (a) 2 boys and 2 girls, (b) at least one boy, (c) no girls, and (d)

at the most 2 girls. Assume equal probabilities for boys and girls.

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20.

m

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A

ultiple-choice tes consi

t

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sts of 8 questions with 3 answers to each

question (of which only one is correct). A student answers each question by

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ling

rol

balan

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a

die

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ced

ch

and

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ecking the first answer if he gets 1 or 2, the second

answer if he gets 3 or

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the

4 and

third answer if he gets 5 or 6. To get a

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distinction, the student must secure at least 75% correct answers. If there is no

negativ m

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e arking, what is the probability that the student secures distinctions?

21.

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One fifth per cent of the blades produced by a blade manufacturing

factory turn out to be defective. The blades are supplied in packets of 10. Use

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Poisson distribution to calculate the approximate number of packets containing

ve, one defective and two

no defecti

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defective blades respectively in a

consignment of 100,000 packets.

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--- Content provided by FirstRanker.com ---

22.

It is known from past experience that in a certain plant there are on the

average 4 industrial accidents per month. Find the probability that in a given

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year there will be less than 4 accidents. Assume Poisson distribution.

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23. Calculate Laspeyre's, Paasche's, Bowley's, Fisher's, Marshall

Edgeworth index number from

ing

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the follow

data:

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--- Content provided by FirstRanker.com ---

Base year

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Current year

Price

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Value

Price

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Value

A

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6

50

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75

B

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8

90

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--- Content provided by⁠ FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

12

80

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15

100

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D

5

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8

30

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--- Content provided by FirstRanker.com ---

10

60

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24.

From the data given below about the treatment of 500 patients suffering

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from a disease, state whether the new treatment is superior to the conventional

trea

e

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tm nt:

Treatment

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No.

of

Patients

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Favourable

Not

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favourable Total

New

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250

40 290

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Conventional

160 50 210

Total

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410

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90 500

(Given for degrees of freedom = 1, chi-square 5 per cent = 3.84)

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25.

300 digits are chosen at random from a set of tables. The frequencies of

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the digits are as follows:

t

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Digi

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0 1

3 4

2

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5

6

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7

8

9

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--- Content provided by FirstRanker.com ---

Frequency

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28 29 36 31 20 35

35

30

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31

25

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Use 2 test to assert the correctness of the hypothesis that the digits were

distributed in equal numbers in the tables from which they were chosen.

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(Given for degrees of freedom = 1, chi-square 5 per cent = 3.84)

26.

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The number of defects per unit in a sam

f 165 units of a

ple o

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manufactured product was found as follows:

Number

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of

defects:

0 1 2 3 4

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Number of units : 107

46

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10

1

1

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Fit a Poisson distribution to the data and test for goodness

27.

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Assume the mean height of soldiers to be 68 inches with a variance of

9 inches. How many solders in a regime

u expect are over

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nt of 1,000 would yo

70 inches tall?

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28.

The weekly wages of 5,000 workmen are normally distributed around a

mean of Rs.70 and with a standard deviation of Rs. 5. Estimate the number of

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workers whose weekly wages will be:

(a)

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between Rs. 70 and Rs. 72,

(b)

between Rs. 69 and Rs. 72,

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(d)

less than Rs. 63, and

(e)

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more than Rs. 80.

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29.

In a distribution exactly normal, 7% of the items are under 35 and 89%

are under 63. What are the mean and standard deviation of the distribution?

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30.

Find the mean and standard deviation of a normal distribution of marks

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in an examination where 58 percent of the candidates obtained marks below 75,

four per cent got above 80 and the rest between 75 and 80.

31.

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A sample of 1600 male students is found to have a mean height of 170

cms. Can it be reasonably regarded as a sample from a la

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on with

rge populati

mean height 173 cms and standard deviation 3.50 cms.

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32.

Fit a trend line to the following data by the free-hand method, semi-

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average method and moving average method.

Year 1995 1996 1997 1998 1999 2000 2001

Sales 65 95 85 115 110 120 130

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33.

The following table gives the sterling assets of the R.B.I. in crores of

--- Content provided by FirstRanker.com ---

rupees:

(a)

Represent the data graphically

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(b)

Fit a straight line trend

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Year: 1996-97 1997-98 1998-99 1999-00 2000-01 2001-02

Assets: 83 92 71 90 169 191

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Also estimate the figures for 1996-97.

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***

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UNIT ? IV

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STATISTICAL APPLICATIONS

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A BRIEF INTRODUCTION TO STATISTICAL APPLICATIONS

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A manager in a business organization ? whether in the top level, or the

middle level, or the bottom level - has to perform an important role of decision

making. For solving any organizational problem ? which most of the times

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happens to be complex in nature -, he has to identify a set of alternatives,

evaluate them and choose the best alternative. The experience, expertise,

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rationality and wisdom gained by the manager over a period of time will

definitely stand in good stead in the evaluation of the alternatives available at his

disposal. He has to consider several factors, sometimes singly and sometimes

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jointly, during the process of decision making. He has to deal with the data of

not only his organization but also of other competing organizations.

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It would be a challenging situation for a manager when he has to face so

many variables operating simultaneously, something internal and something

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external. Among them, he has to identify the important variables or the

dominating factors and he should be able to distinguish one factor from the

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other. He should be able to find which factors have similar characteristics and

which factors stand apart. He should be able to know which factors have an inter

play with each other and which factors remain independent. It would be

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advantageous to him to know whether there is any clear pattern followed by the

variables under consideration. At times he may be required to have a good idea

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of the values that the variables would assume in future occasions. The task of a

manager becomes all the more difficult in view of the risks and uncertainties

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surrounding the future events. It is imperative on the part of a manager to

understand the impact of various policies and programmes on the development

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of the organization as well as the environment. Also he should be able to

understand the impact of several of the environmental factors on his

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organization. Sometimes a manager has to take a single stage decision and at

times he is called for to take a multistage decision on the basis of various factors

operating in a situation.

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Statistical analysis is a tool for a manager in the process of decision

making by means of the data on hand. All managerial activities involve an

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analysis of data. Statistical approach would enable a manager to have a scientific

guess of the future events also. Statistical methods are systematic and built by

several experts on firmly established theories and consequently they would

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enable a manager to overcome the uncertainties associated with future

occasions. However, statistical tools have their shortcomings too. The

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limitations do not reflect on the subject. Rather they shall be traced to the

methods of data collection and recording of data. Even with highly sophisticated

statistical methods, one may not arrive at valid conclusions if the data collected

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are devoid of representative character.

In any practical problem, one has to see whether the assumptions are

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reasonable or not, whether the data represents a wide spectrum, whether the data

is adequate, whether all the conditions for the statistical tests have been fulfilled,

etc. If one takes care of these aspects, it would be possible to arrive at better

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alternatives and more reliable solutions, thereby avoiding future shocks. While it

is true that a statistical analysis, by itself, cannot solve all the problems faced by

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an organization, it will definitely enable a manager to comprehend the ground

realities of the situation. It will for sure provide a foresight in the identification

of the crucial variables and the key areas so that he can locate a set of possible

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solutions within his ambit. A manager has to have a proper blend of the

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statistical theories and practical wisdom and he shall always strive for a holistic

approach to solve any organizational problem. A manager has to provide some

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safe-guarding measures against the limitations of the statistical tools. In the

process he will be able to draw valid inferences thereby providing a clue as to

the direction in which the organization shall move in future. He will be ably

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guided by the statistical results in the formulation of appropriate strategies for

the organization. Further, he can prepare the organization to face the possible

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problems of business fluctuations in future and minimize the risks with the help

of the early warning signals indicated by the relevant statistical tools.

A marketing manager of a company or a manager in a service

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organization will have occasions to come across the general public and

consumers with several social and psychological variables which are difficult to

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be measured and quantified.

Depending on the situation and the requirement, a manager may have to

deal with the data of just one variable (univariate data), or data on two variables

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(bivariate data) or data concerning several simultaneous variables (multivariate

data).

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The unit on hand addresses itself to the role of a manager as a decision

maker with the help of data available with him. Different statistical techniques

which are suitable for different requirements are presented in this unit in a

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simple style. A manager shall know the strengths and weaknesses of various

statistical tools. He shall know which statistical tool would be the most

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appropriate in a particular context so that the organization will derive the

maximum benefit out of it.

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The interpretation of the results from statistical analysis occupies an

important place. Statistics is concerned with the aggregates and not just the

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individual data items or isolated measurements of certain variables. Therefore

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the conclusions from a statistical study will be valid for a majority of the objects

and normal situations only. There are always extreme cases in any problem and

ealt with separately. St

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they have to be d

atistical tools will enable a manager to

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identify such outliers (abnormal cases or extreme variables) in a problem. A

manager has to evaluate the statistical inferences, interpret them in the proper

context and apply them in

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ons.

appropriate situati

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While in an actual research problem, one has to handle a large quantum

of data, it is not possible to treat such voluminous data by a beginner in the

subject. Keeping this point in mi

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, any

nd

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numerical example in the present unit is

based on a few data items only. It would be worthwhile to the budding managers

to make a start in solving statistical problems by practicing the ones furnished in

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this unit.

The candid

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are su

ates

ggested to use hand calculators for solving

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statistical problems. There will be frequent occasions to use Statistical Tables of

F-values furnished in this unit. The candidates are suggested to have with them a

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copy of the tables for easy, ready reference. The books and articles listed under

the references may be consulted for further study or applications of statistical

techniques in relevant research areas.

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UNIT IV

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RRE

1. CO

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LATION AND REGRESSION ANALYSIS

Lesson Outline

?

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The concept of correlation

?

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Determination of simple correlation coefficient

?

Properties of correlation coefficient

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?

The concept of rank correlation

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?

Determination of rank correlation coefficient

?

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The concept of regression

?

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The principle of least squares

?

Normal equations

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?

Determination of regression equations

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Learning Objectives

After reading this lesson you should be able to

-

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understand the concept of correlation

-

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calculate simple correlation coefficient

-

understand the properties of correlation coefficient

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-

understand the concept of rank correlation

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-

calculate rank correlation coefficient

-

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resolve ties in ranks

- understand the concept of regression

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- determine regression equations

- understand the managerial applications of correlation and regression

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SIMPLE COR E

R LATION

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orrelation

C

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Correlation means the average relationship between two or more

variables. When changes in the values of a variable affect the values of another

ariab

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v

le, we say that there is a correlation between the two variables. The two

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variables may move in the same direction or in opposite directions. Simply

because of the presence of correlation

le

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between two variab s, we cannot jump to

the conclusion that there is a cause-effect relationship between them.

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Sometimes, it may be due to chance also.

Simple correlation

We say that the correlation is simple if the comparison involves two variables

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only.

S OF CORRELATION

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TYPE

Positive correlation

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If two variables x and y move in the same direction, we say that there is a

positive correlation between them. In this case, when the value of one variable

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increases, the value of the other variable also increases and when the value of

one variable decreases, the value of the other variable also decreases. Eg. The

age and height of a child.

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Negative correlation

If two variables x and y move in opposite directions, we say that there is a

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negative correlation between them. i.e., when the value of one variable

increases, the value of the other variable decreases and vice versa. Eg. The price

and demand of a normal good.

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The following diagrams illustrate positive and negative correlations

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between x and y.

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Positive Correlation

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Negative Correlation

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Perfect positive correlation

If changes in two vari

les a

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ab

re in the same direction and the changes are

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in equal proportion, we say that there is a perfect positive correlation between

them.

Perfect negative correlation

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If changes in two variables are in opposite directions and the absolute

values of changes are in equal proportion, we say that there is a perfect negative

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correlation between them.

y

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--- Content provided by‌ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

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x

Perfect Positive Correlation

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Perfect Negative Correlation

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Zero correlation

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If there is no relationship between the two variables, then the variables are

said to be independent. In this case the correlation between the two variables is

zero.

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Zero correlation

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relation

Linear cor

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If the quantum of change in one variable always bears a constant ratio to

the quantum of change in the other variable, we say that the two variables have a

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linear correlation between them.

Coefficient of correlation

The coefficient of correlation between two variables X, Y is a measure of the

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degree of association (i.e., strength of relationship) between them. The

coefficient of correlation is usually denoted by `r'.

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Karl Pearson's Coefficient of Simple Correlation:

Let N denote t

ber of pairs of obs

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he num

ervations of two variables X and Y.

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The correlation coefficient r between X and Y is defined by

N XY - ( X ) (Y )

r =

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N X -( X )2 N Y -(Y )2

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2

2

This formula is suitable for solving problems with hand calculators. To apply

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Y,

2

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2

this formula, we have to calculate X,

XY, X , Y .

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Properties of Correlation Coefficient

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Let r denote the correlation coefficient between two variables. r is interpreted

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using the following properties:

1.

The value of r ranges from ?

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o 0.0 or fr

1.0 t

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om 0.0 to 1.0

2.

A value of r = 1.0 indicates

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there exis

that

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rfect posi

ts pe

tive correlation

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between the two variables.

3.

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A value of r = - 1.0 indicates that there exists perfect negative correlation

between the two variables.

4.

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A value r = 0.0 indicates zero correlation i.e., It shows that there is no

correlation at all between the two variables.

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5.

A positive value of r shows a positive correlation between the two

variables.

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6.

A negative value of r shows a negative correlation between the two

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variables.

7.

A value of r = 0.9 and above indicates a very high degree of positive

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correlation between the two variables.

8.

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A value of - 0.9 r > - 1.0 shows a very high degree of negative

e

correlation b tween the two variables.

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9.

n

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For a reaso ably high degree of positive correlation, we require r to be

from 0.75 to 1.0.

10.

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A value of r from 0.6 to 0.75 may be taken as a moderate degree of

positive correlation.

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Problem 1

The following are data on Advertising Expenditure (in Rupees Thousand) and

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Sales (Rupees In lakhs) in a com any.

p

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Advertising

Expenditure :

18

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19 20 21 22 23

Sales : 17 17

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18 19 19 19

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Determine the correlation coefficient between them and interpret the result.

Solution: We have N = 6. Calculate X, Y, XY, X2, Y2 as follows:

X

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Y

XY

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2

X

2

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Y

18

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17

306

324

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289

19

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17

323

361

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289

20

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18

360

400

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324

21

--- Content provided by‍ FirstRanker.com ---

19

399

441

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361

22

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19

418

484

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361

23

--- Content provided by FirstRanker.com ---

19

437

529

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361

Total

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:123

109 2243 2539 1985

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The correlation coefficient r bet

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ween the two variables is calculated as o

f llows:

N XY - ( X )(Y )

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r =

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2

N X - ( X )2

2

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2

N Y - (Y )

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6 ? 2243 -123?109

r =

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6 ? 2539 - (123)2 6?1985 - (109)2

= (13458 ? 13407) / {(15234- 15129) (11910- 11881)}

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=51/{105 29} = 51/ (10.247 X 5.365) = 51/ 54.975 = 0.9277

Interpretation

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The value of r is 0.92. It shows that there is a high, positive correlation

between the two variables `Advertising Expenditure' and `Sales'. This

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provides a basis to consider some functional relationship between them.

Problem 2

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Consider the following data on two variables X and Y.

X

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: 12

14

18 23

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24

27

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Y : 18 13

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12 30 25 10

Determine the correlation coefficient between the two variables and interpret the

result.

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Solution:

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e

W have N = 6. Calculate X, Y, XY, X2, Y2 as follows:

X

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Y

XY

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2

X

2

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Y

12

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18

216

144

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324

14

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13

182

6

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--- Content provided by FirstRanker.com ---

169

18

12

--- Content provided by‍ FirstRanker.com ---

216

--- Content provided by‌ FirstRanker.com ---

324

--- Content provided by‍ FirstRanker.com ---

144

23

--- Content provided by FirstRanker.com ---

30

690

--- Content provided by‍ FirstRanker.com ---

529

--- Content provided by⁠ FirstRanker.com ---

900

24

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

600

576

--- Content provided by‌ FirstRanker.com ---

625

--- Content provided by FirstRanker.com ---

27

10

--- Content provided by⁠ FirstRanker.com ---

270

--- Content provided by‌ FirstRanker.com ---

729

100

--- Content provided by‌ FirstRanker.com ---

Total : 118 108

2174

--- Content provided by‌ FirstRanker.com ---

24 8

9

2262

--- Content provided by‌ FirstRanker.com ---

The correlation coefficient between the two variables is r =

--- Content provided by‍ FirstRanker.com ---

{6 X 2174 ? (118 X 108)} / { (6 X 249 -

8 1182) (

2262

--- Content provided by FirstRanker.com ---

6 X

- 1082) }

--- Content provided by FirstRanker.com ---

= (13044 ? 12744) / {(14988- 13924) (13572- 11664)}

=300 / {1064 1908} = 300 / (32.62 X 43.68) = 300 / 1424.84 = 0.2105

re

--- Content provided by‌ FirstRanker.com ---

Interp tation

--- Content provided by FirstRanker.com ---

The value of r is 0.21. Even though it is positive, the value of r is very less.

Hence we conclude that there is no correlation between the two variables X

and Y. Consequently we cannot construct any functional relational

--- Content provided by FirstRanker.com ---

relationship between them.

--- Content provided by‍ FirstRanker.com ---

Problem 3

Consider the following data on supply and price. Determine the correlation

coefficient between the two variables and interpret the result.

--- Content provided by‍ FirstRanker.com ---

Supply : 11 13 17 18 22 24 26 28

--- Content provided by‍ FirstRanker.com ---

Price : 25 32 26 25 20 17 11 10

Determine the correlation coefficient between the two variables and interpret the

--- Content provided by‍ FirstRanker.com ---

result.

Solution:

--- Content provided by⁠ FirstRanker.com ---

We have N = 8. Take X = Supply and Y = Price.

Calculate X, Y, XY, X2, Y2 as follows:

--- Content provided by‍ FirstRanker.com ---

X

Y

--- Content provided by‌ FirstRanker.com ---

XY

X2

--- Content provided by⁠ FirstRanker.com ---

Y2

11

25

--- Content provided by⁠ FirstRanker.com ---

275

121

--- Content provided by FirstRanker.com ---

625

13

32

--- Content provided by‍ FirstRanker.com ---

416

169

--- Content provided by FirstRanker.com ---

1024

17

26

--- Content provided by⁠ FirstRanker.com ---

442

289

--- Content provided by‍ FirstRanker.com ---

676

18

25

--- Content provided by FirstRanker.com ---

450

324

--- Content provided by FirstRanker.com ---

625

22

20

--- Content provided by FirstRanker.com ---

440

484

--- Content provided by‍ FirstRanker.com ---

400

24

17

--- Content provided by FirstRanker.com ---

408

576

--- Content provided by‌ FirstRanker.com ---

289

26

11

--- Content provided by⁠ FirstRanker.com ---

286

676

--- Content provided by‍ FirstRanker.com ---

121

28

10

--- Content provided by FirstRanker.com ---

280

784

--- Content provided by‌ FirstRanker.com ---

100

Total: 159

166

--- Content provided by FirstRanker.com ---

2997

3423

--- Content provided by⁠ FirstRanker.com ---

3860

--- Content provided by‌ FirstRanker.com ---

The correlation coefficient between the two variables is r =

{8 X 2997 ? (159 X 166)} / { (8 X 3423 - 1592) (8 X 3860 - 1662) }

--- Content provided by‌ FirstRanker.com ---

= (23976 ? 26394) / {

84- 2

(273

--- Content provided by FirstRanker.com ---

5281) (30880- 27566)}

= - 2418 / {2103 3

--- Content provided by‍ FirstRanker.com ---

- 24

314} =

86 X 5

--- Content provided by FirstRanker.com ---

18 / (45.

7.57)

--- Content provided by FirstRanker.com ---

= - 2418 / 2640.16 = - 0.9159

Interpretation

--- Content provided by FirstRanker.com ---

The value of r is - 0.92. The negative sign in r shows that the two variables

--- Content provided by FirstRanker.com ---

move in opposite directions. The absolute value of r is 0.92 which is very

high. Therefore we conclude that there is high negative correlation between

--- Content provided by⁠ FirstRanker.com ---

the two variables `Supply' and `Price'.

Problem 4

--- Content provided by‌ FirstRanker.com ---

Consider the following data on income and savings in Rs. thousand.

Income : 50 51 52 55 56 58 60 62 65 66

--- Content provided by⁠ FirstRanker.com ---

Savings : 10 11 13 14 15 15 16 16 17 17

Determine the correlation coefficient between the two variables and interpret the

result.

--- Content provided by‌ FirstRanker.com ---

Solution:

--- Content provided by⁠ FirstRanker.com ---

We have N = 10. Take X = Income and Y = Savings.

Calculate X, Y, XY, X2, Y2 as follows:

X

--- Content provided by‍ FirstRanker.com ---

2

XY

X2

--- Content provided by‍ FirstRanker.com ---

Y

50

--- Content provided by‍ FirstRanker.com ---

10

500

2500

--- Content provided by⁠ FirstRanker.com ---

100

--- Content provided by FirstRanker.com ---

51

11

561

--- Content provided by‍ FirstRanker.com ---

2601

121

--- Content provided by‌ FirstRanker.com ---

52

13

676

--- Content provided by⁠ FirstRanker.com ---

2704

169

--- Content provided by FirstRanker.com ---

55

14

770

--- Content provided by FirstRanker.com ---

3025

196

--- Content provided by‌ FirstRanker.com ---

56

15

840

--- Content provided by FirstRanker.com ---

3136

225

--- Content provided by FirstRanker.com ---

58

15

870

--- Content provided by‌ FirstRanker.com ---

3364

225

--- Content provided by FirstRanker.com ---

60

16

960

--- Content provided by FirstRanker.com ---

3600

256

--- Content provided by FirstRanker.com ---

62

16

992

--- Content provided by FirstRanker.com ---

3844

256

--- Content provided by FirstRanker.com ---

65

17

1105

--- Content provided by⁠ FirstRanker.com ---

4225

289

--- Content provided by⁠ FirstRanker.com ---

66

17

1122

--- Content provided by FirstRanker.com ---

4356

289

--- Content provided by FirstRanker.com ---

Total: 575

144

8396

--- Content provided by‍ FirstRanker.com ---

33355

2126

--- Content provided by FirstRanker.com ---

The correlation coefficient between the two variables is r =

{10 X 8396 ? (575 X 144)} / {(10 X 33355 - 5752) (10 X 2126 - 1442)}

= (83960 ? 82800) / {(333550- 3 0

--- Content provided by FirstRanker.com ---

3 625) 2

( 126 -

--- Content provided by‌ FirstRanker.com ---

0 20 3

7 6)

}

--- Content provided by‌ FirstRanker.com ---

= 1160 / {2925 524} = 1160 / (54.08 X 22.89)

= 1160 / 1237.89 = 0.9371

--- Content provided by FirstRanker.com ---

Interpretation

The value of r is 0.93. The positive sign in r shows that the two variables

--- Content provided by‌ FirstRanker.com ---

move in the same direction. The value of r is very high. Th

e

--- Content provided by‌ FirstRanker.com ---

erefor we

conclude that there is high positive correlation between the two variables

`Income' and `Savings'. As a result, e

--- Content provided by‌ FirstRanker.com ---

w can construct a functional

relationshi betw

--- Content provided by‌ FirstRanker.com ---

p

een them.

--- Content provided by‌ FirstRanker.com ---

RANK CORRELATION

Spearman's Rank Correlation Coefficient

--- Content provided by‌ FirstRanker.com ---

If ranks can be assigned to pairs of observations for two variables X and Y, then

the corre

e ranks is

--- Content provided by FirstRanker.com ---

lation between th

called the rank correlation coefficient. It

--- Content provided by‍ FirstRanker.com ---

is usually denoted by the symbol (rho). It is given by the formula

2

6 D

--- Content provided by‍ FirstRanker.com ---

= 1-

--- Content provided by⁠ FirstRanker.com ---

3

N - N

where D = i

--- Content provided by⁠ FirstRanker.com ---

d fference between t e

h corresponding rank

--- Content provided by FirstRanker.com ---

s of X and Y

--- Content provided by FirstRanker.com ---

= R - R

X

--- Content provided by FirstRanker.com ---

Y

and N is the total number of pairs of observations of X and Y.

--- Content provided by‍ FirstRanker.com ---

Problem 5

Alpha Recruiting Agency short listed 10 candidates for final selection. They

--- Content provided by‌ FirstRanker.com ---

were examined in

t

wri ten and oral communication skills. They were ranked as

--- Content provided by FirstRanker.com ---

follows:

--- Content provided by FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

Candidate's Serial No.

1 2

3

--- Content provided by FirstRanker.com ---

4 5

6

--- Content provided by⁠ FirstRanker.com ---

7

8

9

--- Content provided by FirstRanker.com ---

10

Rank in written

--- Content provided by FirstRanker.com ---

8 7

2

10 3

--- Content provided by⁠ FirstRanker.com ---

5

1

--- Content provided by‌ FirstRanker.com ---

9

6

4

--- Content provided by‌ FirstRanker.com ---

communication

Rank in oral communication 10 7

--- Content provided by FirstRanker.com ---

2

6 5

4

--- Content provided by‌ FirstRanker.com ---

1

9

--- Content provided by⁠ FirstRanker.com ---

8

3

--- Content provided by⁠ FirstRanker.com ---

Find out whether there is any correlation between th

w

--- Content provided by⁠ FirstRanker.com ---

e r

n

itte

--- Content provided by⁠ FirstRanker.com ---

d

an

--- Content provided by⁠ FirstRanker.com ---

l

ora

communication skills of the o

--- Content provided by‌ FirstRanker.com ---

sh

ist

--- Content provided by FirstRanker.com ---

rt l

a

ed c

--- Content provided by FirstRanker.com ---

a

ndid tes.

--- Content provided by‌ FirstRanker.com ---

Solution:

Take X = Wr

--- Content provided by⁠ FirstRanker.com ---

n sk

itten communicatio

--- Content provided by FirstRanker.com ---

ill and Y = Oral communication skill.

ANK OF X: R

--- Content provided by FirstRanker.com ---

R

1

--- Content provided by⁠ FirstRanker.com ---

RANK OF Y: R2

D=R1- R2

D2

--- Content provided by‌ FirstRanker.com ---

8

10

--- Content provided by‍ FirstRanker.com ---

- 2

4

7

--- Content provided by FirstRanker.com ---

7

0

--- Content provided by⁠ FirstRanker.com ---

0

2

2

--- Content provided by FirstRanker.com ---

0

0

--- Content provided by FirstRanker.com ---

10

6

--- Content provided by FirstRanker.com ---

4

16

--- Content provided by‌ FirstRanker.com ---

3

5

- 2

--- Content provided by⁠ FirstRanker.com ---

4

5

--- Content provided by⁠ FirstRanker.com ---

4

1

--- Content provided by‍ FirstRanker.com ---

1

1

--- Content provided by‍ FirstRanker.com ---

1

0

--- Content provided by‌ FirstRanker.com ---

0

9

--- Content provided by‌ FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

0

0

--- Content provided by‍ FirstRanker.com ---

6

8

- 2

--- Content provided by⁠ FirstRanker.com ---

4

4

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

1

1

--- Content provided by‌ FirstRanker.com ---

Total: 30

We hav

= 10. The rank c

--- Content provided by‌ FirstRanker.com ---

e N

orrelation coefficient is

--- Content provided by FirstRanker.com ---

= 1 - {6 D2

/ (N3 ? N)} = 1 ? {6 x 30 / (1000 ? 10)} = 1 ? (180 / 990)

= 1 ? 0.18 = 0.82

--- Content provided by‍ FirstRanker.com ---

Inference:

--- Content provided by⁠ FirstRanker.com ---

From the value of r, it is inferred that there is a high, positive rank correlation

between the written and oral communication skills of the short listed candidates.

--- Content provided by⁠ FirstRanker.com ---

Problem 6

The following are the ranks obtained by 10 workers in ABC Company on the

basis of their length of service and efficiency.

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

Ranking as per service 1

2

3

--- Content provided by FirstRanker.com ---

4

5

--- Content provided by⁠ FirstRanker.com ---

6 7

8

9

--- Content provided by FirstRanker.com ---

10

Rank as per efficiency

--- Content provided by‌ FirstRanker.com ---

2

3

6

--- Content provided by⁠ FirstRanker.com ---

5

1

--- Content provided by‌ FirstRanker.com ---

10 7

9

8

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

Find out whether there is any correlation between the ranks obtained by the

workers as per the two criteria.

--- Content provided by‌ FirstRanker.com ---

Solution:

Take X = Length of service and Y = Efficiency.

--- Content provided by⁠ FirstRanker.com ---

Rank of X: R1

Rank of Y: R2

--- Content provided by FirstRanker.com ---

D= R1- R2

D2

--- Content provided by‌ FirstRanker.com ---

1

2

- 1

--- Content provided by‌ FirstRanker.com ---

1

2

--- Content provided by⁠ FirstRanker.com ---

3

- 1

1

--- Content provided by FirstRanker.com ---

3

6

--- Content provided by FirstRanker.com ---

- 3

9

4

--- Content provided by‌ FirstRanker.com ---

5

- 1

--- Content provided by‍ FirstRanker.com ---

1

5

1

--- Content provided by FirstRanker.com ---

4

16

--- Content provided by‌ FirstRanker.com ---

6

10

- 4

--- Content provided by FirstRanker.com ---

16

7

--- Content provided by FirstRanker.com ---

7

0

0

--- Content provided by FirstRanker.com ---

8

9

--- Content provided by‌ FirstRanker.com ---

- 1

1

9

--- Content provided by‍ FirstRanker.com ---

8

1

--- Content provided by‍ FirstRanker.com ---

1

10

--- Content provided by‍ FirstRanker.com ---

4

6

--- Content provided by‌ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

Total

--- Content provided by‍ FirstRanker.com ---

82

We have N = 10. The rank correlation coefficient is

--- Content provided by FirstRanker.com ---

= 1 - {6 D2 / (N3 ? N)} = 1 ? { 6 x 82 / (1000 ? 10) } = 1 ? (492 / 990)

= 1

.497 =

--- Content provided by FirstRanker.com ---

? 0

503

--- Content provided by‌ FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

Inference:

The rank correlation coefficient is not high.

--- Content provided by‌ FirstRanker.com ---

Pro

(C

--- Content provided by⁠ FirstRanker.com ---

blem 7

onversion of s

--- Content provided by FirstRanker.com ---

res into ranks

co

)

--- Content provided by FirstRanker.com ---

Calculate the rank correlation to determine the relationship between equity

shares and preference shares given by the fol

--- Content provided by FirstRanker.com ---

ing data on

low

their price.

--- Content provided by FirstRanker.com ---

Equity share

--- Content provided by FirstRanker.com ---

90.0 92.4 98.5 98.3

95.4 91.3 98.0

92.0

--- Content provided by‍ FirstRanker.com ---

Preference share

76.0 74.2 75.0 77.4

--- Content provided by FirstRanker.com ---

78.3 78.8 73.2

76.5

--- Content provided by‍ FirstRanker.com ---

Solution:

From the given data on share price, we have to find out the ranks for equity

--- Content provided by‍ FirstRanker.com ---

shares and pr

ares.

eference sh

--- Content provided by‍ FirstRanker.com ---

Step 1. First, consider the equity shares and arrange them in descending order

--- Content provided by FirstRanker.com ---

of their price as 1,2,...,8. We have the following ranks:

Equity share

98.5

--- Content provided by⁠ FirstRanker.com ---

98.3 98.0

95.4 92.4

--- Content provided by⁠ FirstRanker.com ---

92.0

91.3 90.0

Rank

--- Content provided by‍ FirstRanker.com ---

1

2

--- Content provided by⁠ FirstRanker.com ---

3

4 5

6

--- Content provided by‌ FirstRanker.com ---

7

8

--- Content provided by‌ FirstRanker.com ---

Step 2. Next, take the preference shares and arrange them in descending order

of their price as 1,2,...,8. We obtain the following ranks:

--- Content provided by FirstRanker.com ---

Preference share

78.8

--- Content provided by‌ FirstRanker.com ---

78.3 77.4 76.5

76.0 75.0

74.2

--- Content provided by⁠ FirstRanker.com ---

73.2

--- Content provided by‍ FirstRanker.com ---

Rank

1

--- Content provided by FirstRanker.com ---

2

3

4

--- Content provided by⁠ FirstRanker.com ---

5

6

--- Content provided by‌ FirstRanker.com ---

7

8

--- Content provided by FirstRanker.com ---

Step 3. Calculation of D2:

Fit the given data with the correct rank. Take X = Equity share and Y =

--- Content provided by‍ FirstRanker.com ---

h

Preference s are. We have the following table:

X

--- Content provided by‍ FirstRanker.com ---

Y

Rank of X: R1

--- Content provided by‌ FirstRanker.com ---

Rank of Y: R2

D=R1- R2

D2

--- Content provided by⁠ FirstRanker.com ---

90.0 76.0

8

--- Content provided by FirstRanker.com ---

5

3

9

--- Content provided by‌ FirstRanker.com ---

92.4 74.2

5

--- Content provided by⁠ FirstRanker.com ---

7

-

2

--- Content provided by FirstRanker.com ---

4

98.5 75.0

--- Content provided by‌ FirstRanker.com ---

1

6

-

--- Content provided by FirstRanker.com ---

5

25

--- Content provided by⁠ FirstRanker.com ---

98.3 77.4

2

3

--- Content provided by FirstRanker.com ---

-

1

--- Content provided by⁠ FirstRanker.com ---

1

95.4 78.3

4

--- Content provided by FirstRanker.com ---

2

2

--- Content provided by‌ FirstRanker.com ---

4

91.3

78.8 7

--- Content provided by⁠ FirstRanker.com ---

1

6 36

--- Content provided by FirstRanker.com ---

98.0 73.2

3

8

--- Content provided by‌ FirstRanker.com ---

-

5

--- Content provided by FirstRanker.com ---

25

92.0

76.5

--- Content provided by FirstRanker.com ---

4

2

4

--- Content provided by⁠ FirstRanker.com ---

Total 108

--- Content provided by FirstRanker.com ---

Step 4. Calculation

of :

e have

--- Content provided by‍ FirstRanker.com ---

W

8. The

--- Content provided by‍ FirstRanker.com ---

N =

correlatio

rank

--- Content provided by FirstRanker.com ---

oefficient i

n c

--- Content provided by FirstRanker.com ---

s

= 1 - { 6 D2

/ (N3 ? N)} = 1 ? { 6 x 108 / (512 ? 8) } = 1 ? (648 / 504)

--- Content provided by⁠ FirstRanker.com ---

= 1 ? 1.29 = - 0.29

--- Content provided by FirstRanker.com ---

Inference:

From the value of , it is inferred that the equity shares and preference shares

under consideration are negatively correlate

--- Content provided by FirstRanker.com ---

d. However, the absolute value of

--- Content provided by‌ FirstRanker.com ---

is 0.29 which is not even moderate.

--- Content provided by FirstRanker.com ---

Problem 8

Three managers evaluate the performance

--- Content provided by⁠ FirstRanker.com ---

an organization

of 10 sales persons in

--- Content provided by FirstRanker.com ---

and award ranks to them as follows:

Sales Person

1 2

--- Content provided by‌ FirstRanker.com ---

3

4 5

--- Content provided by FirstRanker.com ---

6

7

8

--- Content provided by FirstRanker.com ---

9

10

--- Content provided by‍ FirstRanker.com ---

Rank awarded by Manager I

8 7

6

--- Content provided by FirstRanker.com ---

1 5

9

--- Content provided by⁠ FirstRanker.com ---

10

2

3

--- Content provided by⁠ FirstRanker.com ---

4

Rank awarded by Manager II

--- Content provided by‍ FirstRanker.com ---

7 8

4

6 5

--- Content provided by FirstRanker.com ---

10

9

--- Content provided by FirstRanker.com ---

3

2

1

--- Content provided by FirstRanker.com ---

Rank awarded by

4 5

--- Content provided by‍ FirstRanker.com ---

1

8 9

10

--- Content provided by‌ FirstRanker.com ---

6

7

--- Content provided by‌ FirstRanker.com ---

3

2

Manager III

--- Content provided by‌ FirstRanker.com ---

Determine which two managers have the nearest approach in the evaluation of

the performance of the sales persons.

--- Content provided by⁠ FirstRanker.com ---

Solution:

Sales

--- Content provided by‍ FirstRanker.com ---

Manager I Manager II

Manager III

--- Content provided by FirstRanker.com ---

(R1- R2) 2

(R1 -R3) 2

(R2-R3) 2

--- Content provided by‌ FirstRanker.com ---

Rank: R1

Rank: R

--- Content provided by‍ FirstRanker.com ---

Person

2

Rank: R3

--- Content provided by‌ FirstRanker.com ---

1

8

--- Content provided by FirstRanker.com ---

7

4

1 16 9

--- Content provided by FirstRanker.com ---

2

7

--- Content provided by⁠ FirstRanker.com ---

8

5

1 4 9

--- Content provided by FirstRanker.com ---

3

6

--- Content provided by FirstRanker.com ---

4

1

4 25 9

--- Content provided by⁠ FirstRanker.com ---

4

1

--- Content provided by FirstRanker.com ---

6

8

25 49 4

--- Content provided by‍ FirstRanker.com ---

5

5

--- Content provided by FirstRanker.com ---

5

9

0 16 16

--- Content provided by FirstRanker.com ---

6

9

--- Content provided by‌ FirstRanker.com ---

10

10

1 1 0

--- Content provided by‍ FirstRanker.com ---

7

10

--- Content provided by⁠ FirstRanker.com ---

9

6

1 16 9

--- Content provided by‌ FirstRanker.com ---

8

2

--- Content provided by⁠ FirstRanker.com ---

3

7

1 25 16

--- Content provided by FirstRanker.com ---

9

3

--- Content provided by⁠ FirstRanker.com ---

2

3

1 0 1

--- Content provided by FirstRanker.com ---

10

4

--- Content provided by FirstRanker.com ---

1

2

9 4 1

--- Content provided by FirstRanker.com ---

Total

44

--- Content provided by‍ FirstRanker.com ---

156 74

We have N = 10. The rank correlation coefficient between mangers I and II is

--- Content provided by‌ FirstRanker.com ---

= 1 - { 6 D2 / (N3 ? N)} = 1 ? { 6 x 44 / (1000 ? 10) } = 1 ? (264 / 990)

--- Content provided by FirstRanker.com ---

= 1 ? 0.27 = 0.73

The rank correlation coefficient between mangers I and III is

1 ? { 6 x 156 / (1000 ? 10) } = 1 ? (936 / 990) = 1 ? 0.95 = 0.05

--- Content provided by FirstRanker.com ---

The rank correlation coefficient between mangers II and III is

1 ? { 6 x 74 / (1000 ? 10) } = 1 ? (444 / 990) = 1 ? 0.44 = 0.56

--- Content provided by FirstRanker.com ---

Inference:

Comparing the 3 values of , it is inferred that Mangers I and II have the

--- Content provided by FirstRanker.com ---

nearest approach in the evaluation of the performance of the sales persons.

--- Content provided by‌ FirstRanker.com ---

Repeated values: Resolving ties in ranks

--- Content provided by‌ FirstRanker.com ---

When ranks are awarded to candidates, it is possible that certain

candidates obtain equal ranks. For example, if two or three, or four candidates

--- Content provided by‍ FirstRanker.com ---

secure equal ranks, a procedure that can be followed to resolve the ties is

described below.

We follow the Average Rank Method. If there are n items, arrange

--- Content provided by FirstRanker.com ---

them in ascending order or descending order and give ranks 1, 2, 3, ..., n. Then

look at those items which have equal values. For such items, take the average

--- Content provided by‍ FirstRanker.com ---

ranks.

If there are two items with equal values, their ranks will be two

consecutive integers, say s and s + 1. Their average is { s + (s+1)} / 2. Assign

--- Content provided by FirstRanker.com ---

this rank to both items. Note that we allow ranks to be fractions also.

--- Content provided by⁠ FirstRanker.com ---

If there are three items with equal values, their ranks w

re

ill be th e

--- Content provided by⁠ FirstRanker.com ---

consecutive integers, say s, s + 1 and s + 2. Their average is { s + (s+1) +

--- Content provided by FirstRanker.com ---

(s+2) } / 3 = (3s + 3) / 3 = s + 1. Assign this rank to all the three items. A similar

procedure is followed if four or more number of items has equal values.

--- Content provided by‍ FirstRanker.com ---

Correction term for when ranks are tied

--- Content provided by‌ FirstRanker.com ---

Consider the formula for rank correlation coefficient. We have

--- Content provided by‍ FirstRanker.com ---

2

= 1- 6 D

3

--- Content provided by⁠ FirstRanker.com ---

N - N

--- Content provided by‌ FirstRanker.com ---

If there is a tie involving m items, we have to add

3

m - m

--- Content provided by FirstRanker.com ---

12

to the term D2 in . We have to add as many terms like (m3 ? m) / 12 as there

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are ties.

Let us calculate the correction terms for certain values of m. These are provided

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in the following table.

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m - m

=

Correction term

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m3

m3 ? m

12

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2 8 6

0.5

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3 27 24

2

4 64 60

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5

5 125 120

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10

Illustrative examples:

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If there is a tie involving 2 items, then the correction term is 0.5

If there are 2 ties involving 2

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s

item each, then the correction

is

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term

0.5 + 0.5 = 1

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there a

If

re 3 ties with 2 items each, then the correction term is

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0.5 + 0.5 + 0.5 = 1.5

If there is a tie involving 3 items, then the correction term is 2

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If there are 2 ties involving 3 items each, then the correction term is 2 + 2 = 4

If there is a tie with 2 items and another tie with 3 items, then the correction

term is 0.5 + 2 = 2.5

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If there are 2 ties with 2 items each and another tie with 3 items, then the

is 0.5 + 0.5 + 2 = 3

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correction term

Problem 9 : Resolving ties in ranks

The following are the details of ratings scored by two popular insurance

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schemes. Determine the rank correlation coefficient between them.

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Scheme I

80 80 83 84 87 87 89 90

Scheme II

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55 56 57 57 57 58 59 60

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Solution:

From the given values, we have to determine the ranks.

Step 1.

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Scheme I in descending order and rank

Arrange the scores for Insurance

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them as 1,2,3,...,8.

Scheme I Score

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90

89

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87

87

84

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83

80

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80

Rank

1

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2

3

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4

5

6

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7

8

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The score 87 appears twice. The corresponding ranks are 3, 4. Their average is

(3 + 4) / 2 = 3.5. Assign this rank to the two equal scores in Scheme I.

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The score 80 appears twice. The corresponding ranks are

Their average is

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7, 8.

(7 + 8) / 2 = 7.5. Assign this rank to the two equal scores in Scheme I.

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The revised ranks for Insurance Scheme I are as follows:

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Scheme I Score

90

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89

87 87

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84

83

80

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80

Rank

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1

2

3.5 3.5

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5

6

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7.5

7.5

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Step 2. Arrange the scores for Insurance Scheme II in descending order and

rank them as 1,2,3,...,8.

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Scheme II Score

60

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59

58

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57

57

57

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56

55

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Rank

1 2

3 4 5 6 7 8

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The score 57 appears thrice. The corresponding ranks are 4, 5, 6.

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Their average is (4 + 5 + 6) / 3 = 15 / 3 = 5. Assign this rank to the three equal

scores in Scheme II.

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The revised ranks for Insurance Scheme II are as follows:

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Scheme II Score

60

59

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58

57

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57

57

56

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55

Rank

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1

2

3

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5

5

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5

7

8

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Step 3. Calculation of D2:

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Assign the revised ranks to the given pairs of values and calculate D2 as follows:

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Scheme I

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Scheme II

Sche e I

m

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Scheme II

D = R1- R2

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D2

Score

Score

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Rank: R1

Rank: R2

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80

55

7.5

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8

- 0.5

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0.25

80

56

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7.5

7

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0.5

0.25

83

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57

6

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5

1

1

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84

57

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5

5

0

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0

87

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57

3.5

5

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- 1.5

2.25

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87

58

3.5

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3

0.5

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0.25

89

59

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2

2

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0

0

90

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60

1

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1

0

0

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Total 4

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Step 4. Calculation of :

We have N = 8.

Since there are 2 ties with 2 items each and another tie with 3 items, the

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correction term is 0.5 + 0.5 + 2 .

The rank correlation coefficient is

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= 1 - [{ 6 D2 + (1/2) + (1/2) +2 }/ (N3 ? N)}]

= 1 ? { 6 (4.+0.5+0.5+2) / (512 ? 8) } = 1 ? (6 x 7 / 504) = 1 - ( 42/504 )

= 1 - 0.083 = 0.917

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Inference:

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It is inferred that the two insurance schemes are highly, positively correlated.

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REGRESSION

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In the pairs of observations, if there is a cause and effect relationship between

the variables X and Y, then the average relationship between these two variables

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is called regression, which means "stepping back" or "return to the average".

The linear relationship giving the best mean value of a variable corresponding to

the other variable is called a regression line or line of the best fit. The

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regression of X on Y is different from the regression of Y on X. Thus, there are

two equations of regression and the two regression lines are given as follows:

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Regression of

Y on X: Y - Y = b ( X - X )

yx

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Regression of X on Y: X - X = b (Y - Y )

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xy

where X , Y are the means of X, Y respectively.

Result:

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Let x, y denote the standard deviations of x, y respectively. We have the

fo

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sult.

llowing re

Y

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X

b = r

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and b = r

yx

xy

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X

Y

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2

r = b b

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and so r = b b

yx xy

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yx xy

Result:

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The coefficient of correlation r between X and Y is the square root of the

product of the b values in the two regression equations. We can find r by this

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way also.

Application

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e

The m thod of regression is very much useful for business forecasting.

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PRINCIPLE OF LEAST SQUARES

e

L t x, y be two variables under consideration. Out of them, let x be an

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independent variable and let y be a dependent variable, depending on x. We

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desire to build a functional relationship between them. For this purpose, the first

nd

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a

foremost requirement is that x, y have a high degree of correlation. If the

correlation coefficient between x and y is moderate or less, we shall o

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n t go

ahead with the task of fitting a functional relationship between them.

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Suppose there is a high degree of corr l

e ation (positive or negative)

between x and y. Suppose it is required to build a linear relations i

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h p between

them i.e., we want a regression of y on x.

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Geometrically speaking, if we plot the corresponding values of x and y

in a 2-dimensional plane and join such points, we shall obtain a straight line.

However, hardly we can expect all the pairs (x, y) to lie on a straight line. We

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can consider several straight lines which are, to some extent, near all the points

(x, y). Consider one line. An observation (x1, y1) may be either above the line of

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consideration or below the line. Project this point on the x-axis. It will meet the

straight line at the point (x1, y1e). Here the theoretical value (or the expected

value) of the variable is y1e while the observed value is y1. When there is a

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difference between the expected an

ved values, there appears an erro

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d obser

r.

This error is E1 = y1 ? ^y . This is positive if (x1, y1) is a point above the line and

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1

negative if (x1, y1) is a point below the line. For the n pairs of observations, we

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have the following n quantities of error:

E1 = y1 ? ^y ,

1

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E2 = y2 ? ^y ,

2

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.

.

.

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En = yn ? ^y .

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n

Some of these quantities are positive while the remaining ones are negative.

However, the squares of all these quantities are positive.

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--- Content provided by⁠ FirstRanker.com ---

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Y

(X1, Y1)

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--- Content provided by FirstRanker.com ---

e

1

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(X2, Y2)

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X

i.e., E2 = (y ? ^ 2

y ) 0, E2 = (y

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2

?

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2

^ 2

1

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1

y ) 0, ..., E2 = (y

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n

?

2

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n

^y ) 0.

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1

2

n

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Hence the sum of squares of errors (SSE) = E21 + E22 + ... + E2n

= (y1 ? ^y 2

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+ (y ? ^y 2

+ ... + (y

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^y 2

1 )

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2

2 )

n ?

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)

0.

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n

Among all those straight lines which are somewhat near to the given

observations

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(x1, y1), (x2, y2), ..., (xn , yn) , we consider that straight line as the ideal one for

which the SSE is the least. Since the ideal straight line giving regression of y on

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x is based on this concept, we call this principle as the principle of least

squares.

Normal equations

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Suppose we have to fit a straight line to the n pairs of observations (x1, y1), (x2,

y2), ...,

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(xn , yn). Suppose the equation of straight line finally comes as

Y = a + b X (1)

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where a, b are constants to be determ ned

i

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a

. M t

m

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he atically speaking, when we

require finding the equation of a straight lin

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distin

e, two

ct points on the straight

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line are sufficient. However, a different approach is followed here. We want to

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include all the observations in our attempt to build a straight line. Then all the n

observed points (x, y) are required to satisfy

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e r

th

elation (1). Consider the

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summation of all such terms. We get

y = (a + b x ) = (a .1 + b x ) = ( a.1) + ( b x ) = a ( 1 ) + b ( x).

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i.e. y = an + b ( x) (2)

To find two quantities a and b, we require two equations. We have

obtained one equation i.e., (2). We need one m re equation. For this purpose,

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o

multiply both sides of (1) by x. We obtain

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x y = ax + bx2 .

Consider the summation of all such terms. We get

x y = (ax + bx2 ) = ( a x) + ( bx2)

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i.e., x y = a ( x ) +

2

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b ( x ) .............. (3)

Equations (2) and (3) are referred to as the normal equations associated

with the regression of y on x. Solving these two equations, we obtain

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2

X Y - X XY

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a =

n X - ( X)2

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2

n X

--- Content provided by FirstRanker.com ---

Y - X Y

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and b =

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- (

--- Content provided by⁠ FirstRanker.com ---

X)2

2

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n

X

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Note: For calculating the coefficient of correlation,

we require X, Y, XY, X2, Y2.

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For calculating the regression of y on x, we require X, Y, XY, X2. Thus,

tabular colu

e

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mn is sam in both the cases with the difference that Y2 is also

required for the coefficient of correlation.

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Next, if we consider the regression line of x on y, we get the equation

X = a + b

Y. The expressions for the coefficients can be got by inter

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n

cha ging th r

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e oles of

X and Y in the previous discussion. Thus, we obtain

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2

Y X - Y XY

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a =

2

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n

2

Y

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- (

--- Content provided by‍ FirstRanker.com ---

Y)

n X

--- Content provided by FirstRanker.com ---

Y - X Y

and b =

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n Y - (

Y)2

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Problem 10

Consider the following data on sales and profit.

X 5 6 7 8 9 10

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11

Y 2 4 5 5 3

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--- Content provided by‌ FirstRanker.com ---

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Determine the regression of profit on sales.

Solution:

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We have N = 7. Take X = Sales, =

Y

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Profit.

Calculate X, Y, XY, X2

o

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as f llows:

X

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--- Content provided by‍ FirstRanker.com ---

XY

X2

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5

2

10

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25

6

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4

24

36

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7

5

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35

49

8

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5

40

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64

9

3

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27

81

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10

8

80

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100

11

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7

77

121

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Total: 56

34

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293

476

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a = {( x2) ( y) ? ( x) ( x y)} / {n ( x2) ? ( x) 2}

= (476 x 34 ? 56 x 293) / ( 7 x 476 - 562 ) = (16184 ? 16408 ) / ( 3332 ? 3136 )

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= - 224 / 196 = ? 1.142

9

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b = {n ( x y) ? ( x) ( y)} / {n ( x2) ? ( x) 2}

= (7 x 293 ? 56 x 34)/ 196 = (2051 ? 1904)/ 196 = 147 /196 = 0.75

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The

is given by the equation

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regression of Y on X

Y = a + b X

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i.e., Y = ? 1.14 + 0.75 X

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Problem 11

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The

are the details of income

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following

and expenditure of 10 households.

Income

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40 70 50 60 80 50 90 40

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60 60

Expenditure 25 60 45 50 45 20 55 30 35 30

Determine the regression of expenditure on income n

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a d estimate the expenditure

when the income is 65.

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Solution:

e

--- Content provided by FirstRanker.com ---

W have N = 10. Take X = Income, Y = Expenditure

Calculate X, Y, XY, X2 as follows:

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X Y XY X2

40 25 1000

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1600

70 60 4200

--- Content provided by FirstRanker.com ---

4900

50 45 2250

2500

--- Content provided by FirstRanker.com ---

60 50 3000

3600

--- Content provided by FirstRanker.com ---

80 45 3600

6400

50 20 1000

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2500

90 55 4950

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8100

40 30 1200

1600

--- Content provided by‌ FirstRanker.com ---

60 35 2100

3600

--- Content provided by‌ FirstRanker.com ---

60 30 1800

--- Content provided by FirstRanker.com ---

3600

Total: 600

395

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25100

38400

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

a = {( x2) (

y) ? ( x) ( x y)} /

2

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{n ( x ) ? ( x) 2}

= ( 38400 x 395 - 600

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100 ) /

x 25

(10 x 38400 - 6002)

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= (15168000 ? 15060000) / (384000 ? 360000) = 108000 / 24000 = 4.5

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b = {n ( x y) ? ( x) ( y)} / {n ( x2) ? ( x) 2}

= ( 10 x 25100 ? 600 x 395) / 24000 = (251000- 237000) / 24000

= 14000 / 24000 = 0.58

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The

ion of Y on X is given by the equation

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regress

Y = a + b X

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i.e., Y = 4.5 + 0.583 X

To estimate the expenditure w en income is 65

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h

:

Take X = 65 in the above equation. Then we get

--- Content provided by FirstRanker.com ---

Y = 4.5 + 0.583 x 65 = 4.5 + 37.895 = 42.395 = 42 (approximately).

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Problem 12

Consider the following data on occupancy rate and profit of a hotel.

Occupancy rate 40

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45

70

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60

70

75

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70 80 95

90

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Profit

50 55 65 70 90 95 105 110 120 125

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Determine the regressions of (i) profit on occupancy rate and

(ii) occupancy rate on profit.

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Solution:

We have N = 10. Take X = Occupancy rate, Y = Profit.

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Note that in Problems 10 and 11, we wanted only one regression line and so we

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did not take Y2 . Now we require two regression lines. Therefore,

calculate X, Y, XY, X2, Y2.

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X Y

XY

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X2

Y2

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40 50

2000

1600

--- Content provided by FirstRanker.com ---

2500

45 55

--- Content provided by‍ FirstRanker.com ---

2475

2025

3025

--- Content provided by⁠ FirstRanker.com ---

70 65

4550

--- Content provided by FirstRanker.com ---

4900

4225

60 70

--- Content provided by‍ FirstRanker.com ---

4200

3600

--- Content provided by‍ FirstRanker.com ---

4900

70 90

6300

--- Content provided by‌ FirstRanker.com ---

4900

8100

--- Content provided by FirstRanker.com ---

75 95

7125

5625

--- Content provided by FirstRanker.com ---

9025

70 105

--- Content provided by‍ FirstRanker.com ---

7350

4900

11025

--- Content provided by FirstRanker.com ---

80 110

8800

--- Content provided by‌ FirstRanker.com ---

6400

12100

95 120

--- Content provided by‍ FirstRanker.com ---

11400

9025

--- Content provided by FirstRanker.com ---

14400

90 125

11250

--- Content provided by‌ FirstRanker.com ---

8100

15625

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Total: 695

885

65450

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51075

84925

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The regression line of Y on X:

Y = a + b X

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where a ={( x2) ( y) ? ( x) ( x y)} / {n ( x2) ? ( x) 2}

and b ={n ( x y) ? ( x) ( y)} / {n ( x2) ? ( x) 2

--- Content provided by‌ FirstRanker.com ---

}

We obtain

--- Content provided by FirstRanker.com ---

a = (51075 x 885 ? 695 x 65450) / (10x51075 - 6952)

= (45201375 ? 45487750)/ (510750 ? 483025)

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= - 286375 / 27725 = - 10.329

b = (10 x 65450 ? 695 x 885) / 27725

= (654500 ? 615075) / 27725 = 39425 / 27725 = 1.422

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So, the regression equation is Y = - 1

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0.329 + 1.422 X

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Next, if we consider the regression line of X on Y,

get the equation X = a + b Y where

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a = {( y2) ( x) ? ( y) ( x y)} / {n ( y2) ? ( y) 2}

and b = {n ( x y) ? ( x) ( y)} / {n ( y2) ? ( y) 2}.

We get

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a = (84925 x 695 ? 885 x 65450) / (10 x 84925 ? 8852)

= (59022875 ? 57923250) / ( 849250 ? 783225) = 1099625 / 66025 = 16.655,

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b = (10 x 65450 ? 695 x 885) / 66025 = (654500 ? 615075) / 66025

= 39425 / 66025 = 0.597

So, the regression equation is X = 16.655 + 0.597 Y

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Note: For the data given in this problem, if we use the formula for r, we get

N XY - ( X ) (Y )

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r =

N X - ( X )2 N Y - (Y )2

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2

2

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= (10 x 65450 ? 695 x 885) / { (10 x 51075 - 6952 ) (10 x 84925 - 8852 ) }

= (654500 ? 615075) / ( 27725 66025 ) = 39425 / 166.508 x 256.95

= 39425 / 42784.23 = 0.9214

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However, once we know the two b values, we can find the coefficient of

correlation r between X and Y as the square root of the product of the two b

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values.

Thus we obtain

r = (1.422 x 0.597) = 0.848934 = 0.9214.

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Note that this agrees with the above value of r.

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QUESTIONS

1.

Explain the aim of `Correlation Analysis'.

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2.

Distinguish between positive and negative correlation.

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State the formula for simple

coefficient.

correlation

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4.

State the properties of the correlation coefficient.

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5.

nk correlation'?

What is `ra

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Explain.

6.

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k

State the formula for ran correlation coefficient.

7.

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Explain how to resolve ties while calculating ranks.

8.

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Explain the concept of regression.

9.

What is the principle of least squares? Explain.

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10.

Explain normal equations in the context of regression analysis.

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11.

State the formulae for the constant term and coefficient in the regression

equation.

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12.

State the relationship between the regression coefficient and correlation

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coefficient.

13.

Explain the managerial uses of Correlation Analysis and Regression

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Analysis.

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UNIT I

2. ANALYSIS OF VARIANCE

Lesson Outline

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?

Definition of ANOVA

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?

Assumptions of ANOVA

?

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n of linear models

Classificatio

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?

ANOVA for one-way classified data

?

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ANOVA table for one-way classified data

?

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Null and Alternative Hypotheses

?

Type I Error

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?

Level of significance

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?

SS, MSS and Variance ratio

?

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Calculation of F value

?

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Table value of F

?

Coding Method

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?

Inference fro ANOVA table

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m

?

Managerial applications of ANOVA

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Learning Objectives

After reading this lesson you should be able to

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-

understand the concept of ANOVA

-

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formulate Null and Alternative Hypotheses

-

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construct ANOVA table for one-way classified data

-

calculate T, N and CF

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-

calculate SS, df and MSS

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-

calculate F value

-

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find the table value of F

-

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draw inference from ANOVA

-

apply coding met

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-

understand the managerial

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of ANOVA

applications

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ANALYSIS OF VARIANCE (ANOVA)
Introduction

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For managerial decision making, sometimes one has to carry out tests of

significance. The analysis of variance is an effective tool for this purpose. The

objective of the analysis of variance is to test the homogeneity of the means of

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different samples.

Definition

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According to R.A. Fisher, "Analysis of variance is the separation of variance

ascribable to one group of causes from the variance ascr

ther groups".

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ibable to o

OVA

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Assumptions of AN

The technique of ANOVA is mainly used for the analysis and interpretation of

data obtained from experiments. This technique is based on three important

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assumptions, namely

1.

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The parent population is normal.

2.

The error component is distributed normally with zero mean and

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constant variance.

3.

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The various effects are additive in nature.

The technique of ANOVA essentially consists of partitioning the total variation

in an experiment into components of

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hese

different sources of variation. T

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sources of variations are due to controlled factors and uncontrolled factors.

Since the variation in the sample data is characterized by means of many

components of variation, it can be symbolically represented in the mathematical

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form called a linear model for the sample data.

Classification of models

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Linear models for the sample data may broadly be classified into three types as

follows:

1.

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Random effect model

2.

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Fixed effect model

3.

Mixed effect model

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In any variance components model, the error component has always

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random effects, since it occurs purely in a random manner. All other

components may be either mi

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random

xed or

.

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Random effect model

A model in which each of the factors has random effect (including error effect)

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is called a random effect model or simply a random model.

Fixed effect model

A model in which each of the factors has fixed effects, buy only the error effect

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is random is called a fixed effect model or simply a fixed model.

Mixed effect model

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A model in which some of the factors have fixed effects and some others have

random effects is called a mixed effect model or simply a mixed model.

In what follows, we shall restrict ourselves to a fixed effect model.

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In a fixed effect model, the main objective is to estimate the effects and

find the measure of variability among each of the factors and finally to find the

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variability among the error effects.

The ANOVA technique is mainly based on the linear model which

depends on the types of data used in the linear model. There are several types of

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data in ANOVA, depending on the number of sources of variation namely,

One-way classified data,

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Two-way classified data,

...

m-way classified data.

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One-way classified data

When the set of observations is distributed over different levels of a single

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factor, then it gives one-way classified data.

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ANOVA for One-way classified data

Let y denote the jth observation corresponding to the ith level of factor A and

i j

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Yij the corresponding random variate.

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Define the linear model for the sample data obtained from the

experiment by the equation

i =1,2, .

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.. , k

y = ? + a + e

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ij

i

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ij

j = 1, 2,..., n

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i

where ? represents the general mean effect which is fixed and which represents

the general condition of the experimental units, a denotes the fixed effect due

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i

to ith level of the factor A (i=1,2,...,k) and hence the variation due to a

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i

(i=1,2,...,k) is said to be control.

The last component of the model e is the random variable. It is called the error

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ij

component and it makes the Yij a random variate. The variation in e is due to

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ij

all the uncontrolled factors and e is independently, identically and normally

ij

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distributed with mean zero and constant variance 2

.

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For the realization of the random variate Yij, consider y defined by

ij

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i =1,2,...,k

y = ? + a + e

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ij

i

ij

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j = 1, 2,..., n

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i

The expected value of the general observation y in the experimental units is

ij

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given by E( y ) = ? for all i = 1, 2,..., k

ij

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i

with y = ? + e , where e is the random error effect due to uncontrolled factors

ij

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i

ij

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ij

(i.e., due to chance only).

Here we may expect ? = ? for all i

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= 1, 2,..., k , if there is no variation due to

i

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control factors. If it is not the case, we have

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? ? for all i =1,2,...,k

i

i e

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. ., ? - ? 0 for all i = 1, 2,..., k

i

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Suppose ? - ? a .

i

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i

Then we have ? ? + a for all i = 1, 2,..., k

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i

i

On substitution for ? in the above equation, the linear model reduces to

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i

i =1,2,...,k

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y = ? + a + e

(1)

ij

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i

ij

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j = 1, 2,..., n

i

The objective of ANOVA is to test the null hypothesis

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H : ? = ? for all i = 1, 2,..., k or H : a = 0 for all i = 1, 2,..., k . For carrying

o

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i

o

i

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out this test, we need to estimate the unknown parameters

? , a for all i =1,2,...,k by the principle of least squares. This can be done by

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i

minimizing the residual sum of squares defined by

E = e2ij

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ij

2

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=

( y - ? - a ) ,

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ij

i

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ij

using (1). The normal equations can be obtained by partially differentiating E

with respect to ? and a for all i = 1, 2,..., k and equating the results to zero.

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i

We obtain

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G = N ? + n a

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(2)

i

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i

i

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and T

?

i = ni

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+ ni ai, i = 1,2,...,k (3)

where N = nk. We see that the number of variables (k+1) is more than the

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number of independent equations (k). So, by the theorem on a system of linear

equations, it follows that unique solution for this system is not possible.

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However, by making the assumption that n a = 0 , we can get a

i

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i

i

unique solution for ? and ai (i = 1,2,...,k). Using this condition in equation (2),

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we get

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G = N ?

G

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.

i .

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e ? = N

Therefore the estimate of ? is given by ? G

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? =

(4)

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N

Again from equation (2), we have

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Ti = ? + a i

ni

T

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,

i

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Hence a =

- ?

i

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ni

Therefore, the estimate of a is given by

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i

? Ti

?

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a =

- ?

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i

ni

T

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G

i.e., ?

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i

a =

-

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(5)

i

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n

N

i

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Substituting the least square estimates of ?

? and ?

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a in the residual sum of

i

squares, we get

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?

2

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E = (y - ? - a$i )

ij

ij

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After carrying out some calculations and using the normal equations (2) and (3)

we obtain

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2

2

2

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G

T

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G

2

E = y -

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i

-

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-

(6)

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ij

N

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n

ij

i

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N

i

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The first ter in the RHS of

m

equation (6) is called the corrected total sum of

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squares while

2

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y is called the uncorrected total sum of squares.

ij

ij

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For measuring the variation due to treatment (controlled factor), we

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consider the null hypothesis that all the treatment effects are equal. i.e.,

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H : ? = ? = ... = ? = ?

o

1

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2

k

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.

i .

e , H : ? = ? for all i = 1, 2,..., k

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o

i

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.

i .

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e , H : ? - ? = 0 for all i = 1, 2,..., k

o

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i

.

i .

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e , H : a = 0

o

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i

Under H , the linear model reduces to

o

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i =1,2,...,k

y = ? + e

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ij

ij

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j = 1, 2,..., n

i

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Proceeding as before, we get the residual sum of squares for this hypothetical

model as

2

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G

2

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E = y -

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(7)

1

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ij

N

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ij

Actually, E contains the variation due to both treatment and error. Therefore a

1

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measure of variation due to treatment can be obtained by " E - E ". Using (6)

1

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and (7), we get

k

2

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2

T

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G

i

E - E =

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(8)

1

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=

n

N

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i 1

i

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The expression in (8) is usually called the corrected treatment sum of squares

k

2

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T

while the term

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i

is called uncorrected treatment sum of squares. Here it

=

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n

i 1

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i

2

G

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may be noted that

is a correction factor (Also called a correction term).

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N

Since E is based on N-k free observations, it has N - k degrees of freedom (df).

Similarly, since E is based on N -1 free observation, E has N -1 degrees of

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1

1

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freedom. So E - E has k -1 degrees of freedom.

1

When actually the null hypothesis is true, if we reject it on the basis of

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the estimated value in ou

tis

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r sta

tical analysis, we will be committing Type ? I

error

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o

. The probability f r committing this error is referre

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e

d to as th level of

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significance, denoted by . The testing of the null hypothesis H may be

o

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carried out by F test. For given , we have

Trss

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TrMSS

dF

F =

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=

: F

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.

k 1,

- N -k

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EMSS

Ess dF

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i.e., It follows F distribution with degrees of freedom k-1 and N-k.

All these values are represented in the form of a table called ANOVA table,

furnished below.

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ANOVA Table for one-way classified data

Source of

Degrees of

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u

S m of Squares

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Mean Squares

Variance ratio

Variation

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freedom

(SS)

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(MS)

F

Between the

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E - E = Q

M

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level of the

1

T

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T

Q

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F =

:

T

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k-1

k

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2

2

T

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G

T

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M =

M

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T

E

factor

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-

k -1

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n

N

F

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i

i

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k 1,

- N -k

(Treatment)

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Q :

E

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Within the level N-k

Q

-

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E

By subtraction

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M =

E

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of factor (Error)

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N - k

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2

G

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Total

N-1

Q = y -

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- -

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ij

N

ij

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tio

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Variance ra

The variance ratio is the ratio of

er variance to the s

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the great

maller variance. It is

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also called the F-coefficient. We have

F = Greater variance / Smaller variance.

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We refer to the table of F values at a desired level of significance . In general,

is taken to be 5 %. The table value is referred to as the theoretical value or the

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expected value. The calculated value is referred to as the observed value.

Inference

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If the observed value of F is less than the expected value of F (i.e., Fo < Fe) for

the given level of significance , then the null hypothesis H is accepted. In

o

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this case, we conclude that there is no significant difference between the

treatment effects.

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On the other hand, if the observed value of F is greater than the expected value

of F (i.e., F > F ) for the given level of significance , then the null hypothesis

o

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e

H is rejected. In this case, we conclude that all the treatment effects are not

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o

equal.

Note: If the calculated value of F and the table value of F are equal, we can try

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some other value of .

Problem 1

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The f llowing are th

o

e details of sales effected by three sales persons in three

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door-to-door campaigns.

Sales person

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Sales in door ? t ? d

o

oor campaign

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A

8

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9

5

10

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B

7

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6

6

9

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C

6

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6

7

5

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Construct an ANOVA table and find out whether there is any significant

difference in the performance of the sales persons.

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Solution:

Method I (Direct method) :

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A =8+9+5+10 = 32

B = 7+6+6+9 = 28

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C = 6+6+7+5 = 24

32

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Sample mean f r

o A : A =

= 8

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4

28

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Sample mean o

f r B : B =

= 7

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4

24

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Sample mean f r

o C : C =

= 6

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4

Total number of sample item = N

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s

o. of items for A + No. of items o

f r B + No.

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of items for C

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4 + 4 + 4

=

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= 12

32 + 28 + 24

84

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Mean of all the samples X =

=

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= 7

12

12

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Sum of squares of deviations for A:

A

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A - A = A - 8

( - )2

A

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A

8

--- Content provided by FirstRanker.com ---

0

0

9

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1

1

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5

-3

9

--- Content provided by FirstRanker.com ---

10

2

--- Content provided by‍ FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

Sum of squares of devia ions

t

for B:

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B

- B = B -

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(B - B)2

B

7

--- Content provided by‍ FirstRanker.com ---

7

0

--- Content provided by FirstRanker.com ---

0

6

-1

--- Content provided by FirstRanker.com ---

1

6

--- Content provided by‍ FirstRanker.com ---

-1

1

--- Content provided by‍ FirstRanker.com ---

2

9

--- Content provided by‌ FirstRanker.com ---

4

6

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Sum of squares of deviations for C:

C

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C - C = C - 6

( - )2

--- Content provided by‍ FirstRanker.com ---

C

C

6

--- Content provided by‌ FirstRanker.com ---

0

0

--- Content provided by‌ FirstRanker.com ---

6

0

0

--- Content provided by‍ FirstRanker.com ---

7

1

--- Content provided by FirstRanker.com ---

1

5

-1

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1

2

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Sum of squares of deviations within

2

2

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2

varieties = ( A - A) + (B - B) + (C - C )

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= 14 + 6 + 2

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= 22

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Sum of squares of deviations for total variance:

Sales person

Sales

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Sales - X = Sales ? 7

(Sales - )2

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--- Content provided by‌ FirstRanker.com ---

A

8

1

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1

A

--- Content provided by FirstRanker.com ---

9

2

4

--- Content provided by‍ FirstRanker.com ---

A

5

--- Content provided by‍ FirstRanker.com ---

- 2

4

A

--- Content provided by FirstRanker.com ---

10

3

--- Content provided by‌ FirstRanker.com ---

9

B

7

--- Content provided by FirstRanker.com ---

0

0

--- Content provided by‌ FirstRanker.com ---

B

6

- 1

--- Content provided by FirstRanker.com ---

1

B

--- Content provided by FirstRanker.com ---

6

- 1

1

--- Content provided by FirstRanker.com ---

B

9

--- Content provided by FirstRanker.com ---

2

4

C

--- Content provided by⁠ FirstRanker.com ---

6

- 1

--- Content provided by FirstRanker.com ---

1

C

6

--- Content provided by‌ FirstRanker.com ---

- 1

1

--- Content provided by‍ FirstRanker.com ---

C

7

0

--- Content provided by‌ FirstRanker.com ---

0

C

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5

2

4

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--- Content provided by FirstRanker.com ---

ANOVA Table

Source of variati n

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o

Degrees of freedom

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Sum of squares of

Variance

deviations

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8

Between varieties

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3 ? 1 = 2

8

= 4

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2

22

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Within varieties

12 ? 3 = 9

22

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= 2.44

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9

Total

12 ? 1 = 11

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Calculation of F value:

Greater Variance

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F =

= 4.00 = 1.6393

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Smaller Variance

2.44

Degrees of freedom for greater variance (df = 2

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1 )

Degrees of freedom for smaller variance (df = 9

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2 )

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Let us take the level of significance as 5%

The table value o

= 4

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f F

.26

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Inference:

The calcula d

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te value of F is less tha the

n

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table value of F. Therefore, the null

hypothesis is accepted. It is concluded that there is no significant difference in

the performance of the sales persons, at 5% level of significance.

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Method II (Short cut method):

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A = 32, B = 28, C = 24.

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T= Sum of all the sample items

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= A+ B + C
= 32 + 28 + 24

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= 84

N = Total number of te

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i ms in all the samples = 4 + 4 + 4 =12

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2

2

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T

84

Correction Factor =

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=

= 588

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N

12

Calculate the su

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m of squares of t e

h observed v lu

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a es as follows:

Sales Person

X

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A

8

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64

A

9

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81

A

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5

25

A

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10

100

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B

7

49

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B

6

--- Content provided by‍ FirstRanker.com ---

36

B

6

--- Content provided by‌ FirstRanker.com ---

36

B

--- Content provided by FirstRanker.com ---

9

81

C

--- Content provided by‌ FirstRanker.com ---

6

36

--- Content provided by‌ FirstRanker.com ---

C

6

36

--- Content provided by FirstRanker.com ---

C

7

--- Content provided by⁠ FirstRanker.com ---

49

C

5

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--- Content provided by⁠ FirstRanker.com ---

618

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2

Sum of squares of deviations for total variance = X - correction factor

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= 618 ? 588 = 30.

Sum of squares of deviations for variance between samples

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( A)2 (B)2 (C)2

=

--- Content provided by FirstRanker.com ---

+

+

- CF

--- Content provided by⁠ FirstRanker.com ---

N

N

--- Content provided by FirstRanker.com ---

N

1

2

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3

2

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2

2

32

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28

24

--- Content provided by‍ FirstRanker.com ---

=

+

+

--- Content provided by‌ FirstRanker.com ---

- 588

4

--- Content provided by‌ FirstRanker.com ---

4

4

1024

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784

576

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=

+

+

--- Content provided by‌ FirstRanker.com ---

- 588

--- Content provided by⁠ FirstRanker.com ---

4

4

4

--- Content provided by FirstRanker.com ---

= 256 +196 +144 - 588
= 8

ANOVA Table

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Source of

Degrees of

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Sum of squares of

Variance

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variation

Freed

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om deviations

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Between varieties

3-1 = 2

8

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8 = 4

2

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Within varieties

12 ? 3 = 9

22

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22 = 2.44

9

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Total

12 ? 1 = 11

30

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It is to be noted that the ANOVA tables in the methods I and II are one and the

same. For the further steps of calculation of F value and drawing inference,

refer to method I.

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Problem 2

The following are the details of plinth areas of ownership apartment flats offered

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y 3

b

housing companies A,B,C. Use analysis of variance to determine whe

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ther

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there is any significant difference in the plinth areas of the apartment flats.

Housing Company

Plinth

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f

area o apartment flats

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A

1500

1430

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1550

1450

--- Content provided by‍ FirstRanker.com ---

B

1450

1550

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1600

1480

--- Content provided by⁠ FirstRanker.com ---

C

1550

1420

--- Content provided by FirstRanker.com ---

1450

1430

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Use analysis of variance to determine whether there is any significant difference

in the plinth areas of the apartment's flats.

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Note: As the given figures are large, working with them will be difficult.

Therefore, we use the following facts:

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i.

Variance ratio is independent of the change of origin.

ii.

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Variance ratio is independent of the change of scale.

In the problem under consideration, the number

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s vary from 1420 to 1600. So

we follow a method a

c lled the coding method. First, let us subtract 1400 from

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each item. We get the following transformed data:

--- Content provided by FirstRanker.com ---

Compa

r

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ny T ansformed

mea ur

s ement

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A

100

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30

150

50

--- Content provided by‍ FirstRanker.com ---

B

50

--- Content provided by FirstRanker.com ---

150

100

80

--- Content provided by FirstRanker.com ---

C

150

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20

50

30

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Next, divide each entry by 10.

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The transformed data are given below.

Compan

a

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y Tr nsf m

or ed

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measurement

A

10

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3

15

--- Content provided by⁠ FirstRanker.com ---

5

B

5

--- Content provided by FirstRanker.com ---

15

10

--- Content provided by FirstRanker.com ---

8

C

15

--- Content provided by FirstRanker.com ---

2

5

--- Content provided by FirstRanker.com ---

3

We work with these transformed data. We have

A=10+3+15+5=33

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B =5+15+10+8=38

C=15+2+5+3=25

--- Content provided by FirstRanker.com ---

T = A+B +C

= 33 + 38 + 25

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= 96

N = Total number of items in all the samples = 4 + 4 + 4 = 12

2

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T

962

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Correction Factor =

=

= 768

--- Content provided by FirstRanker.com ---

N

12

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Calculate the sum of squares of the b

o served values as follows:

Compa y

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n X

X2

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--- Content provided by‌ FirstRanker.com ---

A

10

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100

A

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3

9

A

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15

225

--- Content provided by‍ FirstRanker.com ---

A

5

25

--- Content provided by‌ FirstRanker.com ---

B

5

--- Content provided by FirstRanker.com ---

25

B

15

--- Content provided by‌ FirstRanker.com ---

225

B

--- Content provided by FirstRanker.com ---

10

100

B

--- Content provided by FirstRanker.com ---

8

64

--- Content provided by‍ FirstRanker.com ---

C

15

225

--- Content provided by‍ FirstRanker.com ---

C

2

--- Content provided by FirstRanker.com ---

4

C

--- Content provided by FirstRanker.com ---

5

25

--- Content provided by FirstRanker.com ---

C

3

--- Content provided by FirstRanker.com ---

1036

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Sum of squares of deviations for total variance =

2

--- Content provided by FirstRanker.com ---

X - correction factor

--- Content provided by‍ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

= 1036 ? 768 = 268

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u

S m of squares of deviations for variance between samples

--- Content provided by⁠ FirstRanker.com ---

( A)2 (B)2 (C)2

=

+

--- Content provided by FirstRanker.com ---

+

- CF

--- Content provided by‌ FirstRanker.com ---

N

N

N

--- Content provided by FirstRanker.com ---

1

2

--- Content provided by FirstRanker.com ---

3

2

2

--- Content provided by‌ FirstRanker.com ---

2

33

--- Content provided by‌ FirstRanker.com ---

38

25

=

--- Content provided by FirstRanker.com ---

+

+

--- Content provided by‍ FirstRanker.com ---

- 768

4

4

--- Content provided by‍ FirstRanker.com ---

4

1089

--- Content provided by‍ FirstRanker.com ---

1444

=

+

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+ 625

--- Content provided by‍ FirstRanker.com ---

- 768

4

4

--- Content provided by⁠ FirstRanker.com ---

4

= 272.25 + 361+156.25 - 768
= 789.5 - 768

--- Content provided by‍ FirstRanker.com ---

= 21.5

ANOVA Table

Source of variation

--- Content provided by‌ FirstRanker.com ---

Degrees of Freedom Sum of squar s

e

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Variance

--- Content provided by⁠ FirstRanker.com ---

of deviations

Between varieties

3-1 = 2

--- Content provided by⁠ FirstRanker.com ---

21.5

21.5

--- Content provided by‍ FirstRanker.com ---

= 10.75

2

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Within varieties

12 ? 3 = 9

--- Content provided by FirstRanker.com ---

264.5

24.65 = 27.38

9

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Total

12 ? 1 = 11

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268

Calculation of F value:

--- Content provided by⁠ FirstRanker.com ---

Greater Variance

27.38

--- Content provided by⁠ FirstRanker.com ---

F =

=

= 2.5470

--- Content provided by FirstRanker.com ---

Smaller Variance

10.75

--- Content provided by FirstRanker.com ---

Degrees of freedom for greater variance (df = 9

1 )

Degrees of freedom for smaller va

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riance (df = 2

--- Content provided by FirstRanker.com ---

2 )

The table value of F at 5% level of significance = 19.

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Inference:

Since the calculated value of F is less than the t b

a le value of F, the null

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hypothesis is accepted and it is concluded that there is no significant difference

in the plinth areas of ownership apartment flats off e

--- Content provided by FirstRanker.com ---

er d by the three companies,

at 5% level of significance.

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Problem 3

A finance manager has collected the following information on the performance

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of three financial schemes.

o

S urce of variation

--- Content provided by FirstRanker.com ---

Degrees of Freedom Sum of sq a

u res of deviations

--- Content provided by‌ FirstRanker.com ---

Treatments

5 15

Residual 2

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25

Total (corrected)

--- Content provided by FirstRanker.com ---

7

40

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Interpret the information obtained by him.

--- Content provided by‌ FirstRanker.com ---

Note: `Treatments' means `Bet een

w

--- Content provided by‌ FirstRanker.com ---

varieties'.

`Residual' means `Within varieties' or `Error'.

--- Content provided by‌ FirstRanker.com ---

Solution:

Number of sc e

--- Content provided by⁠ FirstRanker.com ---

h mes = 3 (since 3 ? 1 = 2)

Total number of sample items = 8 (since 8 ? 1 = 7)

Let us calculate the variance.

--- Content provided by‍ FirstRanker.com ---

15

Variance between varieties =

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= 7.5

2

25

--- Content provided by‍ FirstRanker.com ---

Variance between varieties =

= 5

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5

Greater Variance

F =

--- Content provided by FirstRanker.com ---

= 7.5 = 1.5

Smaller Variance

--- Content provided by‌ FirstRanker.com ---

5

Degrees of freedom for greater variance (df = 2

1 )

--- Content provided by‍ FirstRanker.com ---

Degrees of freedom for smaller variance (df = 5

2 )

--- Content provided by FirstRanker.com ---

The total value of F at 5% level of significance = 5.79

Inference:

--- Content provided by FirstRanker.com ---

Since the calculated value of F is less than the table value of F, we accept the

null-hypothesis and conclude that there is no significant difference in the

--- Content provided by FirstRanker.com ---

performance of the three financial schemes.

QUESTIONS

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1. Define analysis of variance.

2.

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State the assumptions in analysis of variance.

3.

Explain the classification of linear m

--- Content provided by‍ FirstRanker.com ---

e

od ls for the sample data.

--- Content provided by FirstRanker.com ---

4.

Explain ANOVA Table.

5.

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Ex lain

p

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how inference is drawn from ANOVA Table.

6.

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Explain the managerial applications of analysis of variance.

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UNIT IV

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NS OF EXPE

3. DESIG

RIMENTS

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Lesson Outline

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?

Definition of design of experiments

?

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Key concepts in the design of experiments

?

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Steps in the design of experiments

?

Replication, Randomization and Blocking

--- Content provided by FirstRanker.com ---

?

Lay out of an experimental design

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?

Data Allocation Table

?

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Completely Randomized Design

?

--- Content provided by FirstRanker.com ---

ANOVA tabl

D

e for CR

--- Content provided by FirstRanker.com ---

?

Working rule for an example

--- Content provided by⁠ FirstRanker.com ---

?

Randomized Block Design

?

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ANOVA table for RBD

?

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Latin Square Design

?

ANOVA table for LSD

--- Content provided by FirstRanker.com ---

?

Managerial applications of experimental designs

--- Content provided by FirstRanker.com ---

Learning Objectives

After reading this lesson you should be able to

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- understand the definition of design of experiments
- understand the key concepts in the design of experiments
- understand the steps in the design of experiments

und

--- Content provided by‌ FirstRanker.com ---

-

erim

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erstand the lay out of an exp

ental design

und

--- Content provided by⁠ FirstRanker.com ---

-

le

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erstand a data allocation tab

- construct ANOVA table for CRD
- draw inference from ANOVA table for CRD
- construct ANOVA tabl

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BD

e for R

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- draw inference from

or RBD

ANOVA table f

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- construct ANOVA table for LSD
- draw inference from ANOVA tabl

LS

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e for

-

--- Content provided by FirstRanker.com ---

s

understand the working rules for solving problem

- understand the managerial applications of experimental designs

--- Content provided by‌ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

DESIGN OF EXPERIMENTS

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. FUN

I

DAMENTALS OF DESIGNS

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Introduction

The theory of design of experiments was originally developed for

--- Content provided by‍ FirstRanker.com ---

agriculture. For example, to determine whic

m

h fertilizer would give

--- Content provided by⁠ FirstRanker.com ---

ore yield of

certain crop, from

--- Content provided by⁠ FirstRanker.com ---

a

among a set of fertilizers. Nowadays the design of

experiments finds its application in the area of management also. While carrying

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ut research for m

o

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anagerial decision making, one may go for descriptive

research or experimental research. The advantage of experimental research is

at it can

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th

be used to establish the cause-effect relationship between the

--- Content provided by FirstRanker.com ---

n.

variables under consideratio

Such a relationship is called a causal

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relationship.

An experiment may be carried out with a control group or without a

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control group, depending on the resources available and the nature of the

subjects involved in the experiment. The researcher has to select different

subjects, put them into several groups and administer treatments to the subjects

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. It would be advisa

within each group

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ble to include a control group wherever

possible so as to increase the level of validity of the inference drawn from the

experim

--- Content provided by⁠ FirstRanker.com ---

ent.

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Definition of the design of experiments

perim

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The design of ex

ents is the logical construction of the experiment

--- Content provided by FirstRanker.com ---

h a w

wit

in

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ell-defined level of uncerta ty involved in the inference drawn.

Key concepts in the design of experiments

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The design of experiments centers around the following three key

concepts:

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(1) Treatments

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(2) Factors

(3) Levels of a treatment factor

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Types of experiments

There are two types of experiments, namely absolute experiment and

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comparative experiment. In an absolute experiment, one takes into account the

absolute value of a certain characteristic. As distinct from this, a comparative

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experiment seeks to compare the effect of two or more objects on some

characteristic of the population under examination. For example, one may think

of the following situations:

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* Comparison of the effect of different fertilizers on a certain crop

* Comparison of the effect of different medicines on a disease

--- Content provided by‍ FirstRanker.com ---

* Comparison of different marketing stra

om

tegies for the pr

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otion of a product

* Comp

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achines in

arison of different m

the production of a certain product

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* Comparison of different met

mobilization

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hods of resource

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Steps in the Design of Experiments

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The design of experiments consists of the following steps:

1.

Statement of the objectives

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2.

Formulation of the statistical hypotheses

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3.

Choice of the treatments

4.

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Choice of the experimental sites

5.

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Replication and levels of variation

6.

Choice of the experimental blocks, if necessary

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7.

haracteristics of the plots undertaken for the experim

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C

ents

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Assignment of treatments to various units

9.

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Recording of data

10.

Statistical analysis of data

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Basic designs

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The following are the basic desi

in sta

gns

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tistical analysis:

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1. Completely Randomized Design (CRD)

2. Randomized Block Design (RBD)

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n (LSD)

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3. Latin Square Desig

m

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Other designs can also be used for drawing inferences fro experiments.

However, they are quite complex and we shall confine ourselves to the above

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three designs.

Basic principles

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The design of experiments is mainly based on the following three basic

principles:

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.

1

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Replication

2.

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Randomization

3.

Blocking or Local Control.

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Replication means the repetition of each treatment a certain

number of times. This will help in reducing the effect due to a possible extreme

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situation (outlier) arising out of a single

m

treat ent. Thus replication will reduce

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the experimental error. Homogeneity is possible only within a replication.

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Randomization means

cation of the treatm

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allo

ents to different

units in a random way. i.e., all the units will have equal chance of allotment of

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treatments. But, w at

h treatment is a tual

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c

ly allotted to a unit will depend on pure

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chance only.

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The basic design is Completely Randomized Design (CRD). In this

esign, the first two principles namely

d

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replication and randomization are used.

There is no necessity of blocking in CRD, because the entire area of experiment

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is assumed to be homogeneous. If it is not so,

en it becom

th

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es necessary to

us experi

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subdivide the non-homogeneo

mental area into homogeneous sub-

groups such that each subgroup has almost the same level of attribute. The

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iding the experim

technique of subdiv

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ental area into groups is called as blocking

or local control and such subgroups are called as Blocks. The RBD and LSD

RD is not a bock design.

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are bock designs. However, C

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II. Completely Randomized Design (CRD)

This design is useful to compare several treatments in an experiment. For

example, suppose there are three training institutes each offering a distinct

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training programm

nd

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e to sales persons a

a manager wants to know which of the

three training programmes would be highly rewarding for his business

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organization. One option for him would be the comparison of the means of the

samples taken two at a time. However, comparison of the sample means may not

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yield accurate results when more than two samples are involved in the

experiment. Because of this reason, the m nager m

a

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ay opt for a completely

ized design. In this design, all

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random

are taken for sim

the samples

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ultaneous

co side

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n

ration and they are examined by m

a single s

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eans of

tatistical test.

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For the application of this design, the first and foremost condition is that

the experime

rea shoul

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ntal a

d be homogeneous in the particular attribute about

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which the experiment is carried out. For

lustra

the purpose of il

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tion, we consider

an example with 3 treatments denoted by A, B, C. A lay out is a pictorial

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representation of assignment of treatments to various experime

T

ntal areas.

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ple design has the following lay out.

exam

Experimental area

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B A B

A A C

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C B A

Data on treatments

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Suppose there are 3 treatments A, B, C and each treatment is used a

certain number of times as illustrated in the following example:

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TREATMENT

E

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NO. OF TIMES TH

TREATMENT IS APPLIED

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A 4

B 3

C 2

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Collect the results on the data arising out of the application of these treatments.

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Supp

e results

ose th

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e attri

on th

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bute pertai

o treatm

ning t

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ent A are 38, 36, 35 and

40. Suppose the results pertaining to treatment B are 26, 30 and 28. Suppose the

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results pertaining to tre

ent C ar

atm

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e 30 and 28. Using these values, a `Data

Allocation Table' is constructed as follows:

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Treatme

ata Allocation

nt D

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A 38 36 35 30

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B 26 30 28

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C 30 28

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The sums of the values for the 3 treatments are denoted by T1, T2 and T3,

respectively. For the above example data, we obtain

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T1 = 38 + 36 + 35 + 30 = 139,

T2 = 26 + 30 + 28 = 84 and

T3 = 30 + 8

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2 = 58.

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Statistical Analysis of CRD

As already mentioned, the experimental units in a CRD are taken in a

single group with the condition that the units forming the group must be

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ogeneous as far as po

hom

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ssible. Suppose there are k treatments in an

experiment. Let the ith treatment be replicated n times. Then the total number

i

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k

of experimental units in the design is n + n + ... + n + ... + n = n = N .

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1

2

i

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k

i

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i=1

The treatments are allocated at random to all the units in the experimental area.

This design provides a one-way classified data with different levels of a single

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factor called treatments. The linear model for CRD is defined by the relation

i =1,2,...,k

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y = ? + a + e

ij

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i

ij

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j = 1, 2,..., n

i

where y is the jth observation of the ith treatment,

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ij

?

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is the general mean effect which is fixed,

a is the fixed effect due to ith treatment and

i

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e is the random error effect which is distributed normally with zero mean and

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ij

constant variance.

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Let y = G be the Grand total of all the observations.

ij

ij

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In y , fix i and vary j. Then the sum gives the ith treatment total, denoted by

ij

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T . i.e., y = T (i=1,2,...,k).

i

ij

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i

j

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Apply the ANOVA for one-way classified data and compute the total

sum of squares (TSS) and treatment sum of squares (TrSS) as follows:

2

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G

2

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TSS = y -

= Q

ij

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N

ij

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2

T

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G

i

TrSS =

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-

= QT

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n

N

i

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i

G2/N is called the correction factor or the correction term.

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The error sum of squares (ESS) can be obtained by subtraction. All these values

are represented in the form of an ANOVA Table provided below.

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ANOVA Table for CRD

Degrees of

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Source of

Sum of Squares

Mean Sum of Variance ratio

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Freedom

Variation

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(SS)

Squares (MSS)

F

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(df)

M

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2

T

T

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G

Q

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F =

:

T

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Treatments k?

1

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i

Q =

-

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T

M =

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T

E

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N

N

T

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-

i

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k 1

i

Fk 1,

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- N -k

Q :

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E

Q

Error N?

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k

E

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M =

-

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E

By subtraction

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N - k

2

G

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Total N?

1

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2

Q = y -

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- -

ij

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N

ij

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Application of ANOVA

Objective of ANOVA:

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We apply ANOVA to find out whether there is any significant difference

in the performance of the treatments. We formulate the following null

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hypothesis:

H0: There is no significant difference in the performance of the

treatments.

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The null hypothesis has to be tested against the following alternative

hypothesis:

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H1: There is a significant difference in the performance of the treatments.

We have to decide whether the null hypothesis has to be accepted or rejected at

a desired level of significance ().

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Inference

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If the observed value of F is less than the expected value of F, i.e., Fo <

F

e, then the null-hypothesis H is accepted for a given level of significance (

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)

o

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and we conclude that the effects due to various treatments do not differ

significantly.

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If the observed value of F is greater than the expected value of F,

i.e., F > F , then the null-hypothesis H is rejected for a given level of

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o

o

significance ( ) and we conclude that the effects due to various treatments

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differ significantly.

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Working rule for an example:

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We have to consider three quantities G, N and the Correction Factor

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(denoted by CF) defined as follows:

G = Sum of the values for all the treatments,

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N = The sum of the number of times each treatment is applied

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The correction factor CF = G2 / N.

Let us consider an example of CRD. Suppose there are 3 treatments A, B, C.

Suppose the number of times the treatment is applied is n1 in the case of A, n2 for

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B and n3 for C. The sums of the values for the 3 treatments are denoted by T1, T2

and T3. With these notations, we have

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N = n1 + n2 + n3,

G = T1 +T2 +T3,

CF = G2/N = ( T1 +T2 +T3 )2 / (n1 + n2 + n3).

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Define the following quantities:

TSS = Sum of the squares of the observed values ? Correction Factor

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T

2

2

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2

r SS = ( T1

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/ n1 + T2 / n2 + T3 / n3 ) ? Correction Factor

ESS = TSS ? Tr SS

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Calculation of the Degrees of Freedom (df):

The df for treatments = No. of treatments ? 1.

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The df for the total = Total no. of times all the treatments have been applied ? 1

= N ? 1 = n1 + n2 + n3 ? 1.

The df for the Error = (Total no. of times all the treatments have been applied -

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No. of treatments) ? 2.

We have the following ANOVA table for this example.

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ANOVA Table for CRD

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Source of

Degrees of

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SS MSS

Variance

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ratio

variation

freedom

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F

Treatment

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3? 1 = 2

Tr SS

Tr SS / df =

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Tr SS / 2

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Error

8? 2 = 6

ESS

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ESS / df =

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ESS / 6

Total

9? 1 = 8

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TSS

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After these steps, carry out the Analysis of Variance and draw the inference.

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Problem 1

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Examine the CRD with the following Data Allocation Table and determine

whether or not the treatments differ significantly.

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Treatment Data Allocation

A 28 36 32 34

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B 40 38 36

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C 32 34

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Solution:

The treatments in the design are A, B and C.

We have

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n1 = The number of times A is applied = 4,

n2 = The number of times B is applied = 3,

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n3 = The number of times C is applied = 2.

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N = n1 + n2 + n3 = 4 + 3 + 2 = 9.

The sums of the values for the 3 treatments are denoted by T1, T2 and T3,

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respectively.

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For the given data on experimental values, we obtain

T1 = 28 + 36 + 32 + 34 = 130,

T2 = 40 + 38 + 36 = 114 and

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T3 = 32 + 34 = 66.

G = T1 + T2 + T3 = 130 + 114 + 66 = 310.

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The correction factor = G2/N = 3102/9 = 10677.8

y2 ij = 282 + 362 + 322 + 342 + 402 + 382 + 362 + 322 + 342

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= 784 + 1296 + 1024 + 1156 + 1600 + 1444 + 1296 + 1024 + 1156

= 10780

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(T2i /n i ) = 1302 / 4 + 1142 / 3 + 662 / 2

= 16900 / 4 + 12996 / 3 + 4356 / 2 = 4225 + 4332 + 2178 = 10735

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The total sum of squares (TSS) and treatment sum of squares (TrSS) are

calculated as follows:

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TSS = y2 ij ? CF = 10780 ? 10677.8 = 102.2

TrSS = T2i /n i ? CF = 10735 ? 10677.8 = 57.2

ESS = TSS ? TrSS

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We apply ANOVA to find out whether there is any significant difference in the

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performance of the treatments. We formulate the following null hypothesis:

H0: There is no significant difference in the performance of the

treatments.

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The null hypothesis has to be tested against the following alternative

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hypothesis:

H1: There is a significant difference in the performance of the treatments.

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We have to decide whether the null hypothesis has to be accepted or rejected at

a desired level of significance ().

ANOVA Table for CRD

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Source of

Degrees of

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SS

MSS = SS/DF

Variance ratio

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variation

freedom

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F

Treatment

3? 1 = 2

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57.2

57.2 / 2 = 28.6

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28.6 / 7.5 = 3.81

Error

8? 2 = 6

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45.0

45 / 6 = 7.5

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Total

9? 1 = 8

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102.2

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In the table, first enter the values of SS for `Total' and `Treatment'. From Total,

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subtract Treatment to obtain SS for `Error'.

i.e., ESS = TSS ? TrSS = 102.2 ? 57.2 = 45.0

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Calculation of F value: F = Greater variance / Smaller variance = 28.6 / 7.5

= 3.81

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Degrees of freedom for greater variance (df1) = 2

Degrees of freedom for smaller variance (df2) = 6

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Table value of F at 5% level of significance = 5.14

Inference:

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Since the calculated value of F is less than the table value of F, the null

hypothesis is accepted and it is concluded that there is no significant difference

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in the treatments A, B and C, at 5% level of significance.

III. Randomized Block Design (RBD)

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In CRD, note that the site is not split into blocks. An improvement of

CRD can be obtained by providing the blocking (local control) measure in the

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experimental design. One such design is Randomized Block Design (RBD). In a

block design, the site is split into different blocks such that each block is

homogeneous in itself, with respect to the particular attribute under experiment.

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The result from a RBD will be better than that from a CRD. While we use one-

way ANOVA in CRD, we use two-way ANOVA in RBD.

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Example of the lay out of RBD:

Experimental area

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Treatment Block 1

Block 2

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Block 3

A 19 16 17

B 16 17 20

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C 23 24 22

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This is an example of a RBD with 3 treatments and 3 blocks.

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Statistical Analysis of RBD

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Suppose there are k treatments each replicated r times. Then the total

number of experimental units is rk. These units are rearranged into r groups

(Blocks) of size k. The local control measure is adopted in this design in order to

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make the units of each group to be homogeneous. The group units in these

blocks are known as plots or cells. The k treatments are allocated at random in

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the k plots of each of the blocks selected randomly one by one. This type of

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homogeneous grouping of experimental units and random allocation of

treatments to randomly selected blocks are two main features of RBD.

The technique of ANOVA for two-way classified data is applicable to an

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experiment with RBD lay out. The data collected from the experiment is

classified according to the levels of two factors namely treatments and blocks.

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The linear model for RBD is defined by the relation

i =1, 2,..., k

y = ? + a + b + e

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i

j

ij

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j =1, 2,..., r

where y is the observation corresponding to ith treatment and jth block,

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ij

?

is the general mean effect which is fixed,

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a is the fixed effect due to ith treatment,

i

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b is the fixed effect due to jth block and

j

e is the random error effect which is distributed normally with zero mean and

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ij

constant variance.

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Applying the method of ANOVA for two-way classified data, the sum of

squares due to treatments, blocks and error can be obtained.

Let y = G be the Grand total of all the rk observations.

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ij

ij

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In y , fix i and vary j. Then the sum gives the ith treatment total, denoted by

ij

T . i.e., y = T (i=1,2,...,k).

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i

ij

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i

j

In y , fix j and vary i. Then the sum gives the jth block total, denoted by

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ij

Bj . i.e., y = Bj (j=1,2,...,r).

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ij

j

2

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G

e

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W take

as the correction factor. The number of treatments is k and the

rk

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numb

s

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er of blocks is r. Various sum of squares are computed as follows.

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2

G

2

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TSS = y -

= Q

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ij

rk

ij

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2

2

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T

G

i

--- Content provided by FirstRanker.com ---

TrSS =

-

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= Q ,

T

r

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rk

i

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2

2

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Bj

G

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BSS =

-

= Q ,

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B

k

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rk

j

ESS = Q - Q - Q = Q

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T

B

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E

All these values are represented in the form of an ANOVA table provided

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below.

ANOVA Table for RBD

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Variance

Source of

Degrees of Sum of Squares Mean Sum of

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ratio

Variation

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Freedom

(SS)

Squares (MSS)

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F

MT

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2

2

F =

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:

T

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G

Q

T

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i

Q =

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-

T

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M =

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M

Treatments

k ? 1

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T

r

rk

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T

k -1

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i

Fk 1,

- (k 1

--- Content provided by⁠ FirstRanker.com ---

- )(r 1

- )

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2

2

B

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G

M

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Blocks

r ? 1

j

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Q =

-

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B

F =

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:

B

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Q

B

k

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rk

B

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=

M

j

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--- Content provided by⁠ FirstRanker.com ---

B

E

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--- Content provided by‍ FirstRanker.com ---

r -1

F

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r 1,

- (k 1

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- )(r 1

- )

(k ? 1)(r ?

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Q :

E

--- Content provided by FirstRanker.com ---

Q

Error

E

--- Content provided by FirstRanker.com ---

M =

--- Content provided by FirstRanker.com ---

E

1)

--- Content provided by‌ FirstRanker.com ---

-

-

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By subtraction

(k 1)(r 1)

2

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G

Total

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(rk ? 1)

2

Q = y -

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--- Content provided by‌ FirstRanker.com ---

ij

rk

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We have to find out whether there is any significant difference in the

performance of the treatments. Also we can determine whether there is any

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significant difference in the performance of different blocks. We formulate the

following two null hypotheses:

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Null hypothesis-1

H01: There is no significant difference in the performance of the treatments.

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Null hypothesis-2

H02: There is no significant difference in the performance of the blocks.

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Each null hypothesis has to be tested against the alternative hypothesis. Even

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though there are two null hypotheses, the important one is the null hypothesis

on the treatments. We have to decide whether to accept or reject the null

hypothesis on the treatments at a desired level of significance ().

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Inference

If the observed value of F is less than the expected value of F, i.e., Fo <

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F

e, then the null-hypothesis H is accepted for a given level of significance (

)

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o

and we conclude that the effects due to various treatments do not differ

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significantly.

If the observed value of F is greater than the expected value of F, i.e.,

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F > F then the null-hypothesis H is rejected for a given level of significance

o

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o

( ) and we conclude that the effects due to various treatments differ

significantly.

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Similarly, the blocks' effects may also be tested, if necessary.

Working rule for an example:

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Consider the following example:

Treatment Block 1

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Block 2

Block 3

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Block 4

A

72 68 70 56

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--- Content provided by FirstRanker.com ---

55 60 62 55

C

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65 70 70 60

In this case, we have

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T1 = 72 + 68 + 70 + 56 = 266,

T2 = 55 + 60 + 62 + 55 = 232,

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T3 = 65 + 70 + 70 + 60 = 265,

T1 + T2 + T3 = 266 + 232 + 265 = 763.

B1 = 72 + 55 + 65 = 192,

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B2 = 68 + 60 + 70 = 198,

B3 = 70 + 62 + 70 = 202,

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B4 = 56 + 55 + 60 = 171,

B1 + B2 + B3 + B4 = 192 + 198 + 202 + 171 = 763.

For easy reference, let us take the number of treatments as t and the number of

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blocks as b. Then we have t = 3 and b = 4.

Calculate Tr SS and BSS as follows:

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T

2

2

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2

2

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r SS = ( T1

/ b + T2 / b +T3 / b + T4 / b ) ? Correction Factor

BSS = ( B 2

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2

2

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2

1

/ t + B2 / t + B3 / + B3 / t ) ? Correction Factor

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After these steps, carry out the Analysis of Variance and draw the inference.

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Problem 2

Analyse the following RBD and determine whether or not the treatments differ

significantly.

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Experimental area

Treatment Block 1

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Block 2

Block 3

A 9 5 7

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B 6 8 5

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C 4 5 8

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Solution:

The treatments in the design are A, B and C. There are 3 blocks namely, Block

1, Block 2 and Block 3.

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We have

n1 = the number of times A is applied = 3,

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n2 = the number of times B is applied = 3,

n3 = the number of times C is applied = 3.

N = n1 + n2 + n3 = 3 + 3 + 3 = 9.

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The sums of the values for the 3 treatments are denoted by T1, T2 and T3,

respectively.

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For the given data on experimental values, we obtain

T1 = 9 + 5 + 7 = 21,

T2 = 6 + 8 + 5 = 19,

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T3 = 4 + 5 + 8 = 17,

T1 + T2 + T3 = 21 + 19 + 17 = 57.

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B1 = 9 + 6 + 4 = 19,

B2 = 5 + 8 + 5 = 18,

B3 = 7 + 5 + 8 = 20,

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B1 +B2 +B3 = 19 + 18 + 20 = 57.

G = T1 + T2 + T3 = 57.

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The correction factor = G2/N = 572 / 9 = 3249 / 9 = 361

y2 ij = 92 + 52 + 72 + 62 + 82 + 52 + 42 + 52 + 82

= 81 + 25 + 49 + 36 + 64 + 25 + 16 + 25 + 64 = 385

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No. of blocks = b = 3

No. of treatments = t = 3

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( T2i / b ) = 212 / 3 + 192 / 3 + 172 / 3

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= 441 / 3 + 361 / 3 + 289 / 3 = 147 + 120.3 + 96.3 = 363.6

( B2j / t ) = 192 / 3 + 182 / 3 + 202 / 3

= 361 / 3 + 324 / 3 + 400 / 3 = 120.3 + 108 + 13.3 = 361.6

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The total sum of squares (TSS), treatment sum of squares (TrSS) and block sum

of squares (BSS) are calculated as follows:

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TSS = y2 ij ? CF = 385 ? 361 = 24

TrSS = (T2i /b) ? CF = 363.6 ? 361 = 2.6

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BSS = (B2j /t) ? CF = 361.6 ? 361 = 0.6

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ESS = TSS ? TrSS ? BSS = 24 ? 2.6 ? 0.6 = 24 ? 3.2 = 20.8

We apply ANOVA to find out whether there is any significant difference

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in the performance of the treatments. We formulate the following null

hypothesis:

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H0: There is no significant difference in the performance of the

treatments.

The null hypothesis has to be tested against the following alternative

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hypothesis:

H1: There is a significant difference in the performance of the treatments.

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We have to decide whether the null hypothesis has to be accepted or rejected at

a desired level of significance ().

ANOVA Table for RBD

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Source of

Degrees of

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SS

MSS = SS/DF

Variance ratio

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variation

freedom

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F

Treatment

3? 1 = 2

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2.6

2.6 / 2 = 1.3

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5.2 / 1.3 = 4.0

Block

3? 1 = 2

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0.6

0.6 / 2 = 0.3

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5.2 / 0.3 = 17.3

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Error

8? 4 = 4

20.8

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20.8 / 4 = 5.2

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Total

9? 1 = 8

24.0

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In the table, first enter the values of SS for `Total', `Treatment' and `Block'.

From Total, subtract (Treatment + Block) to obtain SS for `Error'.

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i.e., ESS = 24.0 - 3.2 = 20.8

Calculation of F value: We consider `Treatment'.

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F = Greater variance / Smaller variance = 5.2 / 1.3 = 4

Degrees of freedom for greater variance (df1) = 4

Degrees of freedom for smaller variance (df2) = 2

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Table value of F at 5% level of significance = 19.25

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Inference:

Since the calculated value of F for the treatments is less than the table

value of F, the null hypothesis is accepted and it is concluded that there is no

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significant difference in the treatments A, B and C at 5% level of significance.

Note: If required, by using the same table, we can also test whether there is any

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significant difference in the blocks, at 5% level of significance.

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IV. Latin Square Design (LSD)

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It was pointed out earlier that RBD is an improvement of CRD, since

RBD provides an error control measure for the elimination of block variation. In

RBD, the source of variation is eliminated in only one direction, namely block

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wise. This idea can be further generalized to improve RBD by eliminating more

sources of variation. One such design with a provision for elimination of two

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sources of variation is `Latin Square Design'. The result from LSD will be better

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than that from a RBD.

Suppose there are n treatments each replicated n times. Then the total

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number of experimental units is

2

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n ? n = n . Let p ? q denote the factors whose

variations are to be eliminated from the experimental error. Then both the

factors P and Q should be related to the variable under study. In that case, these

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two factors are control factors of variation.

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Therefore, the total number of level combinations of the two factors

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is

2

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n ? n = n . Now the experimental units are so chosen that each unit contains

b

different level com inations of these two factors. Further the 2

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n experimental

units are arranged in the form of an n? n array so that there are n rows and n

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columns of the 2

n units. Then each unit belongs to different row-column

combination. i.e., the two factors P and Q become the rows and columns of the

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design. Though it is not necessary that the two factors P and Q should always be

called as rows and columns, it has become a convention to define LSD by means

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of two factors, namely rows and columns.

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After the experimental units are obtained, the n treatments are

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allocated to the 2

n units such that each treatment occurs once and only once in

each row and each column. This ensures that each treatment is replicated n

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times. If a two-way table is formed with the levels of the factor P (rows) and the

levels of the factor Q (columns), then the n treatments should be allocated to the

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2

n units such that each treatment occurs once and only once in each level of the

factor P and each level of the factor Q. Such an arrangement is called a Latin

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Square Design of order n? n .

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Example of lay out of LSD

Example 1:

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Experimental area

A B C

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B C A

C A B

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In this design, the first row consists of the experiments A, B, C, in this

order. The second row is got by a cyclic permutation of the first row elements.

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The third row is got by a cyclic permutation of the second row elements.

Example 2:

Experimental area

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A B C

C A B

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B C A

In this design, the first row consists of the experiments A, B, C in this

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order. The third row is got by a cyclic permutation of the first row elements. The

second row is got by a cyclic permutation of the third row elements.

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Example 3:

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Experimental area

A B C D

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B C D A

C D A B

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D A B C

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In this design, the first row consists of the experiments A, B, C, D in this

order. The second row is got by a cyclic permutation of the first row elements.

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The third row is got by a cyclic permutation of the second row elements. The

fourth row is got by a cyclic permutation of the third row elements.

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Example 4:

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Suppose there are 5 treatments denoted by A, B, C, D, E. Then the

following arrangement of the treatments is a Latin Square Design of order 5? 5 .

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Factor Q (Column)

Column

Q

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Q

Q

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Q

2

3

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4

5

--- Content provided by FirstRanker.com ---

Q

Row .

1

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P

1

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A B C D E

P

2

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B C D E A

P

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3

C D E A B

Factor P

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P

4

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D E A B C

P

5

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E A B C D

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Note that every treatment appears in each row and column exactly once.

In the lay out of LSD, apart from indicating the treatment, the

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experimental value also has to be mentioned in each cell.

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Statistical Analysis of LSD

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In LSD, we have to consider three factors namely rows, columns and

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treatments. Therefore, the data collected from this design must be analyzed as a

three-way classified data. For this purpose, actually there must be 3

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n

observations, since there are three factors each with n-levels. However, because

of the particular allocation of the treatment to each cell, there is only one

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observation per cell, instead of n-observations per cell, according to a three-way

classified data. Consequently, there is no interaction between any of the factors

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namely rows, columns and treatments. Hence the appropriate linear model for

LSD is defined by the relation

--- Content provided by FirstRanker.com ---

y

= ? + r + c + t + e (i, j,k =1,2,...,n

--- Content provided by‌ FirstRanker.com ---

ijk

i

j

--- Content provided by FirstRanker.com ---

k

ijk

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)

where y is the general observation corresponding to ith row, jth column and kth

ijk

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treatment,

?

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is the general mean effect which is fixed,

r is the fixed effect due to ith row,

i

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c is the fixed effect due to jth column,

j

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t is the fixed effect due to kth treatment and

k

e is the random error effect which is distributed normally with zero mean and

--- Content provided by FirstRanker.com ---

ijk

constant variance.

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Application of ANOVA:

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The analysis here is similar to the analysis of two-way classified data.

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First of all, the data is arranged in a row-column table. Let y denote

ij

the observation corresponding to ith row and jth column in the table.

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In y , fix i and vary j. Then the sum gives the ith row total, denoted by Ri .

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ij

i.e., y = R (i=1,2,...,n)

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.

ij

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i

j

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In y , fix j and vary i. Then the sum gives the jth column total, denoted by

ij

Cj . i.e., y = Cj (j=1,2,...n).

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ij

j

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Let

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T = kth treatment total (k=1,2,...,n).

k

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We have R = C = T = G which is the Grand total of all the

i

j

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k

i

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j

k

2

--- Content provided by FirstRanker.com ---

G

2

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n observations. The correction factor CF is defined by CF =

where

N

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2

N = n is the total number of observations. We have

--- Content provided by FirstRanker.com ---

y = G .

ij

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ij

Various sums of squares are computed through the CF as follows:

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2

G

2

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2

TSS = y -

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which has (n -1) dF

ij

2

--- Content provided by‍ FirstRanker.com ---

n

ij

--- Content provided by FirstRanker.com ---

2

2

R

--- Content provided by‍ FirstRanker.com ---

G

i

--- Content provided by⁠ FirstRanker.com ---

RSS =

-

which has (n -1) dF

--- Content provided by FirstRanker.com ---

2

n

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n

i

2

--- Content provided by FirstRanker.com ---

2

C j

--- Content provided by⁠ FirstRanker.com ---

G

CSS =

-

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which has (n -1) dF

2

--- Content provided by FirstRanker.com ---

n

n

j

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2

2

--- Content provided by FirstRanker.com ---

T

G

k

--- Content provided by FirstRanker.com ---

TSS =

-

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which has (n -1) dF

2

n

--- Content provided by⁠ FirstRanker.com ---

n

k

--- Content provided by‌ FirstRanker.com ---

ESS = TSS - RSS - CSS - TrSS

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which has (n-1)(n-2) dF.

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All these values are represented in the form of an ANOVA Table below.

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ANOVA Table for n? n Latin Square Design

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Degrees

Variance

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Source of

Sum of Squares

Mean Sum of

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of

ratio

--- Content provided by FirstRanker.com ---

Variation

(SS)

Squares (MSS)

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Freedom

F

--- Content provided by FirstRanker.com ---

M R

2

2

--- Content provided by FirstRanker.com ---

F =

:

--- Content provided by FirstRanker.com ---

R

G

Q

--- Content provided by‍ FirstRanker.com ---

R

Rows (n-1)

--- Content provided by FirstRanker.com ---

i

Q =

-

--- Content provided by‌ FirstRanker.com ---

R

M =

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R

2

n

--- Content provided by FirstRanker.com ---

n

R

--- Content provided by‌ FirstRanker.com ---

n -1

i

Fn 1,

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- (n 1

- )(n-2)

--- Content provided by‍ FirstRanker.com ---

M

Columns

(n-1)

--- Content provided by‍ FirstRanker.com ---

C

2

--- Content provided by‍ FirstRanker.com ---

2

F

C

--- Content provided by FirstRanker.com ---

=

:

--- Content provided by‍ FirstRanker.com ---

G

Q

C

--- Content provided by FirstRanker.com ---

j

Q =

--- Content provided by⁠ FirstRanker.com ---

-

c

M =

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M E

--- Content provided by FirstRanker.com ---

C

2

--- Content provided by FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

n

n

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n -1

j

--- Content provided by‌ FirstRanker.com ---

Fn 1,

- (n 1

- )(n-2)

--- Content provided by FirstRanker.com ---

M

Treatments

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(n-1)

T

2

--- Content provided by‍ FirstRanker.com ---

2

F =

--- Content provided by‌ FirstRanker.com ---

:

T

G

--- Content provided by‌ FirstRanker.com ---

Q

T

--- Content provided by‌ FirstRanker.com ---

k

Q =

-

--- Content provided by FirstRanker.com ---

T

M =

--- Content provided by FirstRanker.com ---

M E

--- Content provided by FirstRanker.com ---

T

2

--- Content provided by FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

n

n

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n -1

k

Fn 1,

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- (n 1

- )(n-2)

--- Content provided by FirstRanker.com ---

Q :

E

Q

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Error (n-1)

(n-2)

--- Content provided by FirstRanker.com ---

E

M =

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E

-

--- Content provided by FirstRanker.com ---

-

By subtraction

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(n 1)(n 2)

2

--- Content provided by‌ FirstRanker.com ---

G

Total

2

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(n - )

2

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1

Q = y -

ij

--- Content provided by FirstRanker.com ---

2

n

ij

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The following hypotheses are formed:

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Null hypothesis-1

H01: There is no significant difference in the performance of the

treatments.

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Null hypothesis-2

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H02: There is no significant difference in the performance of the rows.

--- Content provided by FirstRanker.com ---

Null hypothesis-3

H03: There is no significant difference in the performance of the

--- Content provided by‌ FirstRanker.com ---

columns.

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Each null hypothesis has to be tested against the alternative hypothesis.

Even though there are three null hypotheses, the important one is the null

hypothesis on the treatments. We have to decide whether to accept or reject the

--- Content provided by FirstRanker.com ---

null hypothesis on the treatments at a desired level of significance ().

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Inference

If the observed value of F is less than the expected value of F, i.e., Fo

--- Content provided by FirstRanker.com ---

< Fe, for a given level of significance , then the null hypothesis of equal

treatment effect is accepted. Otherwise, it is rejected.

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Problem 3

Examine the following experimental values on the output due to four

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different training methods A, B, C and D for sales persons and find out whether

there is any significant difference in the training methods.

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A

B

C

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D

28

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20

32

28

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B

C

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D

A

36

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30

28

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20

C

D

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A

B

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25

30

22

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35

D

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A

B

C

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30

26

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36

28

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Solution:

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In this design, there are 4 treatments A, B, C and D. In the lay out of the

design, each treatment appears exactly once in each row as well as each column.

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Therefore this design is LSD. The name of the treatment and the observed value

under that treatment are specified together in each cell.

R1 = first row elements = 28 + 20 + 32 + 28 = 108

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R2 = second row elements = 36 + 30 + 28 + 20 = 114

R3 = third row elements = 25 + 30 + 22 + 35 = 112

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R4 = fourth row elements = 30 + 26 + 36 + 28 = 120

C1 = first column elements = 28 + 36 + 25 + 30 = 119

C2 = second column elements = 20 + 30 + 30 + 26 = 106

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C3 = third column elements = 32 + 28 + 22 + 36 = 118

C4 = fourth column elements = 28 + 20 + 35 + 28 = 111

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From the given table, rewrite the experimental values for each treatment

separately as follows:

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Treatment

A

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B

C

D

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28

20

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32

28

20

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36

30

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28

22

35

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25

30

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26

36

28

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T1 = A = 28 + 20 + 22 + 26 = 96

T2 = B = 20 + 36 + 35 + 36 = 127

T3 = C = 32 + 30 + 25 + 28 = 115

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T4 = D =

0

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28 + 28 + 3

1

+ 30 = 1 6

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G = T1

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2

+ T + T3 + T3

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+

= 96

1

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127 + 1 5 + 116 = 454

n = No. of treatments = 4

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N = n2 = 16

Correction Factor = G2/N = 4542 / 16 = 206116 / 16 = 12882.25

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The total sum of squares (TSS), Row sum of squares (RSS), Column sum of

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squares (CSS) and Treatment sum of squares (TrSS) are calculated as follows:

TSS = y2 ij ? Correction Factor

RSS = ( R 2

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i

/ n ) ? Correction Factor

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CSS = (C 2

j

/ nj ) ? Correction Factor

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TrSS = (T2k /n) ? Correction Factor

y2 ij =282 +202+322+282+362+302 +282 +202 +252 +302 +222 +352 +302 +262

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+362+282

=784+400+1024+784+1296+900+784+400+625+900+484+1225+900+676+12

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96+784

=13262

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TSS = y2 ij ? CF = 13262 ? 12882.25 = 379.75

RSS = R 2

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2

2

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2

1

/ 4 + R2 / 4 + R3 / 4 + R4 / 4 ? CF

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= 108 2

2

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2

2

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/ 4 + 114 / 4 + 112 / 4 + 120 / 4 ? 12882.25

= 11664 / 4 + 12996 / 4 + 12544 / 4 + 14400 / 4 ? 12882.25

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= 2916 + 3249 + 3136 + 3600 ? 12882.25 = 12901 ? 12882.25 = 18.75

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CSS = C 2

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2

2

2

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1

/ 4 + C2 / 4 + C3 / 4 + C4 / 4 ? CF

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= 119 2

2

2

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/ 4 + 106 / 4 + 118 / 4 + 111 / 4 ? 12882.25

= 14161 / 4 + 11236 / 4 + 13924 / 4 + 12321/ 4 ? 12882.25

= 3540.25 + 2809 + 3481 + 3080.25 ? 12882.25 = 12910.5 ?12882.25

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= 28.25

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T

2

2

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2

2

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rSS = T1

/ 4 + T2 / 4 + T3 / 4 + T4 / 4 ? CF

= 96 2

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2

2

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/ 4 + 127 / 4 + 115 / 4 + 1162 / 4 ? 12882.25

= 9216 / 4 + 16129 / 4 + 13225 / 4 +13456/ 4 ? 12882.25

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= 2304 + 4032.25+ 3306.25 + 3364 ? 12882.25 = 13006.5 ? 12882.25

= 124.25

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ESS = Error sum of squares = TSS ? RSS ? CSS ? TrSS

= 379.75 ? (18.75 + 28.25 + 124.25 ) = 379.75 ?171.25 = 208.50

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We apply ANOVA to find out whether there is any significant difference in the

performance of the treatments. We formulate the following null hypothesis:

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H0: There is no significant difference in the training methods.

The null hypothesis has to be tested against the following alternative

hypothesis:

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H1: There is a significant difference in the training methods.

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We have to decide whether the null hypothesis has to be accepted or rejected at

a desired level of significance ().

We have the following ANOVA Table.

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ANOVA Table for LSD

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Source of

Degrees of

Sum of Mean Sum of

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Variance ratio

Variation

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Freedom

Squares Squares (MSS)

F

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(SS)

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Row

4 ? 1 = 3

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18.75

18.75 / 3 = 6.25 34.75 / 6.25 = 5.56

Column

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4 ? 1 = 3

28.25

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28.25 / 3 = 9.42 34.75 / 9.42 = 3.69

Treatment

4 ? 1 = 3

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124.25 124.25 / 3 = 41.42 41.42 / 34.75 = 1.19

Error

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3 x 2 = 6

208.50

208.50 / 6 = 34.75

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Total

16 ? 1 = 15

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379.75

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Calculation of F value: We consider `Treatment'.

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F = Greater variance / Smaller variance = 41.42 / 34.75 = 1.19

Degrees of freedom for greater variance (df1) = 3

Degrees of freedom for smaller variance (df2) = 6

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Table value of F at 5% level of significance = 4.76

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Inference:

Since the calculated value of F for the treatments is less than the table

value of F, the null hypothesis is accepted and it is concluded that there is no

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significant difference in the training methods A, B, C and D, at 5% level of

significance.

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Problem 4

Examine the following production values got from four different

machines A, B, C and D and determine whether there is any significant

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difference in the machines.

A

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D

C

B

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131

129

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126

126

C

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B

A

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D

125

125

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127

124

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D C B A

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125 120 123 126

B

A

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D

C

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123

126

127

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121

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Solution :

In this design, there are 4 treatments A, B, C and D. In the lay out of the

design, each treatment appears exactly once in each row as well as each column.

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Therefore this design is LSD.

Since the entries in the design are large, we will follow the coding method.

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Subtract 120 from each entry. We get the following LSD.

A

D

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C

B

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11

9

6

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6

C

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B

A

D

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5

5

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7

4

D

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C

B

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A

5

0

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3

6

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B

A

D

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C

3

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6

7

1

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R1 = first row elements = 11 + 9 + 6 + 6 = 32

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R2 = second row elements = 5 + 5 + 7 + 4 = 21

R3 = third row elements = 5 + 0 + 3 + 6 = 14

R4 = fourth row elements = 3 + 6 + 7 + 1 = 17

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C1 = first column elements = 11 + 5 + 5 + 3 = 24

C2 = second column elements = 9+ 5 + 0 + 6 = 20

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C3 = third column elements = 6 + 7 + 3 + 7 = 23

C4 = fourth column elements = 6 + 4 + 6 + 1 = 17

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From the given table, rewrite the experimental values for each treatment

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separately as follows:

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Treatment

A

B

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C

D

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11

6

6

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9

7

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5

5

4

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6

3

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0

5

6

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3

1

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7

T1 = A = 11 +7 + 6 + 6 = 30

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T2 = B = 6 +5 + 3 + 3 = 17

T3 = C = 6 + 5 + 0 + 1 = 12

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T4 = D = 9 + 4 + 5 + 7 = 25

G = T1 + T2 + T3 + T3 = 30 + 17 + 12 + 25 = 84

n = No. of treatments = 4

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N = n2 = 16

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Correction Factor = G2/N = 842 / 16 = 7056 / 16 = 441

y2 ij =112 +92+62+62+52+52 +72 +42 +52 +02 +32 +62 +32 +62 +72+12

=121+81+36+36+25+25+49+16+25+0+9+36+9+36+49+1 = 554

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The total sum of squares (TSS), Row sum of squares (RSS), Column sum of

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squares (CSS) and Treatment sum of squares (TrSS) are calculated as follows:

TSS = y2 ij ? CF = 554 ? 441 = 113

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RSS = R 2

2

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2

2

1

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/ 4 + R2 / 4 + R3 / 4 + R4 / 4 ? CF

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= 32 2

2

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2

2

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/ 4 + 21 / 4 + 14 / 4 + 17 / 4 ? 441

= 1024 / 4 + 441 / 4 + 196 / 4 + 289 / 4 ? 441

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= 256 + 110.25 + 49 + 72.25 ? 441 = 487.5 ? 441 = 46.5

CSS = C 2

2

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2

2

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1

/ 4 + C2 / 4 + C3 / 4 + C4 / 4 ? CF

= 24 2

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2

2

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2

/ 4 + 20 / 4 + 23 / 4 + 17 / 4 ? 441

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= 576/ 4 + 400 / 4 + 529 / 4 + 289 / 4 ? 441

= 144 + 100 + 132.25 + 72.25 ? 441 = 448.5 ? 441 = 7.5

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T

2

2

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2

2

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rSS = T1

/ 4 + T2 / 4 + T3 / 4 + T4 / 4 ? CF

= 30 2

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2

2

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/ 4 + 17 / 4 + 12 / 4 + 252 / 4 ? 441

= 900 / 4 + 289 / 4 + 144 / 4 + 625 / 4 ? 441

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= 225 + 72.25+ 36 + 156.25 ? 441 = 489.5 ? 441 = 48.5

ESS = TSS ? RSS ? CSS ? TrSS

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= 113 ? (46.5 + 7.5 + 48.5 ) = 113 ?102.5 = 10.5

We formulate the following null hypothesis:

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H0: There is no significant difference in the performance of the

machines.

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The null hypothesis has to be tested against the following alternative

hypothesis:

H1: There is a significant difference in the performance of the machines.

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We have to decide whether the null hypothesis has to be accepted or rejected at

a desired level of significance ().

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We have the following ANOVA Table.

ANOVA Table for LSD

Sum of

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Variance ratio

Source of

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Degrees of

Mean Sum of

Squares

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F

Variation

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Freedom

Squares (MSS)

(SS)

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Row

4 ? 1 = 3

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46.5

46.5 / 3 = 15.50 15.50 / 1.75 =

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8.857

Column

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4 ? 1 = 3

7.5

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7.5 / 3 = 2.50

2.50 / 1.75 =

1.429

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Treatment

4 ? 1 = 3

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48.5

48.5 / 3 = 16.17

16.17 / 1.75 =

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9.240

Error

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3 x 2 = 6

10.5

10.5 / 6 = 1.75

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Total

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16 ? 1 = 15

113.0

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Calculation of F value: We consider `Treatment'.

F = Greater variance / Smaller variance = 16.17 / 1.75 = 9.240

Degrees of freedom for greater variance (df1) = 3

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Degrees of freedom for smaller variance (df2) = 6

Table value of F at 5% level of significance = 4.76

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Inference:

Since the calculated value of F for the treatments is greater than the table

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value of F, the null hypothesis is rejected and the alternative hypothesis is

accepted. It is concluded that there is a significant difference in the performance

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of the machines A, B, C and D at 5% level of significance.

Problem 5

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The financial manager of a company obtained the following details on

the LSD concerning the resources mobilized through 4 different schemes.

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Source of

Degrees of

SS

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Variation

Freedom

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Row

3

270

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Column

3

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150

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Treatment

3

1380

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Error

6

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156

Total

15

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1956

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Examine the data and find out whether there is any significant difference in the

schemes.

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Solution :

ANOVA Table for LSD

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Sum of

Variance ratio

Source of

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Degrees of

Mean Sum of

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Squares

F

Variation

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Freedom

Squares (MSS)

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(SS)

Row

3

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270

270 / 3 = 90

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90 / 26 = 3.462

Column

3

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150

150 / 3 = 50

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50 / 26 = 1.923

Treatment

3

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1380

1380 / 3 = 460

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460 / 26 = 17.692

Error

6

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156

156 / 6 = 26

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Total

15

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1956

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Null hypothesis:

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H0: There is no significant difference in the performance of the schemes.

Alternative hypothesis:

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H1: There is a significant difference in the performance of the schemes.

Calculation of F value: We consider `Treatment'.

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F = Greater variance / Smaller variance = 460 / 26 = 17.692

Degrees of freedom for greater variance (df1) = 3

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Degrees of freedom for smaller variance (df2) = 6

Table value of F at 5% level of significance = 4.76

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Inference:

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Since the calculated value of F for the treatments is greater than the table

value of F, the null hypothesis is rejected and the alternative hypothesis is

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accepted. It is concluded that there is a significant difference in the financial

schemes A, B, C and D, at 5% level of significance.

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QUESTIONS

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1.

What is an experimental design? Explain.

2.

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Explain the key concepts in experimental design.

3.

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Explain the steps in experimental design.

4.

Explain the terms Replication, Randomization and Local Control.

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5.

What is meant by the lay out of an experimental design? Explain with an

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example.

6.

What is a data allocation table? Give an example.

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7.

Describe a Completely Randomized Design.

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8.

Describe a Randomized Block Design.

9.

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Describe a Latin Square Design.

10.

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Explain the construction of a lay out of a Latin Square Design.

11.

Explain the managerial application of an experimental design.

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UNIT IV

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4. PARTIAL AND MULTIPLE CORRELATION

Lesson Outline

?

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The concept of partial correlation

?

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The concept of multiple correlation

Learning Objectives

After reading this lesson you should be able to

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-

determine partial correlation coefficient

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-

determine multiple correlation coefficient

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I. PARTIAL CORRELATION

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Simple correlation is a measure of the relationship between a dependent

variable and another independent variable. For example, if the performance of a

sales person depends only on the training that he has received, then the

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relationship between the training and the sales performance is measured by the

simple correlation coefficient r. However, a dependent variable may depend on

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several variables. For example, the yarn produced in a factory may depend on

the efficiency of the machine, the quality of cotton, the efficiency of workers,

etc. It becomes necessary to have a measure of relationship in such complex

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situations. Partial correlation is used for this purpose. The technique of partial

correlation proves useful when one has to develop a model with 3 to 5 variables.

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Suppose Y is a dependent variable, depending on n other variables X1,

X2, ..., Xn.. Partial correlation is a measure of the relationship between Y and

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any one of the variables X1, X2,...,Xn, as if the other variables have been

eliminated from the situation.

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The partial correlation coefficient is defined in terms of simple

correlation coefficients as follows:

Let r12. 3 denote the correlation of X1 and X2 by eliminating the effect of X3.

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Let r12 be the simple correlation coefficient between X1 and X2.

Let r13 be the simple correlation coefficient between X1 and X3.

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Let r23 be the simple correlation coefficient between X2 and X3.

Then we have

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r12 - r13 r23

r12.3 =

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2

2

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(1- r 13) (1- r 23)

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r13 - r12 r32

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Similarly, r13.2 =

2

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2

(1- r 12) (1- r 32)

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r23 - r 21 r13

and r32.1 =

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2

(1- r 21) (1- r 13)

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Problem 1

Given that r12 = 0.6, r13 = 0.58, r23 = 0.70 determine the partial correlation

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coefficient r12.3

Solution:

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We have

0.6 0

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- .58 0

x .70

=

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2

(1 (

- 0.58) ) (1 (

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- 0.70) )

0.6 0

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- .406

=

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(1 0

- .3364)(1 0

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- .49)

0.194

=

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0.6636x 0.51

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0.194

=

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0.194

=

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=

0.8146 0

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x .7141 0.5817 0.3335

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Problem 2

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If r12 = 0.75, r13 = 0.80, r23 = 0.70, find the partial correlation coefficient r13.2

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Solution:

We have

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r13 - r12 r32

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r13.2 =

2

--- Content provided by FirstRanker.com ---

2

(1 - r 12) (1 - r 32)

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0.8 - 0.75X 0.70

=

--- Content provided by‍ FirstRanker.com ---

2

2

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(1- (0.75) ) (1- (0.70) )

0.8 - 0.525

--- Content provided by‌ FirstRanker.com ---

(1- 0.5625) (1- 0.49)

0.275

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

(0.4375) (0.51)

0.275

0.275

--- Content provided by‌ FirstRanker.com ---

=

=

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= 0.5823

0.6614 X 0.7141

0.4723

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II. MULTIPLE CORRELATION

When the value of a variable is influenced by another variable, the

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relationship between them is a simple correlation. In a real life situation, a

variable may be influenced by many other variables. For example, the sales

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achieved for a product may depend on the income of the consumers, the price,

the quality of the product, sales promotion techniques, the channels of

distribution, etc. In this case, we have to consider the joint influence of several

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independent variables on the dependent variable. Multiple correlations arise in

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this context.

Suppose Y is a dependent variable, which is influenced by n other

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variables X1, X2, ...,Xn. The multiple correlation is a measure of the relationship

between Y and X1, X2,..., Xn considered together.

The multiple correlation coefficients are denoted by the letter R. The

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dependent variable is denoted by X1. The independent variables are denoted by

X2, X3, X4,..., etc.

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Meaning of Notations:

R1.23 denotes the multiple correlation of the dependent variable X1 with two

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independent variables X2 and X3 . It is a measure of the relationship that X1 has

with X2 and X3 .

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R2.13 is the multiple correlation of the dependent variable X2 with two

independent variables X1 and X3.

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R3.12 is the multiple correlation of the dependent variable X3 with two

independent variables X1 and X2.

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R1.234 is the multiple correlation of the dependent variable X1 with three

independent variables X2 , X3 and X4.

Coefficient of Multiple Linear Correlations

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The coefficient of multiple linear correlation is given in terms of the partial

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correlation coefficients as follows:

2

2

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r 12 + r 13 - 2 12

r 13

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r 23

r

1.

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R 23 =

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2

1 - r 23

2

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2

r 2 1 + r 23 - 2 21

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r 23

r 13

r

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R2.13 =

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2

1 - r 13

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2

2

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--- Content provided by FirstRanker.com ---

31 + r

32 - 2 3

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r 1 32

r

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12

r

R3.12 =

--- Content provided by‍ FirstRanker.com ---

1 - r 12

Properties of the coefficient of multiple linear correlations:

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--- Content provided by⁠ FirstRanker.com ---

The coefficient of multiple linear correlations R is a non-negative

quantity. It varies between 0 and 1.

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2.

R1.23 = R1.32

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R2.13 = R2.31

R3.12 = R3.21, etc.

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3.

R1.23 |r12|,

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R1.32 |r13|, etc.

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Problem 3

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If the simple correlation coefficients have the values r12 = 0.6, r13 = 0.65,

r23 = 0.8, find the multiple correlation coefficient R1.23

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Solution:

2

--- Content provided by FirstRanker.com ---

2

r 12 + r 13 - 2 12

r 13

--- Content provided by FirstRanker.com ---

r 23

r

--- Content provided by FirstRanker.com ---

We have 1.

R 23 =

--- Content provided by‍ FirstRanker.com ---

2

1 - r 23

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2

2

--- Content provided by‌ FirstRanker.com ---

(0.6) + (0.65) - 2x0.6x0.65x0.8

=

--- Content provided by‌ FirstRanker.com ---

2

1 - (0.8)

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0.36+ 0.4225- 0.624

--- Content provided by‍ FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

1 - 0.64

0.7825- 0.624

--- Content provided by FirstRanker.com ---

0.36

0.1585

=

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0.36

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= 0.4403 = 0.6636

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Problem 4

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Given that r21 = 0.7, r23 = 0.85 and r13 = 0.75, determine R2.13

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Solution:

2

--- Content provided by‍ FirstRanker.com ---

2

r 2 1 + r 23 - 2 21

--- Content provided by FirstRanker.com ---

r 23

r 13

r

--- Content provided by FirstRanker.com ---

We have R2.13 =

--- Content provided by⁠ FirstRanker.com ---

2

1 - r 13

2

--- Content provided by FirstRanker.com ---

2

(0.7) + (0.85) - 2 x0.7x0.85x0.75

--- Content provided by‍ FirstRanker.com ---

=

2

--- Content provided by‍ FirstRanker.com ---

1 - (0.75)

0.49+ 0.7225- 0.8925

--- Content provided by‍ FirstRanker.com ---

=

1 - 0.5625

--- Content provided by‍ FirstRanker.com ---

1.2125- 0.8925

=

--- Content provided by FirstRanker.com ---

0.4375

0.32

--- Content provided by FirstRanker.com ---

=

= 0.7314 =0.8552

--- Content provided by‌ FirstRanker.com ---

0.4375

QUESTIONS

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1.

Explain partial correlation.

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2.

Explain multiple correlations.

3.

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State the properties of the coefficient of multiple linear correlations.

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UNIT IV

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5. DISCRIMINATE ANALYSIS

Lesson Outline

?

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An overview of Matrix Theory

?

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The objective of Discriminate Analysis

?

The concept of Discriminant Function

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?

Determination of Discriminant Function

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?

Pooled covariance matrix

Learning Objectives

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After reading this lesson you should be able to

-

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understand the basic concepts in Matrix Theory

- understand the objective of Discriminate Analysis

-

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understand Discriminant Function

-

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calculate the Discriminant Function

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PART ? I: AN OVERVIEW OF MATRIX THEORY

First, let us have an overview of matrix theory required for discriminate

analysis.

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A matrix is a rectangular or square array of numbers. The matrix

--- Content provided by FirstRanker.com ---

a

a

--- Content provided by‌ FirstRanker.com ---

a

11

--- Content provided by FirstRanker.com ---

12

1n

a

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a

a

--- Content provided by FirstRanker.com ---

21

22

2n

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--- Content provided by‍ FirstRanker.com ---

a

a

a

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1

m

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m2

mn

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is a rectangular matrix with m rows and n columns. We say that it is a matrix of

type m? n . A matrix with n rows and n columns is called a square matrix. We

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say that it is a matrix of type n? n .

A matrix with just one row is called a row matrix or a row vector.

Eg:

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(a a

a

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1

2

--- Content provided by‍ FirstRanker.com ---

n )

A matrix with just one column is called a column matrix or a column vector.

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b

1

--- Content provided by‍ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

Eg:

2

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--- Content provided by FirstRanker.com ---

b

m

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A matrix in which all the entries are zero is called a zero matrix.

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Addition of two matrices is accomplished by the addition of the numbers

in the corresponding places in the two matrices. Thus we have

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a

a b

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b

a + b

a + b

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11

12

--- Content provided by FirstRanker.com ---

11

12

11

--- Content provided by FirstRanker.com ---

11

12

--- Content provided by⁠ FirstRanker.com ---

12

+

=

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--- Content provided by⁠ FirstRanker.com ---

a

a

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b

b

--- Content provided by FirstRanker.com ---

a + b

a + b

21

--- Content provided by FirstRanker.com ---

22

21

--- Content provided by FirstRanker.com ---

22

21

21

--- Content provided by FirstRanker.com ---

22

22

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Multiplication of a matrix by a scalar is accomplished by multiplying each

element in the matrix by that scalar. Thus we have

--- Content provided by FirstRanker.com ---

a

a

--- Content provided by FirstRanker.com ---

ka

ka

11

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12

11

--- Content provided by‍ FirstRanker.com ---

12

k

=

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--- Content provided by FirstRanker.com ---

a

a

ka

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ka

21

--- Content provided by FirstRanker.com ---

22

21

22

--- Content provided by FirstRanker.com ---

k (a

a

--- Content provided by FirstRanker.com ---

a

= ka ka

ka

--- Content provided by‌ FirstRanker.com ---

1

2

--- Content provided by⁠ FirstRanker.com ---

n )

( 1

2

--- Content provided by FirstRanker.com ---

n )

b kb

--- Content provided by FirstRanker.com ---

1

1

--- Content provided by⁠ FirstRanker.com ---

b

kb

--- Content provided by⁠ FirstRanker.com ---

2

2

k

--- Content provided by‌ FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

b

kb

--- Content provided by FirstRanker.com ---

m

m

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When a matrix A of type m? n and a matrix B of type n ? p are multiplied, we

obtain a matrix C of type m ? p . To get the element in the ith row, jth column of

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C, consider the elements of the ith row in A and the elements in the jth column of

B, multiply the corresponding elements and take the sum. Thus, we have

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a

a b

--- Content provided by‌ FirstRanker.com ---

b

a b + a b

a b + a b

--- Content provided by FirstRanker.com ---

11

12

--- Content provided by FirstRanker.com ---

11

12

11 11

--- Content provided by⁠ FirstRanker.com ---

12 21

11 12

--- Content provided by FirstRanker.com ---

12 22

=

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

a

a

b

--- Content provided by‌ FirstRanker.com ---

b

a b + a b

--- Content provided by FirstRanker.com ---

a b + a b

21

22 21

--- Content provided by FirstRanker.com ---

22

21 11

--- Content provided by‍ FirstRanker.com ---

22 21

21 12

22 22

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1 0

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The matrix I =

is called the identity matrix of order 2. Similarly we can

0 1

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consider identity ma

s

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trice of higher order. The identity matrix has the following

property: If the ma

A and I are of type

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trices

n? n , then A I = I A = A.

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a b

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Consider a square matrix of order 2. Denote it by A =

. The

c d

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a

b

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determinant of A = det A =

= ad ? bc. If it is zero, we say that A is a

c

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d

singular matrix. If it is not zero, we say that A is a non-singular matrix. When

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ad - bc 0 , A has a multiplicative inverse, denoted by

1

A- with the property

--- Content provided by‍ FirstRanker.com ---

that

1

--- Content provided by FirstRanker.com ---

-

1

AA

--- Content provided by‌ FirstRanker.com ---

A-

=

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A = I .

We have

-

--- Content provided by⁠ FirstRanker.com ---

-

1

--- Content provided by‍ FirstRanker.com ---

d

b

1

--- Content provided by‍ FirstRanker.com ---

A =

--- Content provided by‍ FirstRanker.com ---

det A -c

a

Note that

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1

a b d

--- Content provided by FirstRanker.com ---

b

- 1 0

=

--- Content provided by FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

ad - bc c d -c

a 0 1

--- Content provided by FirstRanker.com ---

A symmetric matrix is the one in which the first row and first column are

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identical; the second row and second column are identical; and so on.

Example:

a h g

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a b

and

--- Content provided by⁠ FirstRanker.com ---

h

b

f

--- Content provided by FirstRanker.com ---

b d

g f

--- Content provided by FirstRanker.com ---

c

are similar matrices.

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PART ? II: DISCRIMINATE ANALYSIS

The objective of discriminate analysis

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The objective of discriminate analysis (also known as discriminant

analysis) is to separate a population (or samples from the population) into two

distinct groups or two distinct conditionalities. After such a separation is made,

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we should be able to discriminate one group against the other. In other words, if

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some sample data is given, it should be possible for us to say with certainty

whether that sample data has come from the first group or the second group. For

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this purpose, a function called `Discriminant function' is constructed. It is a

linear function and it is used to describe the differences between two groups.

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It is to be noted that the concept of discriminant function is applicable

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when there are more than 2 distinct groups also. However, we restrict ourselves

to a situation of two distinct groups only. The discriminant function is the linear

combination of the observations from the two groups which minimizes the

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distance between the mean vectors of the two groups after some transformation

of the vectors. Suppose we consider 2 variables both taking values under two

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different conditions denoted by condition I and condition II. Suppose there are

m samples for each variable under condition I and n samples for each variable

under condition II.

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Let the values of the samples be as follows:

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Condition I

Condition II

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Variable 1

Variable 2

--- Content provided by‌ FirstRanker.com ---

Variable 1

Variable 2

p

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q

1

--- Content provided by FirstRanker.com ---

1

1

1

--- Content provided by FirstRanker.com ---

p

q

--- Content provided by‌ FirstRanker.com ---

2

2

2

--- Content provided by⁠ FirstRanker.com ---

2

M

--- Content provided by‌ FirstRanker.com ---

M

M

M

--- Content provided by FirstRanker.com ---

p

q

--- Content provided by‌ FirstRanker.com ---

m

m

n

--- Content provided by FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

Determine the means of the samples for the two variables under the two

conditions.

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Let

p be the mean of the values of variable 1 under condition I.

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Let

q be the mean of the values of variable 2 under condition I.

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Let

be the mean of the values of variable 1 under condition II.

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Let be the mean of the values of variable 2 under condition II.

Let y , y denote the column vectors whose entries are the mean values

--- Content provided by FirstRanker.com ---

1

2

under conditions I, II respectively.

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i.e.,

y =

,

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y =

--- Content provided by⁠ FirstRanker.com ---

1

2

--- Content provided by FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

( p - )

Calculate the column vector y - y =

--- Content provided by FirstRanker.com ---

. The pooled covariance matrix

1

--- Content provided by FirstRanker.com ---

2
(q - )

S is obtained as follows:

--- Content provided by‍ FirstRanker.com ---

m

(

p -p +

--- Content provided by‍ FirstRanker.com ---

-

p -p q -q + - -

--- Content provided by FirstRanker.com ---

i

) n ( j )

m

--- Content provided by⁠ FirstRanker.com ---

n

2

--- Content provided by‍ FirstRanker.com ---

2

( i )( i )

( j )( j )

--- Content provided by FirstRanker.com ---

1

i 1

--- Content provided by⁠ FirstRanker.com ---

=

j 1

=

--- Content provided by‍ FirstRanker.com ---

i 1

=

--- Content provided by FirstRanker.com ---

j 1

=

S

--- Content provided by‌ FirstRanker.com ---

= m+n-2 m
(

p -p)(q -q) n

--- Content provided by FirstRanker.com ---

+ (

--- Content provided by FirstRanker.com ---

- )( -)

m (

q -q) n

--- Content provided by⁠ FirstRanker.com ---

+ -

i

--- Content provided by‍ FirstRanker.com ---

i

j

j

--- Content provided by⁠ FirstRanker.com ---

i

( j )2

--- Content provided by FirstRanker.com ---

2

i 1

--- Content provided by⁠ FirstRanker.com ---

=

j 1

=

--- Content provided by‍ FirstRanker.com ---

i 1

=

--- Content provided by‌ FirstRanker.com ---

j 1

=

--- Content provided by‍ FirstRanker.com ---

a b

1

--- Content provided by‍ FirstRanker.com ---

d

b

-

--- Content provided by FirstRanker.com ---

Note that the inverse of the matrix

is

--- Content provided by‌ FirstRanker.com ---

, provided

c d

ad - bc -c

--- Content provided by FirstRanker.com ---

a

ad - bc 0 .

--- Content provided by‍ FirstRanker.com ---

Calculate the inverse of the matrix S. Denote it by

1

S - . Find the matrix product

--- Content provided by FirstRanker.com ---

-1

S ( y - y ) . The result is a column vector order 2. Denote it by and the

--- Content provided by‌ FirstRanker.com ---

1

2

--- Content provided by FirstRanker.com ---

entries by and ? . Then = ?

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Fisher's discriminant function Z is obtained as

Z = y + ? y .

1

--- Content provided by FirstRanker.com ---

Application:

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Given an observation of the attributes, we can use the discriminant function

to decide whether it arose from condition I or condition II.

Problem

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A tourism manager adopts two different strategies. Under each strategy, the

number of tourists and the profits earned (in thousands of rupees) are as

--- Content provided by FirstRanker.com ---

recorded below.

Strategy I

No. of tourists

--- Content provided by FirstRanker.com ---

Profit earned

30

--- Content provided by⁠ FirstRanker.com ---

60

32

64

--- Content provided by⁠ FirstRanker.com ---

30

65

--- Content provided by FirstRanker.com ---

38

61

40

--- Content provided by‍ FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

Strategy II

No. of tourists

Profit earned

--- Content provided by FirstRanker.com ---

38

55

--- Content provided by⁠ FirstRanker.com ---

40

61

37

--- Content provided by FirstRanker.com ---

57

36

--- Content provided by⁠ FirstRanker.com ---

55

46

58

--- Content provided by‍ FirstRanker.com ---

41

61

--- Content provided by FirstRanker.com ---

42

59

--- Content provided by‍ FirstRanker.com ---

Construct Fisher's discriminant function and examine whether the strategies

provide an effective tool of discrimination of the tourist operations.

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Solution:

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The given values are plotted in a graph. One point belonging to Strategy

I seems to be an outlier as it is closer to the points of Strategy II. The other

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points seem to fall in two clusters. We shall examine this phenomenon by means

of Fisher's discriminant function.

--- Content provided by‍ FirstRanker.com ---

We have

38 55

1

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

p

30

q 60

--- Content provided by‍ FirstRanker.com ---

40

61

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

1

2

--- Content provided by‌ FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

p

32

--- Content provided by⁠ FirstRanker.com ---

q

64

--- Content provided by FirstRanker.com ---

37 57

2

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

3

3

--- Content provided by FirstRanker.com ---

p = 30 , q = 65 ,

= 36 , = 55

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

3

4

--- Content provided by‌ FirstRanker.com ---

p

38

--- Content provided by FirstRanker.com ---

q

61

--- Content provided by‍ FirstRanker.com ---

46 58

4

--- Content provided by‍ FirstRanker.com ---

4

5

--- Content provided by‍ FirstRanker.com ---

5

p

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

q 65

41

--- Content provided by‍ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

61

5

--- Content provided by FirstRanker.com ---

5

6

--- Content provided by FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

7

The means of the above 4 columns are obtained as

170

--- Content provided by‍ FirstRanker.com ---

315

280

--- Content provided by FirstRanker.com ---

406

p =

= 34, q =

--- Content provided by⁠ FirstRanker.com ---

= 63, =

= 40, =

--- Content provided by FirstRanker.com ---

= 58

5

5

--- Content provided by FirstRanker.com ---

7

7

--- Content provided by‌ FirstRanker.com ---

y

= column vector containing the mean values under strategy I

--- Content provided by‍ FirstRanker.com ---

1

p 34

--- Content provided by FirstRanker.com ---

=

=

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q 63

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y

= column vector containing the mean values under strategy II

2

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--- Content provided by‍ FirstRanker.com ---

=

=

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--- Content provided by‌ FirstRanker.com ---

Therefore we get

34 40 6

-

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y - y =

-

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=

1

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63 58 5

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--- Content provided by‌ FirstRanker.com ---

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Calculation of p - p , q - q etc.,

i

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i

p - p

q - q

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i

i

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P q

( p - p)2 ( p - p )(q - q )

i

--- Content provided by‍ FirstRanker.com ---

i

(q - q

--- Content provided by‍ FirstRanker.com ---

i

)2

i

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= p - 34 = q - 63

30

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60

- 4

- 3

--- Content provided by‍ FirstRanker.com ---

16

12

--- Content provided by‍ FirstRanker.com ---

9

32

64

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- 2

1

--- Content provided by FirstRanker.com ---

4

- 2

1

--- Content provided by‍ FirstRanker.com ---

30

65

--- Content provided by FirstRanker.com ---

- 4

2

16

--- Content provided by‍ FirstRanker.com ---

- 8

4

--- Content provided by‍ FirstRanker.com ---

38

61

4

--- Content provided by FirstRanker.com ---

- 2

16

--- Content provided by FirstRanker.com ---

- 8

4

40

--- Content provided by FirstRanker.com ---

65

6

--- Content provided by‍ FirstRanker.com ---

2

36

12

--- Content provided by FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

88

6

22

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Calculation of - , - , etc.,

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j

j

-

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-

j

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( - )2 ( - )( - )

j

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j

( -

j

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)2

j

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= - 40 = - 58

38

55

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- 2

- 3

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4

6

9

--- Content provided by FirstRanker.com ---

40

61

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0

3

0

--- Content provided by FirstRanker.com ---

0

9

--- Content provided by FirstRanker.com ---

37

57

- 3

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- 1

9

--- Content provided by‍ FirstRanker.com ---

3

1

36

--- Content provided by‌ FirstRanker.com ---

55

- 4

--- Content provided by⁠ FirstRanker.com ---

- 3

16

12

--- Content provided by‌ FirstRanker.com ---

9

46

--- Content provided by‍ FirstRanker.com ---

58

6

0

--- Content provided by‌ FirstRanker.com ---

36

0

--- Content provided by‌ FirstRanker.com ---

0

41

61

--- Content provided by FirstRanker.com ---

1

3

--- Content provided by⁠ FirstRanker.com ---

1

3

9

--- Content provided by‌ FirstRanker.com ---

42

59

--- Content provided by FirstRanker.com ---

2

1

4

--- Content provided by FirstRanker.com ---

2

1

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70

26

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--- Content provided by‌ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

( p - p + - = 88 + 70 = 158

i

)

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7 ( j )2

2

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i 1

=

j 1

--- Content provided by FirstRanker.com ---

=

5

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( p - p q -q + - - = 6 + 26 = 32

i

)( i

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) 7 ( j

)( j )

--- Content provided by FirstRanker.com ---

i 1

=

j 1

--- Content provided by⁠ FirstRanker.com ---

=

5

--- Content provided by⁠ FirstRanker.com ---

(q -q + - = 22 + 38 = 60

i

)

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7 ( j )2

2

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i 1

=

j 1

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=

m + n ? 2 = 5 + 7 ? 2 = 10.

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The pooled covariance matrix

1 158

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32

15.8

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3.2

S =

=

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10 32 60 3.2

6

det S = 94.8 ? 10.24 = 84.56

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1

6

--- Content provided by FirstRanker.com ---

3

- .2 0.071

0.038

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-

1

--- Content provided by‍ FirstRanker.com ---

S - =

=

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84.56 3.2

--- Content provided by FirstRanker.com ---

-

15.8 0

- .038 0.187

--- Content provided by‌ FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

=

= S-

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( y - y

1

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2 )

?

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0.071 -0.038 -6 0.616

-

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=

=

--- Content provided by FirstRanker.com ---

0.038

-

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0.187 5

1

.163

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Fisher's discriminant function is obtained as

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Z = y + ? y

1

2

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= 0.616

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-

y +1.161y

1

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2

where y denotes the number of tourists and y is the profit earned

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1

2

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Inference

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We evaluate the discriminant function for the data given in the problem.

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Strategy I

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No. of tourists

Profit earned

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Z

(y1)

(y2)

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30

60

--- Content provided by‍ FirstRanker.com ---

51.3

32

64

--- Content provided by⁠ FirstRanker.com ---

54.72

30

--- Content provided by FirstRanker.com ---

65

57.12

38

--- Content provided by‍ FirstRanker.com ---

61

47.54

--- Content provided by⁠ FirstRanker.com ---

40

65

50.96

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Strategy II

No. of tourists

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Profit earned

Z

(y1)

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(y2)

38

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55

40.56

40

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61

46.30

--- Content provided by FirstRanker.com ---

37

57

43.50

--- Content provided by FirstRanker.com ---

36

55

--- Content provided by‍ FirstRanker.com ---

41.79

46

58

--- Content provided by‍ FirstRanker.com ---

39.12

41

--- Content provided by‍ FirstRanker.com ---

61

45.69

42

--- Content provided by‍ FirstRanker.com ---

59

42.75

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By referring to the projected values of the discriminant function, it is seen that

the discrimination function is able to separate the two strategies.

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QUESTIONS

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1.

Explain the objective of discriminate analysis.

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2.

Briefly describe how discriminate analysis is carried out.

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UNIT IV

6. CLUSTER ANALYSIS

Lesson Outline

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?

The objective of cluster analysis

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?

Cluster analysis for qualitative data

?

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Resemblance matrix

?

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Simple matching coefficient

?

Pessimistic, moderate, optimistic estimates of similarity

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?

Object-attribute incidence matrix

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?

Matching coefficient matrix

?

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Cluster analysis for quantitative data

?

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Hierarchical cluster analysis

?

Euclidean distance matrix

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?

Dendogram

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Learning Objectives

After reading this lesson you should be able to

- understand the objective of cluster analysis

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- perform cluster analysis for qualitative data

- perform cluster analysis for quantitative data

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- understand resemblance matrix

- determine simple matching coefficient

- understand the properties of simple matching coefficient

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- determine pessimistic, moderate, optimistic estimates of similarity

- understand object-attribute incidence matrix

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- understand matching coefficient matrix

- find out Euclidean distance matrix

- construct Dendogram

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THE OBJECTIVE OF CLUSTER ANALYSIS

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A cluster means a group of objects which remain together as far as a

certain characteristic is concerned. When several objects are examined

systematically, the cluster analysis seeks to put similar objects in the same

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cluster and dissimilar objects in different clusters so that each object will be

allotted to one and only one cluster. Thus, it is a method for estimation of

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similarities among multivariate data. Similarity or dissimilarity is concerned

with a certain attribute like magnitude, direction, shape, distance, colour, smell,

taste, performance, etc.

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Thus, it is to be seen that objects with similar description are pooled together to

form a single cluster and objects with dissimilar properties will contribute to

distinct clusters. For this purpose, given a set of objects, one has to determine

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which objects in that set are similar and which objects are dissimilar.

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Method of Cluster Analysis

Cluster analysis is a complex task. However, we can have a broad outline of

this analysis. One has to carry out the following steps:

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1.

Identify the objects that are required to be put in different clusters.

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2.

Prepare a list of attributes possessed by the objects under consideration.

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If they are too many, identify the important ones with the help of experts.

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3.

Identify the common attributes possessed by two or more objects.

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Find out the attributes which are present in one object and absent in other

objects.

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5.

Evolve a measure of similarity or dissimilarity. In other words, evolve a

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measure of "togetherness" or "standing apart".

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6.

Apply a standard algorithm to separate the objects into different clusters.

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Applications of Cluster Analysis

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The concept of cluster analysis has applications in a variety of areas. A few

examples are listed below:

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1. A marketing manager can use it to find out which brands of products are

perceived to be similar by the consumers.

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2. A doctor can apply this method to find out which diseases follow the same

pattern of occurrence.

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3. An agriculturist may use it to determine which parts of his land are similar as

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regards the cultivating crop.

4. Once a set of objects have been put in different clusters, the top level

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management can take a policy decision as to which cluster has to be paid more

attention and which cluster needs less attention, etc. Thus it will help the

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management in the decision on market segmentation.

In short, cluster analysis finds applications in so many contexts.

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I. Method of Cluster Analysis for Qualitative Data

We consider a case of binary attributes. They have two states, namely

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present or absent. Suppose we have to evolve a measure of resemblance

between two objects P and Q. Suppose we take into consideration certain pre-

determined attributes. If a certain attribute is present in an object, we will

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indicate it by 1 and if that attribute is absent we indicate it by 0. Count the

number of attributes which are present in both the objects, which are absent in

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both the objects and which are present in one object but not in the other. We

use the following notations.

a

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=

Number of attributes present in both P and Q,

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b

=

Number of attributes present in P but not in Q,

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c

=

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Number of attributes present in Q but not in P,

d

=

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Number of attributes absent in both P and Q.

Among these quantities, a and d are counts for matched pairs of attributes while

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b and c are counts for unmatched pairs of attributes.

Resemblance matrix of two objects

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The resemblance matrix of two objects P and Q consists of the values a, b, c, d

as its entries. It is shown below.

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Q

1

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P 1

a

b

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--- Content provided by‍ FirstRanker.com ---

c

d

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Simple matching coefficient

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We consider a similarity coefficient called simple matching coefficient

C(P,Q), defined as the ratio of the matched pairs of attributes to the total number

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a + d

of attributes. i.e., C ( P,Q) =

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a + b + c + d

Properties of simple matching coefficient

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1.

The denominator in C(P,Q) shows that the simple matching coefficient

gives equal weight for the unmatched pairs of attributes as well as the matched

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pairs.

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2.

The minimum value of C(P,Q) is 0.

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3.

The maximum value of C(P,Q) is 1.

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4.

A value of C(P,Q) = 1 indicates perfect similarity between the objects P

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and Q. This occurs when there are no unmatched pairs of attributes. i.e., b = c =

0.

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5.

A value of C(P,Q) = 0 indicates maximum dissimilarity between the

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objects P and Q. This occurs when there are no matched pairs of attributes. i.e., a

= d = 0.

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6.

C(P,Q) = C(Q,P).

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Using C(P,Q), we can estimate the percentage of similarity between P

and Q.

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8. C(P,P) = 1 since b = c = 0.

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Illustrative Problem 1

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A tourist is interested in evaluating two tourist spots P, Q with regard to

their similarity and dissimilarity. He considers 10 attributes of the tourist spots

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and collects the following data matrix:

Attribute

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Tourist Spot 1

Tourist Spot 2

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1

1

1

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2

0

--- Content provided by FirstRanker.com ---

0

3

1

--- Content provided by‍ FirstRanker.com ---

1

4

--- Content provided by⁠ FirstRanker.com ---

0

0

5

--- Content provided by‌ FirstRanker.com ---

0

1

--- Content provided by FirstRanker.com ---

6

1

1

--- Content provided by‌ FirstRanker.com ---

7

1

--- Content provided by‍ FirstRanker.com ---

1

8

1

--- Content provided by FirstRanker.com ---

1

9

--- Content provided by FirstRanker.com ---

1

0

10

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1

1

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Determine whether the two tourist spots are similar or not.

Solution:

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We obtain the following resemblance matrix.

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Q

1

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P

1

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a = 6

b = 1

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0

c = 1

d = 2

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We obtain the similarity coefficient as

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(

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a + d

C P,Q) = a +b +c + d

6 + 2

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6 +1+1+ 2

8

=

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= 0.8

10

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Inference

It is estimated that there is 80% similarity between the two tourist spots P and Q.

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Matching coefficient with correction term

The correction term in the matching coefficient can be defined in several

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ways. We consider two specific approaches.

(a) Rogers and Tanimoto coefficient of matching

By giving double weight for unmatched pairs of attributes, the matching

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coefficient with correction term is defined as

(

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a + d

C P,Q) =

.

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a + d + 2(b + c)

Perfect similarity between P and Q occurs when b = c = 0. In this case, C(P,Q)

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= 1.

Maximum dissimilarity between P and Q occurs when a = d = 0. In this case,

C(P,Q) = 0.

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(b) Sokal and Sneath coefficient of matching

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By giving double weight for matched pairs of attributes, the matching

coefficient with correction term is defined as

2(a + d )

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C(P, Q) =

.

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2(a + d ) + b + c

Perfect similarity between P and Q occurs when b = c = 0. In this case, C(P,Q) =

1.

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Maximum dissimilarity between P and Q occurs when a = d = 0. In this case,

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C(P,Q) = 0.

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Example

If we adopt Rogers and Tanimoto principle in the above problem, we get

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6 + 2

8

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C(P, Q) =

=

= 0.67 .

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6 + 2 + 2(1+1)

12

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So the estimate of similarity between P and Q is 67%

If we adopt Sokal and Sneath principle in the above example, we get

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2(6 + 2)

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16

C(P, Q) =

=

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= 0.89.

2(6 + 2) +1+1

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18

Thus, the similarity between P and Q is estimated as 89%

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Comparison of the three coefficients of similarity:

One can verify the following relation:

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a + d

a + d

+

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2(a d )

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.

a + d + 2(b + c)

a + b + c + d

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2(a + d ) + b + c

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i.e., Rogers-Tanimoto Coefficient Simple matching Coefficient Sokal-

Sneath Coefficient.

It is observed that Rogers and Tanimoto principle provides a pessimistic

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estimate of similarity. On the other hand, Sokal and Sneath principle gives an

optimistic estimate of similarity. The simple matching coefficient (without any

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correction term) gives a moderate estimate of similarity.

Clustering through object-attribute incidence matrix

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Consider a set of objects. Enumerate the attributes of the objects. Not all

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the attributes will be present in all the objects. The object-attribute incidence

matrix consists of the entries 0 and 1. If a certain attribute is present in an

object, the corresponding place in the matrix is marked by 1; otherwise it is

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marked by 0. This matrix is useful in separating the objects into clusters.

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Illustrative Problem 2

An expert of fashion designs identifies six fashions and five important

attributes of fashions. He obtains the following object-attribute incidence

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matrix.

Object

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1

2 3 4 5

6

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1 0 0 0 0 1

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Attribute 1

0 0 0 1 1 0

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0 1 0 0 1 0

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4

0 1 0 1 0 0

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5

1 0 1 0 0 1

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Separate the objects into two clusters.

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Solution:

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Method I: By examination of the entries in the object-attribute incidence

matrix

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Denote the 6 objects by O , O , O , O , O , O and the 5 attributes

1

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2

3

4

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5

6

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by A , A , A , A , A .

1

2

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3

4

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5

Consider the object O . Attributes A and A are present in object O and the

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1

1

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5

1

other 3 attributes are absent in it. Compare other objects with object O and

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1

find which object possesses similar attributes. For this, consider the colum s of

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n

the matrix. It is noticed that columns 1 and 6 in the matrix are identical. i.e.,

Attributes A and A are present in both the objects O and O . All the other

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1

5

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1

6

attributes are absent in both the objects. So the objects O and O can be put in

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1

6

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a cluster. Denote this cluster by{O ,O .

1

6}

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The remaining objects are O ,O ,O , O . Consider the columns 2,3,4,5 in

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2

3

4

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5

the matrix. No other column is id

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mn 2. The object

entical to colu

O possesses

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2

the attributes A and A . Identify other objects which possess at least one of

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3

4

these attributes. Objects O possess attribute A . So put the objects O and O

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4

4

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2

4

in a cluster. Denote this cluster by{O ,O .

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2

4}

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The remaining objects are O and O . The object O possesses only the

3

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5

3

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attribute A and the same is possessed by objects O and O . So the object O

5

1

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6

3

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is closer to the cluster {O ,O rather than the cluster{O ,O . So enlarge the

2

4}

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1

6}

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cluster {O ,O by including the object O . Thus we get the cluster{O ,O ,O .

1

6

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3}

1

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6}

3

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The remaining object is O . It possesses attributes A and A . These

5

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2

3

attributes are absent in the objects O , O , O . Attribute A in present in object

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1

6

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3

3

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O and attribute A is present in object O . So enlarge the cluster {O ,O by

2

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4}

2

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2

4

including the object O . In this way we get the cluster{O ,O ,O .

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2

4

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5}

5

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Result: Thus we obtain the following two clusters.

Cluster I: {O ,O ,O and

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1

3

6}

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Cluster II: {O ,O ,O .

2

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4

5}

The attributes present in cluster I are absent in cluster II and vice verse.

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Method II: Application of simple matching coefficient

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Calculate the matching coefficients of pairs of distinct objects. Since there are 6

objects, we have (6 x 5) / 2 = 15 such pairs. Tabulate the results as follows:

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Counts of matched and unmatched pairs of attributes

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Ordered pairs

a

b

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c

D

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Simple matching coefficient

of objects

= (a+b)/(a+b+c+d)

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O ,

O

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0

2

2

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1

0.2

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1

2

O , O

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1

1

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0

3

0.8

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1

3

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O , O

0

2

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2

1

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0.2

1

4

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O , O

0

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2

2

1

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0.2

1

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5

O , O

2

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0

0

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3

1.0

1

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6

O ,

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O

0

2

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1

2

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0.4

2

3

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O , O

1

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1

1

2

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0.6

2

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4

O ,

O

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1

1

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1

2

0.6

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2

5

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O ,

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O

0

1

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2

2

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0.4

2

6

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O , O

0

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1

2

2

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0.4

3

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4

O , O

0

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1

2

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2

0.4

3

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5

O , O

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1

0

1

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3

0.8

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3

6

O , O

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1

1

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1

2

0.6

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4

5

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O , O

0

2

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2

1

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0.2

4

6

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O , O

0

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2

2

1

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0.2

5

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6

We form the matching coefficient matrix for the objects under

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consideration by entering the simple matching coefficients against the pairs of

objects. It is a symmetric matrix since C(P,Q) = C(Q,P). In the present problem,

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we get the following matrix.

Object

1

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2 3 4 5

6

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1 0.2 0.8 0.2 0.2 1

Object 1

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0.2 1 0.4 0.6 0.6 0.4

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2

0.8 0.4 1 0.4 0.4 0.8

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3

0.2 0.6 0.4 1 0.6 0.2

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0.2 0.6 0.4 0.6 1 0.2

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5

1 0.4 0.8 0.2 0.2 1

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Consider the matching coefficients of pairs of distinct objects. Here there are 15

such pairs. The maximum among them is 1 = C( O , O ). Thus O and O have

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1

6

1

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6

the maximum similarity. Therefore, they can be put in a cluster. The next

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maximum matching coefficient is 0.8 possessed by the pairs ( O , O ) and

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1

3

( O , O ). Therefore the objects O , O , O can be clubbed together. The next

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3

6

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1

3

6

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maximum matching coefficient is 0.6 possessed by the pairs

( O , O ), ( O , O ) and ( O , O ). So the objects O , O , O can be considered

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2

4

2

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5

4

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5

2

4

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5

together. Since we have exhausted all the objects, the process is now complete.

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Result: Thus we have arrived at Cluster I: {O ,O ,O and Cluster II:

1

3

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6}

{O ,O ,O .

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2

4

5}

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II. Method of Cluster Analysis for Quantitative Data Hierarchical Cluster

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Analysis

The aim of the hierarchical cluster analysis is to put the given objects

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into various clusters and to arrange the clusters in a hierarchical order. A cluster

will consist of similar objects. Dissimilar objects will be put into different

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clusters. The clusters so formed will be arranged such that two clusters which

contain somewhat similar objects will be grouped together. Two clusters which

contain extremely dissimilar objects will stand apart in the hierarchical order.

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Steps in hierarchical cluster analysis

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The hierarchical cluster analysis comprises of the following steps.

1.

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Collect the necessary data in a matrix form. The columns in the matrix

denote the objects taken for examination and the rows denote the attributes that

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describe the objects. This matrix is called the data matrix.

2.

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Standardize the data matrix.

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Use the data matrix or the standardized data matrix to determine the

values of "resemblance coefficient". It is measure of similarities among pairs of

objects.

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By means of the values of the resemblance coefficient, construct a

diagram called a dendogram. It is a tree-like structure. A tree will exhibit the

different clusters into which the given set of objects is decomposed. The tree

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will indicate the hierarchy of similarities among different pairs of objects. This

is the reason for calling the method as hierarchical cluster analysis.

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Illustrative problem 3

A marketing manager wishes to examine the sales performance of 4 sales

persons P,Q,R,S in his division by means of cluster analysis. Records indicating

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their performance in the past 6 months are collected in the following table.

Unit: Rs. In lakhs

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Sales Performance

Month

P Q R S

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January

20

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22

25

23

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February

22

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23

27

24

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March

24

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24

28

25

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April

19

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21

22

20

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May

20

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22

24

21

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June

21

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23

25

24

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Help the manager in arranging the sales persons in a hierarchical order

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according to their sales performance.

Solution:

First we construct a Euclidean distance matrix. This matrix is formed

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by entering the Euclidean distances against the pairs of objects. In our context,

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Euclidean distance does not refer to any geographical distance. It is a relative

measure of the performance of two sales persons over the given period of time.

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It will indicate which two sales persons are similar in their performance and

which two sales persons are extremely different in their performance.

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Assume that there are n data values for each sales person. Denote the

sales data of two persons by vectors P and Q as follows:

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P = ( X , X ,..., X

1

2

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n )

Q = (Y ,Y ,...,Y

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1

2

n )

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Then the Euclidean distance between them is denoted by d(P,Q) and is

defined by the following relation:

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2

2

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2

d(P,Q) =

( X -Y + X -Y +...+ X -Y

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1

1 )

( 2 2)

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( n n )

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(1)

Note that d(P,P) = 0 and d(Q,P) = d(P,Q). In the problem under consideration, n

= 6. For the 4 sales persons P,Q,R,S, we have to calculate the 6 quantities

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d(P,Q), d(P,R), d(P,S), d(Q,R), d(Q,S), d(R,S). We have

P = (20, 22, 24,19, 20, 2 )

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1

Q = (22, 23, 24, 21, 22, 23)

R = (25, 27, 28, 22, 24, 25)

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S = (23, 24, 25, 20, 21, 24)

Using formula (1), calculate the Euclidean distances. We obtain

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2

d (P,Q) = (20 - 22)2 + (22 - 23)2 + (24 - 24)2 + (19 - 2 )2

1 + (20 - 22)2 + (21- 23)

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= ( 2

- )2 + (- )2

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1 + (0)2 + ( 2

- )2 + (-2)2 + (-2)2

= 4 +1+ 0 + 4 + 4 + 4

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= 17
= 4.1

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correct to 1 place of decimals. Next we get

2

d (P, R) = (20 - 25)2 + (22 - 27)2 + (24 - 28)2 + (19 - 22)2 + (20 - 24)2 + (21- 25)

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= ( 5

- )2 + ( 5

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- )2 + ( 4

- )2 + ( 3

- )2 + (-4)2 + (-4)2

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= 25 + 25 +16 + 9 +16 +16
= 107
=10.3

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2

d P S = (

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- )2 + ( - )2 + ( - )2 + ( - )2 + ( - )2

( , )

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20 23

22 24

24 25

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19 20

20 21 + (21- 24)

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= 9 + 4 +1+1+1+ 9
= 25
= 5

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2

d Q R = (

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- )2 + ( - )2 + ( - )2 + ( - )2 + ( - )2

( , )

22 25

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23 27

24 28

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21 22

22 24

+ (23- 25)

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= 9 +16 +16 +1+ 4 + 4
= 50
= 7.1

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d Q S = (

- )2 + ( - )2 + ( - )2 + ( - )2 + ( - )2 + ( - )2

( , )

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22 23

23 24

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24 25

21 20

22 21

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23 24

= 1+1+1+1+1+1
= 6

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= 2.4

2

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d R S = (

- )2 + ( - )2 + ( - )2 + ( - )2 + ( - )2

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( , )

25 23

27 24

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28 25

22 20

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24 21 + (25 - 24)

= 4 + 9 + 9 + 4 + 9 +1
= 36
= 6

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The following Euclidean distance matrix is got for the sales persons

P,Q,R and S.

P

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Q

R

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S

P -

4.1 10.3

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5

Q 4.1

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-

7.1

2.4

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R 10.3

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7.1

-

6

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S 5

2.4

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6

-

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Determination of Dendogram:

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We adopt a procedure called single linkage clustering method (SLINK).

This is based on the concept of nearest neighbours.

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Consider the distance between different persons. They are d(P,Q),

d(P,R), d(P,S), d(Q,R), d(Q,S), d(R,S). i.e., 4.1, 10.3, 5, 7.1, 2.4, 6

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The minimum among them is 2.4 = d(Q,S). Thus Q and S are the nearest

neighbours. Therefore, Q and S are selected to form a cluster at the first level,

denoted by {Q,S}. Next, we have to add another object to the list {Q,S}. The

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remaining elements are P and R. We have to decide whether P should be added

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to the list {Q,S} or R should be added. So we have to determine which among

P, R is nearer to the set {Q,S}. We consider the quantities

d ((Q, S ), P) = Minimum d (Q, P), d (S, P)

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= Minimum [4.1,5] = 4.1

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d ((Q, S ), R) = Minimum d (Q, R), d (S, R)

= Minimum [7.1,6] = 6

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Among these two quantities, we find Minimum [d((Q,S),P), d((Q,S),R)] =

Minimum [4.1,6] = 4.1 = d((Q,S),P).

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Therefore, P is nearer to the cluster {Q,S} rather than R. Consequently P

is attached with the set {Q,S} and so we obtain the cluster {{Q,S}, P}. This is

the cluster at the second level. If there are other objects remaining, we have to

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repeat the above procedure. In the present case, there is only one object

remaining i.e., R. We add R to the cluster ((Q,S),P) to form the cluster at the

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third level. We note that

d ((Q, S ), P), R = Minimum d (Q, R), d (S, R), d (P, R)

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= Minimum [7.1,6,10. ]

3 = 6

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Using these values, we obtain the following diagram:

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Dendogram

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Inference

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It is seen that sales persons Q, S are similar in their performance over the

given period of time. The next sales person somewhat similar to them is P. The

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sales person R stands apart.

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QUESTIONS

1.

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Explain the objective of cluster analysis.

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2.

Briefly describe how cluster analysis is carried out.

3.

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State the properties of simple matching coefficient.

4.

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Describe the methods of obtaining pessimistic, moderate and optimistic

estimates of the similarity between two objects.

5.

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Explain object-attribute incidence matrix.

6.

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Explain matching coefficient matrix.

7.

What are the steps in hierarchical cluster analysis?

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8.

What is Euclidean distance matrix? Explain.

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9.

What is a dendogram? Explain.

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UNIT IV

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7. FACTOR ANALYSIS AND CONJOINT ANALYSIS

Lesson Outline

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?

Factor Analysis

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?

Conjoint Analysis

?

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Steps in Development of Conjoint Analysis

?

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Applications of Conjoint Analysis

?

Advantages and disadvantages of Conjoint Analysis

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?

Illustrative problems

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?

Multi-factor evaluation approach in Conjoint Analysis

?

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Two-factor evaluation approach in Conjoint Analysis

Learning Objectives

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After reading this lesson you should be able to

- understand the concept of Factor Analysis

- understand the managerial applications of Factor Analysis

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- understand the concept of Conjoint Analysis

- apply rating scale technique in Conjoint Analysis

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- apply ranking method in Conjoint Analysis

- apply mini-max scaling method in Conjoint Analysis

- understand Multi-factor evaluation approach

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- understand Two-factor evaluation approach

- understand the managerial applications of Conjoint Analysis

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PART I - FACTOR ANALYSIS

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In a real life situation, several variables are operating. Some variables

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may be highly correlated among themselves. For example, if manager of a

restaurant has to analyse six attributes of a new product. He undertakes a sample

survey and finds out the responses of potential consumers. He obtains the

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following attribute correlation matrix.

Attribute

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1

2

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3

4

5

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6

Attribute

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1 1.00 0.05 0.10 0.95 0.20 0.02

2 0.05 1.00 0.15 0.10 0.60 0.85

3 0.10 0.15 1.00 0.50 0.55 0.10

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4 0.95 0.10 0.50 1.00 0.12 0.08

5 0.20 0.60 0.55 0.12 1.00 0.80

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6 0.02 0.85 0.10 0.08 0.80 1.00

Attribute Correlation Matrix

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We try to group the attributes by their correlations. The high correlation

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values are observed for the following attributes:

Attributes 1, 4 with a very high correlation coefficient of 0.95.

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Attributes 2, 4 with a high correlation coefficient of 0.85.

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Attributes 3, 4 with a high correlation coefficient of 0.85.

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As a result, it is seen that not all the attributes are independent. The attributes

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1 and 4 have mutual influence on each other while the attributes 2, 5 and 6 have

mutual influence among themselves. As far as attribute 3 is concerned, it has

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little correlation with the attributes 1, 2 and 6. Even with the other attributes 4

and 5, its correlation is not high. However, we can say that attribute 3 is

somewhat closer to the variables 4 and 5 rather than the attributes 1, 2 and 6.

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Thus, from the given list of 6 attributes, it is possible to find out 2 or 3 common

factors as follows:

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I.

1) The common features of the attributes 1,3,4 will give a factor

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2) The common features of the attributes 2, 5, 6 will give a factor

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or

II.

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1) The common features of the attributes 1,4 will give a factor

2) The common features of the attributes 2,5,6 will give a factor

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3) The attribute 3 can be considered to be an independent factor

The factor analysis is a multivariate method. It is a statistical technique

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to identify the underlying factors among a large number of interdependent

variables. It seeks to extract common factor variances from a given set of

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observations. It splits a number of attributes or variables into a smaller group of

uncorrelated factors. It determines which variables belong together. This method

is suitable for the cases with a number of variables having a high degree of

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correlation.

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In the above example, we would like to filter down the attributes 1, 4

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into a single attribute. Also we would like to do the same for the attributes 2, 5,

6. If a set of attributes (variables) A1, A2, ..., Ak filter down to an attribute Ai

(1 i k), we say that these attributes are loaded on the factor Ai or saturated

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with the factor Ai. Sometimes, more than one factor also may be identified.

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Basic concepts in factor analysis

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The following are the key concepts on which factor analysis is based.

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Factor: A factor plays a fundamental role among a set of attributes or variables.

These variables can be filtered down to the factor. A factor represents the

combined effect of a set of attributes. Either there may be one such factor or

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several such factors in a real life problem based on the complexity of the

situation and the number of variables operating.

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Factor loading: A factor loading is a value that explains how closely the

variables are related to the factor. It is the correlation between the factor and the

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variable. While interpreting a factor, the absolute value of the factor is taken into

account.

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Communality: It is a measure of how much each variable is accounted for by

the underlying factors together. It is the sum of the squares of the loadings of the

variable on the common factors. If A,B,C,... are the factors, then the

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communality of a variable is computed using the relation

h2 = ( The factor loading of the variable with respect to factor A)2 +

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( The factor loading of the variable with respect to factor B)2 +

( The factor loading of the variable with respect to factor C)2 + .....

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Eigen value: The sum of the squared values of factor loadings pertaining to a

factor is called an Eigen value. It is a measure of the relative importance of each

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factor under consideration.

Total Sum of Squares (TSS)

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It is the sum of the Eigen values of all the factors.

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Application of Factor Analysis:

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1. Model building for new product development:

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As pointed out earlier, a real life situation is highly complex and it

consists of several variables. A model for the real life situation can be built by

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incorporating as many features of the situation as possible. But then, with a

multitude of features, it is very difficult to build such a highly idealistic model.

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A practical way is to identify the important variables and incorporate them in the

model. Factor analysis seeks to identify those variables which are highly

correlated among themselves and find a common factor which can be taken as a

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representative of those variables. Based on the factor loading, some of variables

can be merged together to give a common factor and then a model can be built

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by incorporating such factors. Identification of the most common features of a

product preferred by the consumers will be helpful in the development of new

products.

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2. Model building for consumers:

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Another application of factor analysis is to carry out a similar exercise

for the respondents instead of the variables themselves. Using the factor loading,

the respondents in a research survey can be sorted out into various groups in

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such a way that the respondents in a group have more or less homogeneous

opinions on the topics of the survey. Thus a model can be constructed on the

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groups of consumers. The results emanating from such an exercise will guide

the management in evolving appropriate strategies towards market

segmentation.

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PART II - CONJOINT ANALYSIS

Introduction

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Everything in the world is undergoing a change. There is a proverb

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saying that "the old order changes, yielding place to new". Due to rapid

advancement in science and technology, there is fast communication across the

world. Consequently, the whole world has shrunk into something like a village

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and thus now-a-days one speaks of the "global village". Under the present set-

up, one can purchase any product of his choice from whatever part of the world

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it may be available. Because of this reason, what was a seller's market a few

years back has transformed into a buyer's market now.

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In a seller's market of yesterday, the manufacturer or the seller could

pass on a product according to his own perceptions and prescriptions. In the

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buyer's market of today, a buyer decides what he should purchase, what should

be the quality of the product, how much to purchase, where to purchase, when to

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purchase, at what cost to purchase, from whom to purchase, etc. A manager is

perplexed at the way a consumer takes a decision on the purchase of a product.

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In this background, conjoint analysis is an effective tool to understand a buyer's

preferences for a good or service.

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Meaning of Conjoint Analysis

A product or service has several attributes. By an attribute, we mean a

characteristic, a property, a feature, a quality, a specification or an aspect. A

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buyer's decision to purchase a good or service is based on not just one attribute

but a combination of several attributes. i.e., he is concerned with a join of

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attributes.

Therefore, finding out the consumer's preferences for individual

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attributes of a product or service may not yield accurate results for a marketing

research problem. In view of this fact, conjoint analysis seeks to find out the

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consumer's preferences for a `join of attributes', i.e., a combination of several

attributes.

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Let us consider an example. Suppose a consumer desires to purchase a

wrist watch. He would take into consideration several attributes of a wrist

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watch, namely the configuration details such as mechanism, size, dial,

appearance, colour and other particulars such as strap, price, durability,

warranty, after-sales service, etc. If a consumer is asked what the important

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aspect among the above list is, he would reply that all attributes are important

for him and so a manager cannot arrive at a decision on the design of a wrist

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watch. Conjoint analysis assumes that the buyer will base his decision not on

just the individual attributes of the product but rather he would consider various

combinations of the attributes, such as

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`mechanism, colour, price, after-sales service',

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or `dial, colour, durability, warranty',

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or `dial, appearance, price, durability', etc.

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This analysis would enable a manager in his decision making process in the

identification of some of the preferred combinations of the features of the

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product.

The rank correlation method seeks to assess the consumer's preferences

for individual attributes. In contrast, the conjoint analysis seeks to assess the

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consumer's preferences for combinations (or groups) of attributes of a product

or a service. This method is also called an `unfolding technique' because

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preferences on groups of attributes unfold from the rankings expressed by the

consumers. Another name for this method is `multi-attribute compositional

model' because it deals with combinations of attributes.

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Steps in the Development of Conjoint Analysis

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The development of conjoint analysis comprises of the following steps:

1.

Collect a list of the attributes (features) of a product or a service.

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2.

For each attribute, fix a certain number of points or marks. The more the

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number of points for an attribute, the more serious the consumers' concern on

that attribute.

3.

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Select a list of combinations of various attributes.

4.

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Decide a mode of presentation of the attributes to the respondents of the

study i.e., whether it should be in written form, or oral form, or a pictorial

representation etc.

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5.

Inform the combinations of the attributes to the prospective customers.

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6.

Request the respondents to rank the combinations, or to rate them on a

suitable scale, or to choose between two different combinations at a time.

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7.

Decide a procedure to aggregate the responses from the consumers. Any

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one of the following procedures may be adopted:

(i). Go by the individual responses of the consumers.

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(ii). Put all the responses together and construct a single utility

function.

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(iii). Split the responses into a certain number of segments such that

within each segment, the preferences would be similar.

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8.

Choose the appropriate technique to analyze the data collected from the

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respondents.

9.

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Identify the most preferred combination of attributes.

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10.

Incorporate the result in designing a new product, construction of an

advertisement copy, etc.

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Applications of Conjoint Analysis

1.

An idea of consumer's preferences for combinations of attributes will be

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useful in designing new products or modification of an existing product.

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2.

A forecast of the profits to be earned by a product or a service.

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3.

A forecast of the market share for the company's product.

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4.

A forecast of the shift in brand loyalty of the consumers.

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5.

A forecast of differences in responses of various segments of the

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product.

6.

Formulation of marketing strategies for the promotion of the product.

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7.

Evaluation of the impact of alternative advertising strategies.

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8.

A forecast of the consumers' reaction to pricing policies.

9.

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A forecast of the consumers' reaction on the channels of distribution.

10.

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Evolving an appropriate marketing mix.

11.

Even though the technique of conjoint analysis was developed for the

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formulation of corporate strategy, this method can be used to have a

comprehensive knowledge of a wide range of areas such as family decision

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making process, pharmaceuticals, tourism development, public transport system,

etc.

Advantages of Conjoint Analysis

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1.

The analysis can be carried out on physical variables.

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2.

Preferences by different individuals can be measured and pooled

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together to arrive at a decision.

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Disadvantages of Conjoint Analysis

1.

When more and more attributes of a product are included in the study,

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the number of combinations of attributes also increases, rendering the study

highly difficult. Consequently, only a few selected attributes can be included in

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the study.

2.

Gathering of information from the respondents will be a tough job.

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3.

Whenever novel combinations of attributes are included, the respondents

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will have difficulty in capturing such combinations.

4.

The psychological measurements of the respondents may not be

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accurate.

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In spite of the above stated disadvantages, conjoint analysis offers more

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scope to the researchers in identifying the consumers' preferences for groups of

attributes.

Illustrative Problem 1 : Application of Rating Scale Technique

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A wrist watch manufacturer desires to find out the combinations of attributes

that a consumer would be interested in. After considering several attributes, the

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manufacturer identifies the following combinations of attributes for carrying out

marketing research.

Combination ? I

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Mechanism, colour, price, after-scales service

Combination ? II

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Dial, colour, durability, warranty

Combination ? III

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Dial, appearance, price, durability

Combination ? IV

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Mechanism, dial, price, warranty

12 respondents are asked to rate the 4 combinations on the following 3-point

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rating scale.

Scale ? 1

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:

Less important

Scale ? 2

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:

Somewhat important

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Scale ? 3

:

Very important

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Their responses are given in the following table:

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Rating of Combination

Respondent Combination I Combination

Combination

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Combination

No.

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II

III

IV

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1

Less

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Somewhat

Very

Somewhat

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important

important

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important

important

2

--- Content provided by FirstRanker.com ---

Somewhat

Very

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Less

Somewhat

important

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important

important

--- Content provided by FirstRanker.com ---

important

3

Somewhat

--- Content provided by‌ FirstRanker.com ---

Less

Somewhat

--- Content provided by⁠ FirstRanker.com ---

Very important

important

important

--- Content provided by‌ FirstRanker.com ---

important

4

--- Content provided by FirstRanker.com ---

Less

Less

Very

--- Content provided by FirstRanker.com ---

Somewhat

important

--- Content provided by FirstRanker.com ---

important

important

important

--- Content provided by‌ FirstRanker.com ---

5

Somewhat

--- Content provided by‌ FirstRanker.com ---

Very

Very

Less important

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important

important

--- Content provided by FirstRanker.com ---

important

6

Somewhat

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Very

Somewhat

--- Content provided by‍ FirstRanker.com ---

Less important

important

important

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important

7

--- Content provided by FirstRanker.com ---

Somewhat

Less

Very

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Less important

important

--- Content provided by FirstRanker.com ---

important

important

8

--- Content provided by‌ FirstRanker.com ---

Very

Somewhat

--- Content provided by FirstRanker.com ---

Less

Somewhat

important

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important

important

--- Content provided by‍ FirstRanker.com ---

important

9

Very

--- Content provided by⁠ FirstRanker.com ---

Less

Somewhat

--- Content provided by⁠ FirstRanker.com ---

Somewhat

important

important

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important

important

--- Content provided by‌ FirstRanker.com ---

10 Somewhat

Very

Less

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Somewhat

important

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important

important

important

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11 Very Somewhat

Very

--- Content provided by⁠ FirstRanker.com ---

Somewhat

important

important

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important

important

--- Content provided by‍ FirstRanker.com ---

12 Very Less

Very

Somewhat

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important

important

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important

important

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Determine the most important and the least important combinations of the

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attributes.

Solution:

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Let us assign scores to the scales as follows:

Sl. No.

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Scale

Score

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1

Less important

1

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2

Somewhat important

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3

3

Very important

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--- Content provided by FirstRanker.com ---

The scores for the four combinations are calculated as follows:

Score for

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No. of

Combination Response

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Total Score

Response

Respondents

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Less important

1

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2

1 X 2 = 2

Somewhat

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I

3

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6

3 X 6 = 18

important

--- Content provided by FirstRanker.com ---

5

4

--- Content provided by FirstRanker.com ---

5 X 4 = 20

Very important

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12

40

Less important

--- Content provided by FirstRanker.com ---

1

5

--- Content provided by FirstRanker.com ---

1 X 5 = 5

Somewhat

II

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3

3

--- Content provided by‌ FirstRanker.com ---

3 X 3 = 9

important

5

--- Content provided by FirstRanker.com ---

4

5 X 4 = 20

--- Content provided by‌ FirstRanker.com ---

Very important

--- Content provided by‍ FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

34

Less important

1

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3

1 X 3 = 3

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Somewhat

III

3

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3

3 X 3 = 9

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important

5

6

--- Content provided by⁠ FirstRanker.com ---

5 X 6 = 30

Very important

--- Content provided by‌ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

12

42

--- Content provided by⁠ FirstRanker.com ---

Less important

--- Content provided by‌ FirstRanker.com ---

1

3

1 X 3 = 3

--- Content provided by‍ FirstRanker.com ---

Somewhat

IV

--- Content provided by‌ FirstRanker.com ---

3

8

3 X 8 = 24

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important

5

--- Content provided by‌ FirstRanker.com ---

1

5 X 1 = 5

Very important

--- Content provided by FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

12

32

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Let us tabulate the scores earned by the four combinations as follows:

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Combination Total

scores

I

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40

II

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34

III

42

--- Content provided by‌ FirstRanker.com ---

IV

32

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Inference:

It is concluded that the consumers consider combination III as the most

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important and combination IV as the least important.

Note: For illustrating the concepts involved, we have taken up 12 respondents in

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the above problem. In actual research work, we should take a large number of

respondents, say 200 or 100. In any case, the number of respondents shall not

be less than 30.

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Illustrative Problem 2:

Application of Ranking Method

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A marketing manager selects four combinations of features of a product

for study. The following are the ranks awarded by 10 respondents. Rank one

means the most important and rank 4 means the least important.

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Respondent

Rank Awarded

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No.

Combination

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I

Combination Combination Combination

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II III IV

--- Content provided by FirstRanker.com ---

1

2

1

--- Content provided by FirstRanker.com ---

3

4

--- Content provided by⁠ FirstRanker.com ---

2

1

4

--- Content provided by FirstRanker.com ---

2

3

--- Content provided by‌ FirstRanker.com ---

3

1

2

--- Content provided by‍ FirstRanker.com ---

3

4

--- Content provided by FirstRanker.com ---

4

3

2

--- Content provided by FirstRanker.com ---

4

1

--- Content provided by‍ FirstRanker.com ---

5

4

1

--- Content provided by⁠ FirstRanker.com ---

2

3

--- Content provided by FirstRanker.com ---

6

1

2

--- Content provided by FirstRanker.com ---

3

4

--- Content provided by‍ FirstRanker.com ---

7

4

3

--- Content provided by‍ FirstRanker.com ---

2

1

--- Content provided by‍ FirstRanker.com ---

8

3

1

--- Content provided by‍ FirstRanker.com ---

2

4

--- Content provided by‍ FirstRanker.com ---

9

3

1

--- Content provided by‍ FirstRanker.com ---

4

2

--- Content provided by FirstRanker.com ---

10

4

1

--- Content provided by‍ FirstRanker.com ---

2

3

--- Content provided by FirstRanker.com ---

Determine the most important and the least important combinations of the

features of the product.

--- Content provided by‌ FirstRanker.com ---

Solution:

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Let us assign scores to the ranks as follows:

Rank Score

--- Content provided by‌ FirstRanker.com ---

1

10

--- Content provided by‌ FirstRanker.com ---

2

8

3

--- Content provided by⁠ FirstRanker.com ---

6

4

--- Content provided by‍ FirstRanker.com ---

The scores for the 4 combinations are calculated as follows:

Score for

--- Content provided by FirstRanker.com ---

No. of

Combination Rank

--- Content provided by FirstRanker.com ---

Total Score

rank

Respondents

--- Content provided by‍ FirstRanker.com ---

1

10

--- Content provided by⁠ FirstRanker.com ---

3

10 X 3 = 30

2

--- Content provided by⁠ FirstRanker.com ---

8

1

--- Content provided by‌ FirstRanker.com ---

8 X 1= 8

I

3

--- Content provided by FirstRanker.com ---

6

3

--- Content provided by⁠ FirstRanker.com ---

6 X 3 = 18

4

4

--- Content provided by‌ FirstRanker.com ---

3

4 X 3 = 12

--- Content provided by FirstRanker.com ---

10

68

--- Content provided by FirstRanker.com ---

1

10

5

--- Content provided by⁠ FirstRanker.com ---

10 X 5 = 50

2

--- Content provided by FirstRanker.com ---

8

3

8 X 3 = 24

--- Content provided by‌ FirstRanker.com ---

II

3

--- Content provided by⁠ FirstRanker.com ---

6

1

6 X 1 = 6

--- Content provided by‍ FirstRanker.com ---

4

4

--- Content provided by‍ FirstRanker.com ---

1

4 X 1 = 4

--- Content provided by FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

10

84

1

--- Content provided by⁠ FirstRanker.com ---

10

Nil

--- Content provided by‌ FirstRanker.com ---

--

2

8

--- Content provided by FirstRanker.com ---

5

8 X 5 = 40

--- Content provided by FirstRanker.com ---

III

3

6

--- Content provided by⁠ FirstRanker.com ---

3

6 X 3 = 18

--- Content provided by‌ FirstRanker.com ---

4

4

2

--- Content provided by FirstRanker.com ---

4 X 2 = 8

--- Content provided by‍ FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

66

1

--- Content provided by⁠ FirstRanker.com ---

10

2

10 X 2 = 20

--- Content provided by‍ FirstRanker.com ---

2

8

--- Content provided by⁠ FirstRanker.com ---

1

8 X 1 = 8

IV

--- Content provided by FirstRanker.com ---

3

6

--- Content provided by‌ FirstRanker.com ---

3

6 X 3 = 18

4

--- Content provided by FirstRanker.com ---

4

4

--- Content provided by‌ FirstRanker.com ---

4 X 4 = 16

--- Content provided by‌ FirstRanker.com ---

62

The final scores for the 4 combinations are as follows:

--- Content provided by FirstRanker.com ---

Combination Score

I

--- Content provided by FirstRanker.com ---

68

II

84

--- Content provided by‌ FirstRanker.com ---

III

--- Content provided by‌ FirstRanker.com ---

66

IV

--- Content provided by‍ FirstRanker.com ---

62

Inference:

--- Content provided by FirstRanker.com ---

It is seen that combination II is the most preferred one by the consumers

and combination IV is the least preferred one.

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Illustrative Problem 3:

--- Content provided by‌ FirstRanker.com ---

Application of Mini-Max Scaling Method

An insurance manager chooses 5 combinations of attributes of a social

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security plan for analysis. He requests 10 respondents to indicate their

perceptions on the importance of the combinations by awarding the minimum

score and the maximum score for each combination in the range of 0 to 100.

--- Content provided by FirstRanker.com ---

The details of the responses are given below. Help the manager in the

identification of the most important and the least important combinations of the

--- Content provided by‍ FirstRanker.com ---

attributes of the social security plan.

Combination Combination Combination Combination Combination

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Respondent

I

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II

III

IV

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V

Number

--- Content provided by⁠ FirstRanker.com ---

Min Max Min Max Min Max Min Max Min Max

1

30

--- Content provided by FirstRanker.com ---

60

45

--- Content provided by‌ FirstRanker.com ---

85

50

70

--- Content provided by FirstRanker.com ---

40

75

--- Content provided by FirstRanker.com ---

50

80

2

--- Content provided by⁠ FirstRanker.com ---

35

65

--- Content provided by⁠ FirstRanker.com ---

50

80

50

--- Content provided by FirstRanker.com ---

80

35

--- Content provided by⁠ FirstRanker.com ---

75

40

75

--- Content provided by‍ FirstRanker.com ---

3

40

--- Content provided by⁠ FirstRanker.com ---

70

35

80

--- Content provided by‍ FirstRanker.com ---

60

80

--- Content provided by FirstRanker.com ---

40

70

50

--- Content provided by‌ FirstRanker.com ---

80

4

--- Content provided by FirstRanker.com ---

40

80

40

--- Content provided by‌ FirstRanker.com ---

80

60

--- Content provided by⁠ FirstRanker.com ---

85

50

75

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60

80

--- Content provided by FirstRanker.com ---

5

30

75

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50

80

--- Content provided by FirstRanker.com ---

60

75

60

--- Content provided by FirstRanker.com ---

75

60

--- Content provided by⁠ FirstRanker.com ---

85

6

35

--- Content provided by‌ FirstRanker.com ---

70

35

--- Content provided by FirstRanker.com ---

85

50

80

--- Content provided by FirstRanker.com ---

40

80

--- Content provided by‌ FirstRanker.com ---

40

80

7

--- Content provided by‌ FirstRanker.com ---

40

80

--- Content provided by‌ FirstRanker.com ---

40

75

45

--- Content provided by⁠ FirstRanker.com ---

75

50

--- Content provided by FirstRanker.com ---

70

40

80

--- Content provided by FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

30

80

--- Content provided by‌ FirstRanker.com ---

40

75

50

--- Content provided by FirstRanker.com ---

80

50

--- Content provided by‍ FirstRanker.com ---

70

60

80

--- Content provided by⁠ FirstRanker.com ---

9

45

--- Content provided by‍ FirstRanker.com ---

75

45

75

--- Content provided by FirstRanker.com ---

50

80

--- Content provided by‍ FirstRanker.com ---

50

80

50

--- Content provided by‌ FirstRanker.com ---

80

10

--- Content provided by⁠ FirstRanker.com ---

55

75

40

--- Content provided by FirstRanker.com ---

85

35

--- Content provided by⁠ FirstRanker.com ---

75

45

80

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40

80

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Solution:

For each combination, consider the minimum score and the maximum

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score separately and calculate the average in each case.

Combination Combination Combination Combination Combination

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I

II

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III

IV

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V

Min Max Min Max Min Max Min Max Min Max

Total

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380 730 420

800

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510 780 460 750 490 800

Average 38

73

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42

80

--- Content provided by FirstRanker.com ---

51

78

46

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75

49

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80

Consider the mean values obtained for the minimum and maximum of each

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combination and calculate the range for each combination as

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Range = Maximum value ? Minimum value

The measure of importance for each combination is calculated as follows:

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Measure of Importance for a combination of attributes

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Range for that combination

=

?100

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Sum of the ranges for all the combinations

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Tabulate the results as follows:

Measure of

Combination

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Max. Value

Min. Value

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Range

Importance

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I

73

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38

35

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21.875

II

80

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42

38

--- Content provided by‍ FirstRanker.com ---

23.750

III

78

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51

27

--- Content provided by FirstRanker.com ---

16.875

IV

75

--- Content provided by‍ FirstRanker.com ---

46

29

--- Content provided by FirstRanker.com ---

18.125

V

80

--- Content provided by FirstRanker.com ---

49

31

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19.375

Sum of the ranges

160

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100.000

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Inference:

It is concluded that combination II is the most important one and

combination III is the least important one.

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APPROACHES FOR CONJOINT ANALYSIS

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The following two approaches are available for conjoint analysis:

i.

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Multi-factor evaluation approach

ii.

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Two-factor evaluation approach

MULTI-FACTOR EVALUATION APPROACH IN CONJOINT

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ANALYSIS

Suppose a researcher has to analyze n factors. It is possible that each factor can

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assume a value in different levels.

Product Profile

A product profile is a description of all the factors under consideration, with any

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one level for each factor.

Suppose, for example, there are 3 factors with the levels given below.

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Factor

1 : 3

levels

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Factor

2 : 2

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levels

Factor

3 : 4

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levels

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Then we have 3? 2? 4 = 24 product profiles. For each respondent in the

research survey, we have to provide 24 data sheets such that each data sheet

contains a distinct profile. In each profile, the respondent is requested to

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indicate his preference for that profile in a rating scale of 0 to 10. A rating of 10

indicates that the respondent's preference for that profile is the highest and a

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rating of 0 means that he is not all interested in the product with that profile.

Example: Consider the product `Refrigerator' with the following factors and

levels:

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Factor 1

:

capacity of 180 liters; 200 liters; 230 liters

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Factor

2 : number

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of doors: either 1 or 2

Factor

3 : Price

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:

Rs. 9000; Rs. 10,000; Rs. 12,000

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Sample profile of the product

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Profile Number :

Capacity

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:

200 liters

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Number of Doors :

1

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Price

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:

Rs. 10,000

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Rating of Respondent:

(in the scale of 0 to 10)

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Steps in Multi-factor Evaluation Approach:

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Identify the factors or features of a product to be analyzed. If they are too

many, select the important ones by discussion with experts.

2.

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Find out the levels for each factor selected in Step 1.

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Design all possible product profiles. If there are n factors with levels L1,

L2,...Ln respectively, then the total number of profiles = L1L2...Ln.

4.

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Select the scaling technique to be adopted for multi-factor evaluation

approach (rating scale or ranking method).

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5.

Select the list of respondents using the standard sampling technique.

6.

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Request each respondent to give his rating scale for all the profiles of the

product. Another way of collecting the responses is to request each respondent

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to award ranks to all the profiles: i.e., rank 1 for the best profile, rank 2 for the

next best profile etc.

7.

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For each factor profile, collect all the responses from all the participating

respondents in the survey work.

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With the rating scale awarded by the respondents, find out the score secured by

each profile.

8.

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Tabulate the results in Step 8. Select the profile with the highest score.

This is the most preferred profile.

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9.

Implement the most preferred profile in the design of a new product.

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TWO-FACTOR EVALUATION APPROACH IN CONJOINT ANALYSIS

When several factors with different levels for each factor have to be

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analyzed, the respondents will have difficulty in evaluating all the profiles in the

multi-factor evaluation approach. Because of this reason, two-factor evaluation

approach is widely used in conjoint analysis.

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Suppose there are several factors to be analyzed with different levels of

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values for each factor, then we consider any two factors at a time with their

levels of values. For each such case, we have a data sheet called a two-factor

table. If there are n factors, then the number of such data sheets

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n n(n -1)

is

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=

.

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2

2

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Let us consider the example of `Refrigerator' described in the multi-

factor approach. For the two factors (i) capacity and (ii) price, we have the

following data sheet.

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Data Sheet (Two Factor Table) No:

Factor: Price of refrigerator

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Factor: Capacity

Price

of Refrigerator

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Rs. 9,000

Rs. 10,000

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Rs. 12,000

180 liters

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200 liters

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230 liters

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In this case, the data sheet is a matrix of 3 rows and 3 columns.

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Therefore, there are 3? 3 = 9 places in the matrix. The respondent has to award

ranks from 1 to 9 in the cells of the matrix. A rank of 1 means the respondent

has the maximum preference for that entry and a rank of 9 means he has the

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least preference for that entry. Compared to multi-factor evaluation approach,

the respondents will find it easy to respond to two-factor evaluation approach

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since only two factors are considered at a time.

Steps in two-factor evaluation approach:

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Identify the factors or features of a product to be analyzed.

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1.

Find out the levels for each factor selected in Step 1.

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2.

Consider all possible pairs of factors. If there are n factors, then the

n n(n -1)

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number of pairs is

=

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. For each pair of factors, prepare a two-factor

2

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2

table, indicating all the levels for the two factors. If L1 and L2 are the respective

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levels for the two factors, then the number of cells in the corresponding table is

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L1L2.

3.

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Select the list of respondents using the standard sampling technique.

4.

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Request each respondent to award ranks for the cells in each two-factor

table. i.e., rank 1 for the best cell, rank 2 for the next best cell, etc.

5.

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For each two-factor table, collect all the responses from all the

participating respondents in the survey work.

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6.

With the ranks awarded by the respondents, find out the score secured by

each cell in each two-factor table.

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7.

Tabulate the results in Step 7. Select the cell with the highest score.

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Identify the two factors and their corresponding levels.

8.

Implement the most preferred combination of the factors and their levels

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in the design of a new product.

Application:

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The two factor approach is useful when a manager goes for market

segmentation to promote his product. The approach will enable the top level

management to evolve a policy decision as to which segment of the market has

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to be concentrated more in order to maximize the profit from the product under

consideration.

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QUESTIONS

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1.

Explain the purpose of `Factor Analysis'.

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2.

What is the objective of `Conjoint Analysis'? Explain.

3.

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State the steps in the development of conjoint analysis.

4.

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State the applications of conjoint analysis.

5.

Enumerate the advantages and disadvantages of conjoint analysis.

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What is a `product profile'? Explain.

7.

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What are the steps in multi-factor evaluation approach in conjoint

analysis?

8.

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What is a `two-factor table'? Explain.

9.

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Explain two-factor evaluation approach in conjoint analysis.

REFERENCES

Green, P.E. and Srinivasan, V., Conjoint Analysis in Consumer Research: Issues

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and Outlook, Journal of Consumer Research, 5, 1978, 103 ? 123.

Green, P.E., Carrol, J. and Goldberg, A General Approach to Product Design

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Optimization via Conjoint Analysis, Journal of Marketing, 43, 1981, 17 ? 35.

Johnson, R.A. and Wichern, D.W., Applied Multivariate Statistical Analysis,

Pearson Education, Delhi, 2005.

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Kanji, G.K., 100 Statistical Tests, Sage Publications, New Delhi, 1994.

Kothari, C.R., Quantitative Techniques, Vikas Publishing House Private Ltd.,

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New Delhi, 1997.

Marrison, D.F., Multivariate Statistical Methods, McGraw Hill, New York,

1986.

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Panneerselvam, R., Research Methodology, Prentice Hall of India, New Delhi,

2004.

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Rencher, A.V., Methods of Multivariate Analysis, Wiley Inter-science, Second

Edition, New Jersey, 2002.

Romesburg, H.C., Cluster Analysis for Researchers, Lifetime Learning

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Publications, Belmont, California, 1984.

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Statistical Table-1: F-values at 1% level of significance

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df1: degrees of freedom for greater variance

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df2: degrees of freedom for smaller variance

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df2/df1

1 2 3 4 5 6 7 8 9 10

1

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4052.1 4999.5 5403.3 5624.5 5763.6 5858.9 5928.3 5981.0 6022.4 6055.8

2

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98.5 99.0 99.1 99.2 99.2 99.3 99.3 99.3 99.3 99.3

3

34.1 30.8 29.4 28.7 28.2 27.9 27.6 27.4 27.3 27.2

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4

21.1 18.0 16.6 15.9 15.5 15.2 14.9 14.7 14.6 14.5

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5

16.2 13.2 12.0 11.3 10.9 10.6 10.4 10.2 10.1 10.0

6

--- Content provided by‍ FirstRanker.com ---

13.7

10.9 9.7 9.1 8.7 8.4 8.2 8.1 7.9 7.8

--- Content provided by⁠ FirstRanker.com ---

7

12.2 9.5 8.4 7.8 7.4 7.1 6.9 6.8 6.7 6.6

8

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11.2 8.6 7.5 7.0 6.6 6.3 6.1 6.0 5.9 5.8

9

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10.5 8.0 6.9 6.4 6.0 5.8 5.6 5.4 5.3 5.2

10

10.0 7.5 6.5 5.9 5.6 5.3 5.2 5.0 4.9 4.8

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11

9.6 7.2 6.2 5.6 5.3 5.0 4.8 4.7 4.6 4.5

--- Content provided by FirstRanker.com ---

12

9.3 6.9 5.9 5.4 5.0 4.8 4.6 4.4 4.3 4.2

13

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9.0 6.7 5.7 5.2 4.8 4.6 4.4 4.3 4.1 4.1

14

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8.8 6.5 5.5 5.0 4.6 4.4 4.2 4.1 4.0 3.9

15

8.6 6.3 5.4 4.8 4.5 4.3 4.1 4.0 3.8 3.8

--- Content provided by‌ FirstRanker.com ---

16

8.5 6.2 5.2 4.7 4.4 4.2 4.0 3.8 3.7 3.6

--- Content provided by FirstRanker.com ---

17

8.4 6.1 5.1 4.6 4.3 4.1 3.9 3.7 3.6 3.5

18

--- Content provided by‌ FirstRanker.com ---

8.2 6.0 5.0 4.5 4.2 4.0 3.8 3.7 3.5 3.5

19

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8.1 5.9 5.0 4.5 4.1 3.9 3.7 3.6 3.5 3.4

20

8.0 5.8 4.9 4.4 4.1 3.8 3.6 3.5 3.4 3.3

--- Content provided by‌ FirstRanker.com ---

21

8.0 5.7 4.8 4.3 4.0 3.8 3.6 3.5 3.3 3.3

--- Content provided by FirstRanker.com ---

22

7.9 5.7 4.8 4.3 3.9 3.7 3.5 3.4 3.3 3.2

23

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7.8 5.6 4.7 4.2 3.9 3.7 3.5 3.4 3.2 3.2

--- Content provided by‌ FirstRanker.com ---

24

7.8 5.6 4.7 4.2 3.8 3.6 3.4 3.3 3.2 3.1

--- Content provided by‌ FirstRanker.com ---

25

7.7 5.5 4.6 4.1 3.8 3.6 3.4 3.3 3.2 3.1

26

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7.7 5.5 4.6 4.1 3.8 3.5 3.4 3.2 3.1 3.0

27

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7.6 5.4 4.6 4.1 3.7 3.5 3.3 3.2 3.1 3.0

28

7.6 5.4 4.5 4.0 3.7 3.5 3.3 3.2 3.1 3.0

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29

7.5 5.4 4.5 4.0 3.7 3.4 3.3 3.1 3.0 3.0

--- Content provided by‍ FirstRanker.com ---

30

7.5 5.3 4.5 4.0 3.6 3.4 3.3 3.1 3.0 2.9

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Statistical Table-2: F-values at 2.5% level of significance

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df1: degrees of freedom for greater variance

df2: degrees of freedom for smaller variance

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--- Content provided by⁠ FirstRanker.com ---

df2/df1

1

2

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3

4

--- Content provided by‍ FirstRanker.com ---

5

6

7

--- Content provided by⁠ FirstRanker.com ---

8

9

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10

1

647.7 799.5 864.1 899.5 921.8 937.1 948.2 956.6

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963.2 968.6

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2

38.5 39.0 39.1 39.2 39.2 39.3 39.3 39.3 39.3 39.3

3

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17.4 16.0 15.4 15.1 14.8 14.7 14.6 14.5 14.4 14.4

4

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12.2

10.6 9.9 9.6 9.3 9.1 9.0 8.9 8.9 8.8

5

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10.0 8.4 7.7 7.3 7.1 6.9 6.8 6.7 6.6 6.6

6

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8.8 7.2 6.5 6.2 5.9 5.8 5.6 5.5 5.5 5.4

7

8.0 6.5 5.8 5.5 5.2 5.1 4.9 4.8 4.8 4.7

--- Content provided by⁠ FirstRanker.com ---

8

7.5 6.0 5.4 5.0 4.8 4.6 4.5 4.4 4.3 4.2

--- Content provided by‌ FirstRanker.com ---

9

7.2 5.7 5.0 4.7 4.4 4.3 4.1 4.1 4.0 3.9

10

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6.9 5.4 4.8 4.4 4.2 4.0 3.9 3.8 3.7 3.7

11

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6.7 5.2 4.6 4.2 4.0 3.8 3.7 3.6 3.5 3.5

12

6.5 5.0 4.4 4.1 3.8 3.7 3.6 3.5 3.4 3.3

--- Content provided by FirstRanker.com ---

13

6.4 4.9 4.3 3.9 3.7 3.6 3.4 3.3 3.3 3.2

--- Content provided by FirstRanker.com ---

14

6.2 4.8 4.2 3.8 3.6 3.5 3.3 3.2 3.2 3.1

15

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6.1 4.7 4.1 3.8 3.5 3.4 3.2 3.1 3.1 3.0

--- Content provided by FirstRanker.com ---

16

6.1 4.6 4.0 3.7 3.5 3.3 3.2 3.1 3.0 2.9

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17

6.0 4.6 4.0 3.6 3.4 3.2 3.1 3.0 2.9 2.9

18

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5.9 4.5 3.9 3.6 3.3 3.2 3.0 3.0 2.9 2.8

19

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5.9 4.5 3.9 3.5 3.3 3.1 3.0 2.9 2.8 2.8

20

5.8 4.4 3.8 3.5 3.2 3.1 3.0 2.9 2.8 2.7

--- Content provided by FirstRanker.com ---

21

5.8 4.4 3.8 3.4 3.2 3.0 2.9 2.8 2.7 2.7

--- Content provided by‍ FirstRanker.com ---

22

5.7 4.3 3.7 3.4 3.2 3.0 2.9 2.8 2.7

--- Content provided by FirstRanker.com ---

2.7

--- Content provided by FirstRanker.com ---

23

5.7 4.3 3.7 3.4 3.1 3.0 2.9 2.8 2.7

2.6

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--- Content provided by⁠ FirstRanker.com ---

5.7 4.3 3.7 3.3 3.1 2.9 2.8 2.7 2.7

2.6

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25

5.6 4.2 3.6 3.3 3.1 2.9 2.8 2.7 2.6

--- Content provided by‌ FirstRanker.com ---

2.6

26

--- Content provided by‍ FirstRanker.com ---

5.6 4.2 3.6 3.3 3.1 2.9 2.8 2.7 2.6

2.5

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27

5.6 4.2 3.6 3.3 3.0 2.9 2.8 2.7 2.6

--- Content provided by FirstRanker.com ---

2.5

28

--- Content provided by FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

2.6

--- Content provided by FirstRanker.com ---

5.6

4.2

3.6

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3.2

3.0

--- Content provided by FirstRanker.com ---

2.9

2.7

2.6

--- Content provided by⁠ FirstRanker.com ---

2.5

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29

5.5 4.2 3.6 3.2 3.0 2.8 2.7 2.6 2.5

2.5

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5.5 4.1 3.5 3.2 3.0 2.8 2.7 2.6 2.5

2.5

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Statistical Table-3: F-values at 5% level of significance

df1: degrees of freedom for greater variance

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df2: degrees of freedom for smaller variance

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df2/df1

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1

2

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3 4

5

6

--- Content provided by‌ FirstRanker.com ---

7

8

--- Content provided by FirstRanker.com ---

9

10

1

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161.4 199.5 215.7 224.5 230.1 233.9 236.7 238.8 240.5 241.8

2

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18.5 19.0 19.1 19.2 19.2 19.3 19.3

19.3

19.3

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19.3

3

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10.1 9.5 9.2 9.1 9.0 8.9

8.8 8.8 8.8 8.7

4

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7.7 6.9 6.5 6.3 6.2 6.1

6.0 6.0 5.9 5.9

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5

6.6 5.7 5.4 5.1 5.0 4.9

4.8 4.8 4.7 4.7

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6

5.9 5.1 4.7 4.5 4.3 4.2

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4.2 4.1 4.0 4.0

7

5.5 4.7 4.3 4.1 3.9 3.8

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3.7 3.7 3.6 3.6

8

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5.3 4.4 4.0

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3.8 3.6 3.5

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3.5 3.4 3.3 3.3

9

5.1 4.2 3.8 3.6 3.4 3.3

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3.2 3.2 3.1 3.1

10

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4.9 4.1 3.7 3.4 3.3 3.2

3.1 3.0 3.0 2.9

11

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4.8 3.9 3.5 3.3 3.2 3.0

3.0 2.9 2.8 2.8

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12

4.7 3.8 3.4 3.2 3.1 2.9

2.9 2.8 2.7 2.7

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13

4.6 3.8 3.4 3.1 3.0 2.9

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2.8 2.7 2.7 2.6

14

4.6 3.7 3.3 3.1 2.9 2.8

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2.7 2.6 2.6 2.6

15

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4.5 3.6 3.2 3.0 2.9 2.7

2.7 2.6 2.5 2.5

16

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4.4 3.6 3.2 3.0 2.8 2.7

2.6 2.5 2.5 2.4

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17

4.4 3.5 3.1 2.9 2.8 2.6

2.6 2.5 2.4 2.4

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18

4.4 3.5 3.1 2.9 2.7 2.6

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2.5 2.5 2.4 2.4

19

4.3 3.5 3.1 2.8 2.7 2.6

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2.5 2.4 2.4 2.3

20

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4.3 3.4 3.0 2.8 2.7 2.5

2.5 2.4 2.3 2.3

21

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4.3

3.4

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3.0

2.8

2.6

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2.5 2.4

2.4

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2.3

2.3

22

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4.3 3.4 3.0 2.8 2.6 2.5

2.4

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2.4 2.3 2.3

23

4.2 3.4 3.0 2.7 2.6 2.5

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2.4

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2.3 2.3 2.2

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24

4.2 3.4 3.0 2.7 2.6 2.5

2.4

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2.3 2.3 2.2

25

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4.2 3.3 2.9 2.7 2.6 2.4

2.4

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2.3 2.2 2.2

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26

4.2 3.3 2.9 2.7 2.5 2.4

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2.3

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2.3 2.2 2.2

27

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4.2 3.3 2.9 2.7 2.5 2.4

2.3

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2.3 2.2 2.2

28

4.1 3.3 2.9 2.7 2.5 2.4

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2.3

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2.2 2.2 2.1

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29

4.1 3.3 2.9 2.7 2.5 2.4

2.3

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2.2 2.2 2.1

30

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4.1 3.3 2.9 2.6 2.5 2.4

2.3

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2.2 2.2 2.1

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Statistical Table-4: F-values at 10% level of significance

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df1: degrees of freedom for greater variance

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df2: degrees of freedom for smaller variance

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df2/df1 1 2 3 4

5

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6

7

8

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9 10

1

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39.8

49.5

53.5 55.8 57.2 58.2 58.9 59.4 59.8 60.1

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2

8.5

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9.0

9.1 9.2 9.2 9.3 9.3 9.3 9.3 9.3

3

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5.5

5.4

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5.3 5.3 5.3 5.2 5.2 5.2 5.2 5.2

4

4.5

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4.3

4.1 4.1 4.0 4.0 3.9 3.9 3.9 3.9

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5

4.0

3.7

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3.6 3.5 3.4 3.4 3.3 3.3 3.3 3.2

6

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3.7

3.4

3.2 3.1 3.1 3.0 3.0 2.9 2.9 2.9

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7

3.5

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3.2

3.0 2.9 2.8 2.8 2.7 2.7 2.7 2.7

8

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3.4

3.1

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2.9 2.8 2.7 2.6 2.6 2.5 2.5 2.5

9

3.3

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3.0

2.8 2.6 2.6 2.5 2.5 2.4 2.4 2.4

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10

3.2

2.9

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2.7 2.6 2.5 2.4 2.4 2.3 2.3 2.3

11

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3.2

2.8

2.6 2.5 2.4 2.3 2.3 2.3 2.2 2.2

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12

3.1

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2.8

2.6 2.4 2.3 2.3 2.2 2.2 2.2 2.1

13

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3.1

2.7

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2.5 2.4 2.3 2.2 2.2 2.1 2.1 2.1

14

3.1

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2.7

2.5 2.3 2.3 2.2 2.1 2.1 2.1 2.0

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15

3.0

2.6

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2.4 2.3 2.2 2.2 2.1 2.1 2.0 2.0

16

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3.0

2.6

2.4 2.3 2.2 2.1 2.1 2.0 2.0 2.0

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17

3.0

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2.6

2.4 2.3 2.2 2.1 2.1 2.0 2.0 2.0

18

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3.0

2.6

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2.4 2.2 2.1 2.1 2.0 2.0 2.0 1.9

19

2.9

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2.6

2.3 2.2 2.1 2.1 2.0 2.0 1.9 1.9

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20

2.9

2.5

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2.3 2.2 2.1 2.0 2.0 1.9 1.9 1.9

21

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2.9

2.5

2.3 2.2 2.1 2.0 2.0 1.9 1.9 1.9

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22

2.9

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2.5

2.3 2.2 2.1 2.0 2.0 1.9 1.9 1.9

23

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2.9

2.5

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2.3 2.2 2.1 2.0 1.9 1.9 1.9 1.8

24

2.9

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2.5

2.3 2.1 2.1 2.0 1.9 1.9 1.9 1.8

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25

2.9

2.5

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2.3 2.1 2.0 2.0 1.9 1.9 1.8 1.8

26

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2.9

2.5

2.3 2.1 2.0 2.0 1.9 1.9 1.8 1.8

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27

2.9

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2.5

2.2 2.1 2.0 2.0 1.9 1.9 1.8 1.8

28

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2.8

2.5

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2.2 2.1 2.0 1.9 1.9 1.9 1.8 1.8

29

2.8

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2.4

2.2 2.1 2.0 1.9 1.9 1.8 1.8 1.8

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30

2.8

2.4

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2.2 2.1 2.0 1.9 1.9 1.8 1.8 1.8

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UNIT V

1. STRUCTURE AND COMPONENTS OF RESEARCH REPORTS

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Lesson Objectives:

What is a Report?

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Characteristics of a good report

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Framework of a Report

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Practical Reports Vs Academic Reports

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Parts of a Research Report

A note on Literature Review

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Learning Objectives:

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After reading this lesson, you should be able to:

Understand the meaning of a research report

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Analyze the components of a good report

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Structure of a report

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Characteristic differences in Research Reporting

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WHAT IS A REPORT?

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A report is a written document on a particular topic, which conveys

information and ideas and may also make recommendations. Reports often form

the basis of crucial decision making. Inaccurate, incomplete and poorly written

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reports fail to achieve their purpose and reflect on the decision, which will

ultimately be made. This will also be the case if the report is excessively long,

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jargonistic and/ or structureless. A good report can be written by keeping the

following features in mind:

1.

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All points in the report should be clear to the intended reader.

2.

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The report should be concise with information kept to a necessary

minimum and arranged logically under various headings and sub-headings.

3.

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All information should be correct and supported by evidence.

4.

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All relevant material should be included in a complete report.

Purpose of Research Report

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1.

Why am I writing this report? Do I want to inform/ explain/

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persuade, or indeed all of these.

2.

Who is going to read this report? Managers/ academicians/

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researchers! What do they already know? What do they need to know? Do any

of them have certain attitudes or prejudices?

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3.

What resources do we have? Do I have access to a computer? Do I

have enough time? Can any of my colleagues help?

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4.

Think about the content of your report ? what am I going to put in it?

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What are my main themes? How much should be the text, and how much should

be the illustrations?

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Framework of a Report

The various frameworks can be used depending on the content of the

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report, but generally the same rules apply. Introduction, method, results and

discussion with references or bibliography at the end, and an abstract at the

beginning could form the framework.

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STRUCTURE OF A REPORT

Structure your writing around the IMR&D framework and you will

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ensure a beginning, middle and end to your report.

I

Introduction Why did I do this research?

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(beginning)

M

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Method

What did I do and how did I go about (middle)

doing it?

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R

Results

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What did I find?

(middle)

AND

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D

Discussion

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What does it all mean?

(end)

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What do I put in the beginning part?

TITLE PAGE

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Title of project, Sub?title (where

appropriate), Date, Author, Organization,

Logo

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BACKGROUND

History(if any) behind project

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ACKNOWLEDGEMENT

Author thanks people and organization who

helped during the project

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SUMMARY(sometimes called A condensed version of a report ? outlines

abstract of the synopsis)

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salient points, emphasizes main conclusions

and (where appropriate) the main

recommendations. N.B this is often

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difficult to write and it is suggested that you

write it last.

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LIST OF CONTENTS

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An at- a ? glance list that tells the reader

what is in the report and what page

number(s) to find it on.

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LIST OF TABLES

As above, specifically for tables.

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LIST OF APPENDICES

As above, specifically for appendices.

INTRODUCTION

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Author sets the scene and states his/ her

intentions.

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AIMS AND OBJECTIVES

AIMS ? general aims of the audit/ project,

broad statement of intent. OBJECTIVES ?

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specific things expected to do/ deliver(e.g.

expected outcomes)

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What do I put in the middle part?

METHOD

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Work steps; what was done ? how, by

whom, when?

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RESULT/FINDINGS

Honest presentation of the findings,

whether these were as expected or not.

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give the facts, including any

inconsistencies or difficulties

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encountered

What do I put in the end part?

DISCUSSION

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Explanation of the results.( you might like to

keep the SWOT analysis in mind and think about

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your project's strengths, weakness, opportunities

and threats, as you write)

CONCLUSIONS

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The author links the results/ findings with the

points made in the introduction and strives to

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reach clear, simply stated and unbiased

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conclusions. Make sure they are fully supported

by evidence and arguments of the main body of

your audit/project.

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RECOMMENDATIONS

The author states what specific actions should be

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taken, by whom and why. They must always be

linked to the future and should always be

realistic. Don't make them unless asked to.

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REFERENCES

A section of a report, which provides full details

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of publications mentioned in the text, or from

which extracts have been quoted.

APPENDIX

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The purpose of an appendix is to supplement the

information contained in the main body of the

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report.

PRACTICAL REPORTS VS. ACADEMIC REPORTS

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Practical Reports:

In the practical world of business or government, a report conveys an

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information and (sometimes) recommendations from a researcher who has

investigated a topic in detail. A report like this will usually be requested

by people who need the information for a specific purpose and their

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request may be written in terms of reference or the brief. whatever the

report, it is important to look at the instruction for what is wanted. A

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report like this differs from an essay in that it is designed to provide

information which will be acted on, rather than to be read by people

interested in the ideas for their own sake. Because of this, it has a different

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structure and layout.

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Academic Reports:

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A report written for an academic course can be thought of as a

simulation. We can imagine that someone wants the report for a practical

purpose, although we are really writing the report as an academic exercise for

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assessment. Theoretical ideas will be more to the front in an academic report

than in a practical one. Sometimes a report seems to serve academic and

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practical purposes. Students on placement with organizations often have to

produce a report for the organization and for assessment on the course.

Although the background work for both will be related, in practice, the report

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the student produces for academic assessment will be different from the report

produced for the organization, because the needs of each are different.

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RESEARCH REPORT: PRELIMINARIES

It is not sensible to leave all your writing until the end. There is always

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the possibility that it will take much longer than you anticipate and you will not

have enough time. There could also be pressure upon available word processors

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as other students try to complete their own reports. It is wise to begin writing up

some aspects of your research as you go along. Remember that you do not have

to write your report in the order than it will be read. Often it is easiest to start

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with the method section. Leave the introduction and the abstract to last. The

use of a word processor makes it very straightforward to modify and rearrange

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what you have written as your research progresses and your ideas change. The

very process of writing will help your ideas to develop. Last but by no means

least, ask someone to proofread your work.

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STRUCTURE OF A RESEARCH REPORT

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A research report has a different structure and layout in comparison to a

project report. A research report is for reference and is often quite a long

document. It has to be clearly structured for the readers to quickly find the

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information wanted. It needs to be planned carefully to make sure that the

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information given in the report is put under correct headings.

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PARTS OF RESEARCH REPORT

Cover sheet: This should contain some or all of the following:

Full title of the report

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Name of the researcher

Name of the unit of which the project is a part

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Name of the institution

Date/Year.

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Title page: Full title of the report.

Your name

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Acknowledgement: A thank you to the people who helped you.

Contents

List of the Tables

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Headings and sub-headings used in the report should be given with their

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page numbers. Each chapter should begin on a new page. Use a consistent

system in dividing the report into parts. The simplest may be to use chapters for

each major part and subdivide these into sections and sub-sections. 1, 2, 3 etc.

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can be used as the numbers for each chapter. The sections of chapter 3 (for

example) would be 3.1, 3.2, 3.3, and so on. For further sub-division of a sub-

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section you may use 3.2.1, 3.2.2, and so on.

Abstract or Summary or Executive Summary or Introduction:

This presents an overview of the whole report. It should let the reader see

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in advance, what is in the report. This includes what you set out to do, how

review of literature is focused and narrowed in your research, the relation of the

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methodology you chose to your objectives, a summary of your findings and

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analysis of the findings

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BODY

Aims and Purpose or Aims and Objectives:

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Why did you do this work? What was the problem you were

investigating? If you are not including review of literature, mention the specific

research/es which is/are relevant to your work.

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Review of Literature

This should help to put your research into a background context and to

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explain its importance. Include only the books and articles which relate directly

to your topic. You need to be analytical and critical, and not just describe the

works that you have read.

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Methodology

Methodology deals with the methods and principles used in an activity,

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in this case research. In the methodology chapter, explain the method/s you used

for the research and why you thought they were the appropriate ones. You may,

for example, be depending mostly upon secondary data or you may have

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collected your own data. You should explain the method of data collection,

materials used, subjects interviewed, or places you visited. Give a detailed

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account of how and when you carried out your research and explain why you

used the particular method/s, rather than other methods. Included in this chapter

should be an examination of ethical issues, if any.

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Results or Findings

What did you find out? Give a clear presentation of your results. Show

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the essential data and calculations here. You may use tables, graphs and figures.

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Analysis and Discussion

Interpret your results. What do you make out of them? How do they

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compare with those of others who have done research in this area? The accuracy

of your measurements/results should be discussed and deficiencies, if any, in the

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research design should be mentioned.

Conclusions

What do you conclude? Summarize briefly the main conclusions which

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you discussed under "Results." Were you able to answer some or all of the

questions which you raised in your aims and objectives? Do not be tempted to

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draw conclusions which are not backed up by your evidence. Note the

deviation/s from expected results and any failure to achieve all that you had

hoped.

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Recommendations

Make your recommendations, if required. The suggestions for action and

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further research should be given.

Appendix

You may not need an appendix, or you may need several. If you have

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used questionnaires, it is usual to include a blank copy in the appendix. You

could include data or calculations, not given in the body, that are necessary, or

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useful, to get the full benefit from your report. There may be maps, drawings,

photographs or plans that you want to include. If you have used special

equipment, you may include information about it.

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The plural of an appendix is appendices. If an appendix or appendices

are needed, design them thoughtfully in a way that your readers find it/them

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convenient to use.

References

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List all the sources which you referred in the body of the report. You

may use the pattern prescribed by American Psychological Association, or any

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other standard pattern recognized internationally.

REVIEW OF LITERATURE

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In the case of small projects, this may not be in the form of a critical review

of the literature, but this is often asked for and is a standard part of larger

projects. Sometimes students are asked to write Review of Literature on a topic

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as a piece of work in its own right. In its simplest form, the review of literature

is a list of relevant books and other sources, each followed by a description and

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comment on its relevance.

The literature review should demonstrate that you have read and analysed

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the literature relevant to your topic. From your readings, you may get ideas

about methods of data collection and analysis. If the review is part of a project,

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you will be required to relate your readings to the issues in the project, and while

describing the readings, you should apply them to your topic. A review should

include only relevant studies. The review should provide the reader with a

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picture of the state of knowledge in the subject.

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Your literature search should establish what previous researches have been

carried out in the subject area. Broadly speaking, there are three kinds of sources

that you should consult:

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1. Introductory material;

2. Journal articles and

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3. Books.

To get an idea about the background of your topic, you may consult one or

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more textbooks at the appropriate time. It is a good practice to review in

cumulative stages - that is, do not think you can do it all at one go. Keep a

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careful record of what you have searched, how you have gone about it, and the

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exact citations and page numbers of your readings. Write notes as you go along.

Record suitable notes on everything you read and note the methods of

investigations. Make sure that you keep a full reference, complete with page

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numbers. You will have to find your own balance between taking notes that are

too long and detailed, and ones too brief to be of any use. It is best to write your

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notes in complete sentences and paragraphs, because research has shown that

you are more likely to understand your notes later if they are written in a way

that other people would understand. Keep your notes from different sources

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and/or about different points on separate index cards or on separate sheets of

paper. You will do mainly basic reading while you are trying to decide on your

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topic. You may scan and make notes on the abstracts or summaries of work in

the area. Then do a more thorough job of reading later on, when you are more

confident of what you are doing. If your project spans several months, it would

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be advisable towards the end to check whether there are any new and recent

references.

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REFERENCES

There are many methods of referencing your work; some of the most

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common ones are the Numbered Style, American Psychological Association

Style and the Harvard Method, with many other variations. Just use the one

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you are most familiar and comfortable with. Details of all the works referred

by you should be given in the reference section.

THE PRESENTATION OF REPORT

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Well-produced, appropriate illustrations enhance the presentability of

a report. With today's computer packages, almost anything is possible.

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However, histograms, bar charts and pie charts are still the three 'staples'.

Readers like illustrated information, because it is easier to absorb and it's more

memorable. Illustrations are useful only when they are easier to understand than

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words or figures and they must be relevant to the text. Use the algorithm

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included to help you decide whether or not to use an illustration. They should

never be included for their own sake, and don't overdo it; too many

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illustrations distract the attention of readers.

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UNIT V

2. TYPES OF REPORTS: CHARACTERISTICS OF GOOD RESEARCH

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REPORT

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Lesson Outline:

Different types of Reports

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Technical Reports

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General Reports

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Reporting Styles

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Characteristics of a Good Report

Learning Objectives:

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After reading this lesson, you should be able to:

o Understand different types of reports

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o Technical Reports and their contents

o General Reports

o Different types of Writing styles

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o Essential characteristics of a Good Report

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Reports vary in length and type. Students' study reports are often called

Term papers, project reports, theses, dissertations depending on the nature of the

report. Reports of researchers are in the form of monographs, research papers,

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research thesis, etc. In business organizations a wide variety of reports are

under use: project reports, annual reports of financial statements, report of

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consulting groups, project proposals etc. News items in daily papers are also

one form of report writing. In this lesson, let us identify different forms of

reports and their major components.

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Types of Reports

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Reports may be categorized broadly as Technical Reports and General

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Reports based on the nature of methods, terms of reference and the extent of in-

depth enquiry made etc. On the basis of usage pattern, the reports may also be

classified as Information oriented reports, decision oriented reports and research

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based reports. Further, reports may also differ based on the communication

situation. For example, the reports may be in the form of Memo, which is

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appropriate for informal situations or for short periods. On the other hand, the

projects that extend over a period of time, often call for project reports. Thus,

there is no standard format of reports. The most important thing that helps in

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classifying the reports is the outline of its purpose and answers for the following

questions:

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What did you do?

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Why did you choose the particular research method that you used?

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What did you learn and what are the implications of what you learned?

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If you are writing a recommendation report, what action are you

recommending in response to what you learned?

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Two types of report formats are described below:

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A Technical Report

A Technical report mainly focuses on methods employed, assumptions

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made while conducting a study, detailed presentation of findings and drawing

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inferences and comparisons with earlier findings based on the type of data

drawn from the empirical work.

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An outline of a Technical Report mostly consists of the following:

Title and Nature of the Study:

Brief title and the nature of work sometimes followed by subtitle indicate more

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appropriately either the method or tools used. Description of objectives of the

study, research design, operational terms, working hypothesis, type of analysis

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and data required should be present.

Abstract

of

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Findings:

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A brief review of the main findings just can be made either in a paragraph or in

one/two pages.

Review of current status:

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A quick review of past observations and contradictions reported, applications

observed and reported are reviewed based on the in-house resources or based on

published observations.

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Sampling and Methods employed

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Specific methods used in the study and their limitations. In the case of

experimental methods, the nature of subjects and control conditions are to be

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specified. In the case of sample studies, details of the sample design i.e., sample

size, sample selection etc are given.

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Data sources and experiment conducted

Sources of data, their characteristics and limitations should be specified. In the

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case of primary survey, the manner in which data has been collected should be

described.

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Analysis of data and tools used.

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The analysis of data and presentation of findings of the study with supporting

data in the form of tables and charts are to be narrated. This constitutes the

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major component of the research report.

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Summary of findings

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A detailed summary of findings of the study and major observations should be

stated. Decision inputs if any, policy implications from the observations should

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be specified.

References

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A brief list of studies conducted on similar lines, either preceding the present

study or conducted under different experimental conditions is listed.

Technical appendices

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These appendices include the design of experiments or questionnaires used in

conducting the study, mathematical derivations, elaboration on particular

techniques of analysis etc.

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General Reports

General reports often relate popular policy issues mostly related to social

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issues. These reports are generally simple, less technical, good use of tables and

charts. Most often they reflect the journalistic style. Example for this type of

report is the "Best B-Schools Survey in Business Magazines". The outline of

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these reports is as follows:

1.

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Major Findings and their implications

2.

Recommendations for Action

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3.

Objectives of the Study

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4.

Method employed for collecting data

5.

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Results

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Writing Styles

There are atleast 3 distinct report writing styles that can be applied by

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students of Business Studies. They are called:

i.

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Conservative

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ii.

Key points

iii.

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Holistic

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i.

Conservative Style

Essentially, the conservative approach takes the best structural elements from

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essay writing and integrates these with appropriate report writing tools. Thus,

headings are used to deliberate upon different sections of the answer. In

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addition, the space is well utilized by ensuring that each paragraph is distinct

(perhaps separated from other paragraphs by leaving two blank lines in

between).

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ii.

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Key Point Style

This style utilizes all of the report writing tools and is thus more overtly `report-

looking'. Use of headings, underlining, margins, diagrams and tables are

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common. Occasionally reporting might even use indentation and dot points. The

important thing to remember is that the tools should be applied in a way that

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adds to the report. The question must be addressed and the tools applied should

assist in doing that. An advantage of this style is the enormous amount of

information that can be delivered relatively quickly.

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iii.

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Holistic Style

The most complex and unusual of the styles, holistic report writing aims to

answer the question from a thematic and integrative perspective. This style of

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report writing requires the researcher to have a strong understanding of the

course and is able to see which outcomes are being targeted by the question.

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Essentials of a Good Report:

Good research report should satisfy some of the following basic characteristics:

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STYLE

Reports should be easy to read and understand. The style of the writer

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should ensure that sentences are succinct and the language used is simple, to the

point and avoiding excessive jargon.

LAYOUT

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A good layout enables the reader to follow the report's intentions,

and aids the communication process. Sections and paragraphs should be given

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headings and sub-headings. You may also consider a system of numbering or

lettering to identify the relative importance of paragraphs and sub-paragraphs.

Bullet points are an option for highlighting important points in your report.

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ACCURACY

Make sure everything you write is factually accurate. If you would

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mislead or misinform, you will be doing a disservice not only to yourself but

also to the readers, and your credibility will be destroyed. Remember to refer to

any information you have used to support your work.

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CLARITY

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Take a break from writing. When you would come back to it, you'll

have the degree of objectivity that you need. Use simple language to express

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your point of view.

READABILITY

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Experts agree that the factors, which affect readability the most, are:

>

Attractive appearance

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>

Non-technical subject matter

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>

Clear and direct style

>

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Short sentences

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>

Short and familiar words

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REVISION

When first draft of the report is completed, it should be put to one side

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atleast for 24 hours. The report should then be read as if with eyes of the intended

reader. It should be checked for spelling and grammatical errors. Remember the

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spell and grammar check on your computer. Use it!

REINFORCEMENT

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Reinforcement usually gets the message across. This old adage is well

known and is used to good effect in all sorts of circumstances e.g., presentations

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- not just report writing.

>

TELL THEM WHAT YOU ARE GOING TO SAY: in the introduction and

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summary you set the scene for what follows in your report.

>

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THEN SAY IT : you spell things out in results/findings

>

THEN TELL THEM WHAT YOU SAID: you remind your readers through the

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discussion what it was all about.

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FEEDBACK MEETING

It is useful to circulate copies of your report prior to the feedback

meeting. Meaningful discussion can then take place during the feedback meeting

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with recommendations for change more likely to be agreed upon which can then

be included in your conclusion. The following questions should be asked at this

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stage to check whether the Report served the purpose:

>

Does the report have impact?

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>

Do the summary /abstract do justice to the report?

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>

Does the introduction encourage the reader to read more?

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Is the content consistent with the purpose of the report?

>

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Have the objectives been met?

>

Is the structure logical and clear?

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>

Have the conclusions been clearly stated?

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>

Are the recommendations based on the conclusions and expressed

clearly and logically?

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UNIT V

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3. FORMAT AND PRESENTATION OF A REPORT

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Lesson Outline:

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Importance of Presentation of a Report

Common Elements of a Format

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Title Page

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Introductory Pages

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Body of the Text

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References

Appendix

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Dos and Don'ts

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Presentation of Reports

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Learning Objectives:

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After reading this Lesson, you should be able to:

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Understand the importance of Format of a Report

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Contents of a Title Page

What should be in Introductory pages

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Contents of a Body Text

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How to report other studies

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Contents of an Appendix

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Dos and Don'ts of a Report

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Any report serves its purpose, if it is finally presented before the

stakeholders of the work. In the case of an MBA student, Project Work

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undertaken in an industrial enterprise and the findings of the study would be

more relevant, if they are presented before the internal managers of the

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company. In the case of reports prepared out of consultancy projects, a

presentation would help the users to interact with the research team and get

clarification on any issue of their interest. Business Reports or Feasibility

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Reports do need a summary presentation, if they have to serve the intended

purpose. Finally, the Research Reports of the scholars would help in achieving

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the intended academic purpose, if they are made public in academic

symposiums, seminars or in Public Viva Voce examinations. Thus, the

presentation of a report goes along with preparation of a good report. Further,

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the use of graphs, charts, citations and pictures draw the attention of readers and

audience of any type. In this lesson, it is intended to provide a general outline

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related to the presentation of any type of report. See Exhibit I

Exhibit I

Common Elements of a Report

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A report may contain some or all of the following, please refer to your

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departmental guidelines.

MEMORANDUM OR COVERING LETTER

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Memorandum Or Covering Letter is a brief note stating the purpose or giving an

explanation that is used when the report is sent to someone within the same

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organization.

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TITLE PAGE

It is addressed to the receiver of a report while giving an explanation for

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it, and is used when the report is for someone who does not belong to the same

organization as the writer. It contains a descriptive heading or name. It may also

contain author's name, position, company's name and so on.

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EXECUTIVE SUMMARY

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Executive Summary summarizes the main contents and is usually of about

300-350 words.

TABLE OF CONTENTS

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Table of Contents consists of a list of the main sections, indicating the page on

which each section begins.

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INTRODUCTION

Informs the reader of what the report is about--aim and purpose, significant

issues, any relevant background information.

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REVIEW OF LITERATURE

Presents critical analysis of the available research to build a base for the present

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study.

METHODOLOGY

Gives details about nature of the study, research design, sample, and tools used

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for data collection and analysis.

RESULTS

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Presents findings of the study.

DISCUSSION

Describes the reasoning and research in detail.

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CONCLUSION/S

Summarizes the main points made in the written work in the light of objectives.

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It often includes an overall answer to the problem/s addressed; or an overall

statement synthesizing the strands of information dealt with.

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RECOMMENDATION/S OR IMPLICATIONS

Gives suggestions related to the issue(s) or problem(s) dealt with. It may

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highlight the applications of the findings under Implications Section.

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REFERENCES

An alphabetical list of all sources referred in the report.

APPENDICES

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Extra information of further details placed after the main body of the text.

FORMATS OF REPORTS

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Before attempting to look into Presentation dimensions of a Report, a quick look

into standard format associated with a Research Report is examined hereunder.

The format generally includes the steps one should follow while writing and

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finalizing their research report.

Different Parts of a Report

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Generally different parts of a report include:

1.

Cover Page / Title Page

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2.

Introductory Pages ( Foreword, Preface, Acknowledgement, Table of

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Contents, List of Tables, List of Illustrations or Figures, Key words /

Abbreviations used etc.)

3.

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Contents of the Report (which generally includes a Macro setting,

Research Problem, Methodology used, Objectives of the study, Review of

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studies, Tools Used for Data Collection and Analysis, Empirical results in

one/two sections, Summary of Observations etc.)

4.

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References (including Appendices, Glossary of terms used, Source data,

Derivations of Formulas for Models used in the analysis etc.)

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Title Page:

The Cover page or Title Page of a Research Report should contain the following

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information:

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1.

Title of the Project / Subject

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2.

Who has conducted the study

3.

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For What purpose

4.

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Organization

5.

Period of submission

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A Model:

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An example of a Summer Project Report conducted by an MBA student

generally follows the following Title Page:

A STUDY ON THE USE OF COMPUTER TECHNOLOGY IN BANKING

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OPERATIONS IN XXX BANK LTD., PONDICHERRY

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A SUMMER PROJECT REPORT

PREPARED BY

Ms. MADAVI LATHA

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Submitted at

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SCHOOL OF MANAGEMENT

PONDICHERRY UNIVERSITY

PONDICHERRY ? 605 014

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2006

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Introductory Pages:

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Introductory pages generally do not constitute the Write up of the Research

work done. These introductory pages basically form the Index of the work

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done. These pages are usually numbered in Roman numerical (eg, I, ii, iii etc).

The introductory pages include the following components

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Foreword

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Preface

Acknowledgements

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Table of Contents

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List of Tables

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List of Figures / Charts

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Foreword is usually one page write up or a citation about the work by

any eminent / popular personality or a specialist in the given field of study.

Generally, the write up includes a brief background on the contemporary issues

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and suitability of the present subject and its timeliness, major highlights of the

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present work, brief background of the author etc. The writer of the Foreword

generally gives the Foreword on his letter head

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Preface is again one/two pages write up by the author of the book /

report stating circumstances under which the present work is taken up,

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importance of the work, major dimensions examined and intended audience for

the given work. The author gives his signature and address at the bottom of the

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page along with date and year of the work

Acknowledgements is a short section, mostly a paragraph. It mostly

consists of sentences giving thanks to all those associated and encouraged to

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carry out the present work. Generally, author takes time to acknowledge the

liberal funding by any funding agency to carry out the work, and agencies which

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had given permission to use their resources. At the end, the author thanks

everybody and gives his signature.

Table of Contents refers to the index of all pages of the said Research

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Report. These contents provide the information about the chapters, sub-

sections, annexure for each chapter, if any, etc. Further, the page numbers of the

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content of the report greatly helps any one to refer to those pages for necessary

details. Most authors use different forms while listing the sub contents. These

include alphabet classification and decimal classification. Examples for both of

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them are given below:

Example of content sheet (alphabet classification)

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An example of Content Sheet with decimal classification

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CONTENTS

Foreword

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i

Preface

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iii

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Acknowledgement

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v

Chapter I (Title of the Chapter) INTRODUCTION

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1.

Macro

Economic

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Background

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1

2. Performance of a specific industry sector

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6

3. Different studies conducted so far

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9

4

Nature

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and

Scope

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17

4.1.

Objectives

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of

the

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study

18

4.2.

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Methodology

adopted

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19

4.2.

a.

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Sampling

Procedure

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adopted

20

4.2.b.

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Year

of

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the

study 20

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Chapter II (Title of the Chapter): Empirical Results I

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1. Test results of H1

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2. Test Results of H2

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3 Test Results of H3

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32

3.1.

Sub

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Hypothesis

of

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H3

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3.2.

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Sub

Hypothesis

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of

H2

37

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Chapter III

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45

Chapter IV

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85

Chapter V (Summary & Conclusions)

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120

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Appendices

132

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References/Bibliography

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135

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Glossary

140

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List of Tables and Charts:

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Details of Charts and Tables given in the research Report are numbered

and presented on separate pages and the lists of such tables and charts are given

on a separate page. Tables are generally numbered either in Arabic numerals or

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in decimal form. In the case of decimal form, it is possible to indicate the

chapter to which the said table belongs. For example, Table 2.1 refers to Table

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1 in Chapter 2.

Executive Summary:

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Most Business Reports or Project works conducted on a specific issue

carry one or two pages of Executive Summary. This summary precedes the

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Chapters of the Regular Research Report. This summary generally contains a

brief description of problem under enquiry, methods used and the findings. A

line about the possible alternatives for decision making would be the last line of

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the Executive Summary.

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BODY OF THE REPORT:

The body of the Report is the most important part of the report. This

body of report may be segmented into a handful of Units or Chapters arranged in

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a sequential order. Research Report often present the Methodology, Objectives

of the study, Data tools, etc in the first or second chapters along with a brief

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background of the study, review of relevant studies. The major findings of the

study are incorporated into two or three chapters based on the major or minor

hypothesis tested or based on the sequence of objectives of the study. Further,

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the chapter plan may also be based likely on different dimensions of the problem

under enquiry.

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Each Chapter may be divided into sections. While the first section may narrate

the descriptive characteristics of the problem under enquiry, the second and

subsequent sections may focus on empirical results based on deeper insights of

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the problem of study. Each chapter based on Research Studies mostly contain

Major Headings, Sub headings, quotations drawn from observations made by

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earlier writers, footnotes and exhibits.

Use of References:

There are two types of reference formatting. The first is the 'in-text' reference

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format, where previous researchers and authors are cited during the building of

arguments in the Introduction and Discussion sections. The second type of

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format is that adopted for the Reference section for writing footnotes or

Bibliography.

Citations in the text

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The names and dates of researchers go in the text as they are mentioned e.g.,

"This idea has been explored in the work of Smith (1992)." It is generally

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unacceptable to refer to authors and previous researchers etc.

Examples of Citing References (Single author)

Duranti (1995) has argued or It has been argued that (Duranti, 1995)

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In the case of more authors,

Moore, Maguire, and Smyth (1992) proposed or It has been proposed that

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(Moore, Macquire, & Smyth, 1992)

For subsequent citations in the same report: Moore et al.(1992) also proposed... or It

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has also been proposed that. . . . (Moore et al., 1992)

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The reference section:

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The report ends with reference section, which comes immediately after the

Recommendations and begins on a new page. It is titled as 'References' in upper

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and lower case letters centered across the page.

Published Journal Articles

Beckerian, D.A. (1993). In search of the typical eyewitness. American

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Psychologist, 48, 574-576.

Gubbay, S.S., Ellis, W., Walton, J.N., and Court, S.D.M. (1965). Clumsy

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children: A study of apraxic and agnosic defects in 21 children. Brain, 88, 295-

312.

Authored Books

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Cone, J.D., and Foster, S.L. (1993). Dissertations and theses from start to finish:

Psychology and related fields. Washington, DC: American Psychological

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Association.

Cone, J.D., and Foster, S.L. (1993). Dissertations and theses from start to finish:

Psychology and related fields (2nd ed.). Washington, DC: American

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Psychological Association.

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APPENDICES:

The purpose of the appendices is to supplement the main body of your text and

provide additional information that may be of interest to the reader.

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There is no major heading for the Appendices. You simply need to

include each one, starting on a new page, numbered, using capital letters, and

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headed with a centered brief descriptive title. For example:

Appendix A: List of stimulus words presented to the participants

Dos and Don'ts of Report Writing

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1. Choose a font size that is not too small or too large; 11 or 12 is a good font

size to use.

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2. Acknowledgment need not be a separate page, except in the final report. In fact,

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you could just drop it altogether for the first- and second-stage reports. Your

guide already knows how much you appreciate his/her support. Express your

gratitude by working harder instead of writing a flowery acknowledgment.

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3. Make sure your paragraphs have some indentation and that it is not too large.

Refer to some text books or journal papers if you are not sure.

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4. If figures, equations, or trends are taken from some reference, the reference must

be cited right there, even if you have cited it earlier.

5. The correct way of referring to a figure is Fig. 4 or Fig. 1.2 (note that there is a

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space after Fig.). The same applies to Section, Equation, etc. (e.g., Sec. 2, Eq.

3.1).

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6. Cite a reference as, for example, "The threshold voltage is a strong function of

the implant dose [1]." Note that there must be a space before the bracket.

7. Follow some standard format while writing references. For example, you could

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look up any IEEE transactions issue and check out the format for journal papers,

books, conference papers, etc.

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8. Do not type references (for that matter, any titles or captions) entirely in capital

letters. The only capital letters required are (i) the first letter of a name, (ii)

acronyms, (iii) the first letter of the title of an article (iv) the first letter of a

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sentence.

9. The order of references is very important. In the list of your references, the first

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reference must be the one which is cited before any other reference, and so on.

Also, every reference in the list must be cited at least once (this also applies to

figures). In handling references and figure numbers, Latex turns out to be far

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better than Word.

12. Many commercial packages allow "screen dump" of figures. While this is useful

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in preparing reports, it is often very wasteful (in terms of toner or ink) since the

background is black. Please see if you can invert the image or use a plotting

program with the raw data such that the background is white.

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13. The following tips may be useful: (a) For Windows, open the file in

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Paint and select Image/Invert Colors. (b) For Linux, open the file in Image

Magick (this can be done by typing display) and then selecting Enhance/Negate.

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14. As far as possible, place each figure close to the part of the text where it is

referred to.

15. A list of figures is not required except for the final project report. It generally

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does not do more than wasting paper.

16. The figures, when viewed together with the caption, must be, as far as possible,

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self-explanatory. There are times when one must say, "see text for details".

However, this is an exception and not a rule.

17. The purpose of a figure caption is simply to state what is being presented in the

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figure. It is not the right place for making comments or comparisons; that should

appear only in the text.

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18. If you are showing comparison of two (or more) quantities, use the same

notation through out the report. For example, suppose you want to compare

measured data with analytical model in four different figures, in each figure,

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make sure that the measured data is represented by the same line type or symbol.

The same should be followed for the analytical model. This makes it easier for

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the reader to focus on the important aspects of the report rather than getting lost

in lines and symbols.

19. If you must resize a plot or a figure, make sure that you do it simultaneously in

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both x and y directions. Otherwise, circles in the original figure will appear as

ellipses, letters will appear too fat or too narrow, and other similar calamities

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will occur.

20. In the beginning of any chapter, you need to add a brief introduction and then

start sections. The same is true about sections and subsections. If you have

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sections that are too small, it only means that there is not enough material to

make a separate section. In that case, do not make a separate section. Include the

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same material in the main section or elsewhere.

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Remember, a short report is perfectly acceptable if you have put in the effort and

covered all important aspects of your work. Adding unnecessary sections and

subsections will create the impression that you are only covering up the lack of

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effort.

22. Do not make one-line paragraphs.

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23. Always add a space after a full stop, comma, colon, etc. Also, leave a space

before opening a bracket. If the sentence ends with a closing bracket, add the

full stop (or comma or semicolon, etc) after the bracket.

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24. Do not add a space before a full stop, comma, colon, etc.

25. Using a hyphen can be tricky. If two (or more) words form a single adjective, a

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hyphen is required; otherwise, it should not be used. For example, (a) A short-

channel device shows a finite output conductance. (b) This is a good example of

mixed-signal simulation. (c)Several devices with short channels were studied.

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26. If you are using Latex, do not use the quotation marks to open. If you do that,

you get "this". Use the single opening quotes (twice) to get "this".

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27. Do not use very informal language. Instead of "This theory should be taken with

a pinch of salt," you might say, "This theory is not convincing," or "It needs

more work to show that this theory applies in all cases."

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28. Do not use "&"; write "and" instead. Do not write "There're" for "There are" etc.

29. If you are describing several items of the same type (e.g., short-channel effects

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in a MOS transistor), use the "list" option; it enhances the clarity of your report.

30. Do not use "bullets" in your report. They are acceptable in a presentation, but

not in a formal report. You may use numerals or letters instead.

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31. Whenever in doubt, look up a text book or a journal paper to verify whether

your grammar and punctuation are correct.

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32. Do a spell check before you print out your document. It always helps.

33. Always write the report so that the reader can easily make out what your

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contribution is. Do not leave the reader guessing in this respect.

34. Above all, be clear. Your report must have a flow, i.e., the reader must be able to

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appreciate continuity in the report. After the first reading, the reader should be

able to understand (a) the overall theme and (b) what is new (if it is a project

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report).

35. Plagiarism is a very serious offense. You simply cannot copy material from an

existing report or paper and put it verbatim in your report. The idea of writing a

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report is to convey in your words what you have understood from the literature.

The above list may seem a little intimidating. However, if you make a

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sincere effort, most of the points are easy to remember and practice. A

supplementary exercise that will help you immensely is that of looking for all

major and minor details when you read an article from a newspaper or a

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magazine, such as grammar, punctuation, organization of the material, etc.

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PRESENTATION OF A REPORT

In this section, we will look into the issues associated with presentation of a

Research Report by the Researcher or principal investigator. While preparing

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for the presentation of a report, the researchers should focus on the following

issues:

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What is the purpose of the report and issues on which the Presentation

has to focus?

2.

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Who are the stakeholders and what are their areas of interest?

3.

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The mode and media of presentation.

4.

Extent of Coverage and depth to address at.

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Time, Place and cost associated with presentation.

6.

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Audio ? Visual aids intended to be used.

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Document Outline

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This post was last modified on 14 March 2022

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