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K.S. KUSHWAHA M.Sc., P.S.C.C., Ph.D. (Statistics) Recipient of Dr. Radha Krishnan Award (1992) Associate Professor (Statistics) Department of Mathematics & Statistics J.N.K.V.V., Jabalpur, M.P. (India)

RAJESH KUMAR M.Sc., P.S.C.C., Ph. D. (Statistics) Principal Scientist (Statistics) Indian Institute of Sugarcane Research Lucknow, U.P. (India)

The book entitled “The Theory of Samples Surveys and Statistical Decisions” is useful to all the P.G. and Ph.D. students and faculty members of statistics, agricultural statistics and engineering, social; science and biological sciences. It is also useful to those students who have to appear in competitive examinations with statistic as a subject in the state P.S.C’s, U.P.S.C., A.S.R.B and I.S.S etc. this book is the outcome of 25 years of teaching experience to U.G., P.G. and Ph.D. students

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Preface   The book entitled “The Theory of Sample Surveys and Statistical Decisions” has been written for all the P.G. & Ph.D. students of “Social Science, Biological Science, Medical Science, Agrl. Science, Agrl. Scientists and other users who have low mathematical background. This book is the outcome of 24 years of teaching experience of U.G., P.G., & Ph.D. students of different disciplines of Agr., Agrl. Engg. and Agrl. Statistics in J.N.K.V.V., Jabalpur. The content of the book covers the syllabus of the subject “The theory of sample surveys and tests of significance” opted by different Agrl. Universities in India. Further more, the book will also be useful to the professionals associated with “Social and Agrl. Science”. The book contains 13 chapters, out of which chapters (1-7) deal with the “The Theory of Sample Surveys” and chapters (8-13) deal with the “Estimation of Unknown Parameters and Tests of Significance”. For them each chapter has been supported with a good varieties of “Numerical Illustrations” to offer more & better the clarity of the subject matter to its readers and users. During the preparation of the manuscript of this book, the author has incorporated the fruitful academic suggestions provided by Dr. B.L. Mishra, the Ex. Prof & Head, Department of Agrl. Economics and Farm Management J.N.K.V.V., and presently, Prof & Head, Department of System Science and Statistics, R.D. University, Jabalpur and Dr. Housala P. Singh, Prof. & Head, School of Studies in Statistics, Vikram University, Ujjain to enrich the book. It is expected to have a good popularity due to its usefulness amongst its readers and users. The author is very much thankful to Dr. Rajesh Kumar, the senior scientist (statistics), of “Indian Sugarcane Research Institute, Lucknow” to share the burden facing in giving this shape of a textbook in the capacity of its co-author. The authors are very much thankful and extends their honourable honor of high order to the Vice Chancellor (JNKVV, Jabalpur) and Director (IISR, Lucknow) for their kind permission to hand over the manuscript to “New India Publishing Agency” New Delhi to publish it in the shape of a book. The author extends his elderly affectionate and caring attitude to a very young man, Mr. Manoj Kr. Maurya for taking keen interest in providing physical facilities to get this manuscript back from Yojana Bhavan, U.P. Units of Govt. of India and also to a very young and distinguished Dr. Sehar of K.G.M.U. Lucknow for providing moral support to the author for getting rid of his bad health condition during (Nov. 2006-Feb. 2007). The author acknowledges very strongly his wife Smt.Veena Kushwaha, daughter Miss Akansha Kushwaha, son Mr. Utkarsh Kushwaha and sister-in-law, Mrs. Seema Maurya who cared the author a lot during the preparation of this manuscript and his bad health. Last but not the least; the author is also very much thankful to the entire staff of New India Publishing Agency, New Delhi for taking keen interest in publishing the manuscript in its present shape. Although, the authors have tried their best in the formulation of the subject matter very nicely but that can be further improved based on the suggestions received from its readers. Therefore, the constructive and academic suggestions received from all sides will always be welcomed and highly appreciated by the authors.

1 Preliminaries on Sample Survey Theory

The use of sampling in making decisions about an aggregate (or population) is possibly as old as civilization itself. Sampling is first broadly classified into two categories known as subjective and objective. Any type of sampling which depends upon the personal judgement or discretion of the sampler himself, is called subjective sampling. But the sampling method which is governed by a sampling rule or is independent of the sampler’s own judgement is known as objective sampling. The main difficulty with subjective sampling is that the sampler is ignorant of the degree of representativeness of his sample or the accuracy of the final estimates of the population values obtained. Objective sampling is further subdivided as non-probabilistic, probabilistic and mixed. In non-probabilistic objective sampling, there is a fixed sampling rule but there is no selection probability attached to the sampling unit e.g. selection of every 10th individual from a list, starting with the first, or selecting every 10th line in a paddy field. If however, the selection of the first individual is made in such a manner that each of the first 10 gets an equal chance of being selected, it becomes a case of mixed sampling, partly probabilistic and partly non-probabilistic). On the other hand, if for each individual unit there is a definite pre-assigned probability of being selected, the sampling is said to be probabilistic sampling.

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2 Methods of Simple Random Sampling

2.1 Simple Random Sampling The simplest and common most method of sampling is simple random sampling. In this procedure, the sample is drawn unit by unit with equal probability of selection for each unit at each draw. It is sometimes referred to as unrestricted random sampling.

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3 Stratified Random Sampling

3.1 Introduction Out of all methods of sampling, the most commonly used procedure in surveys is the stratified random sampling. When population units are of heterogeneous nature and if we select a sample using SRSWOR scheme then the sample selected may not be the representative sample and the estimate worked out from that sample will not be the reliable estimate of the population parameter. The estimate of sampling variance of the sample estimator obtained from that sampling scheme may happen to be very high leading to get a less precise estimate of the parameter. In such type of population, to overcome these draw backs, we use another sampling scheme known as stratified random sampling which is defined as follows.

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4 Ratio, Product and Regression Methods of Estimation

4.1 Introduction In many surveys, information on an auxiliary (supplementary, ancillary or apriori) variable x, which is highly (positively or negatively)correlated with the variable y under study, is readily available and can be used for improving the sampling design. Stratified sampling and probability proportional to size (p.p.s) schemes are two examples of improved sampling designs, in which the use of data on auxiliary variable has been made. However in both the schemes, it implies that the information on auxiliary variable on individual sampling units is available prior to presentation of sampling design. In case, data on auxiliary variable for individual sampling units are not available but only the aggregate value for all units of auxiliary variable is available, the two schemes can not be used. In such a situation, the aggregate data on auxiliary variable can still be used at the time of estimation of parameters under consideration, provided the data on auxiliary variable for the sampled units can be easily obtained at the time of recording the values of the study variable. Such types of estimation are based on ratio, product, difference and regression methods of estimation.

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5 Cluster Sampling

5.1 Introduction In random sampling it is assumed in prior that the population is divisible into a finite number of distinct and identifiable units and these units are known as sampling units. The smallest unit into which the population can be divide, is called an elementary unit (element) of the population. A group of such units is known as cluster. When such cluster is treated as sampling unit, then the sampling procedure adopted is known as cluster sampling. If the entire area containing the population under study is divided into smaller area segments and each element in the population belongs to one and only one such area segment, the sampling is sometimes known as area sampling. Generally, identification and location of an element requires considerable time. However, once an element is located, the time taken for surveying a few neighbouring elements is small. Thus, the main function in cluster sampling is to specify clusters or to divide the population into appropriate number of clusters. Clusters are generally made up of neighbouring elements and hence elements within a cluster happen to be of similar nature. As a matter of fact, the number of elements in a cluster should be small and the number of clusters should be large. The required number of clusters can be selected either by equal or unequal probabilities of selection. All the units in the selected clusters are enumerated completely i.e. complete enumeration is made in all the selected clusters.

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6 Systematic Random Sampling

6.1 Introduction So far we have considered the methods of sampling in which the successive units (elements or clusters) were selected with the help of random numbers. Now, we shall consider a new method of sample selection in which only the first unit of the sample is selected with the help of random number, the rest being selected automatically according to a predetermined pattern. The method is known as systematic sampling. It is operationally more convenient than simple random sampling and which at the same time ensures for each unit, equal probability of inclusion in the sample. The systematic sampling can also be termed as systematic random sampling because of the first unit of the sample being chosen randomly. The pattern usually followed in selecting a systematic sample is a simple pattern involving regular spacing of units. Thus, suppose a population consists of N units and is serially numbered from 1 to N. Suppose further that N is expressable as the product of two integers k and n so that N=nk. Draw a random number i such that 1 < i < k and select the unit with the corresponding serial number and every kth unit in the population there after i.e. the sample consists the units bearing the serial numbers as

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7 Multistage Sampling

7.1 Introduction In chapter 5, the cluster sampling has been discussed in which clusters were considered as sampling units and all the elements in the selected clusters were enumerated completely. It has been stated there that the cluster sampling is economical under certain circumstance but the method restricts the spread of sample over the population which results generally in increased sampling variance of the estimator under consideration. It is, therefore, logical to expect that the efficiency of the estimator will be increased by distributing the elements over a large numbers of clusters. Hence instead of observing all the units in the selected clusters, one may like to observe only few elements of the selected clusters. This will involve the sampling work to be done also within the selected clusters. If we do so, the process of selection will involve sampling work to be done at two stage, one for selecting the clusters where sampling units are the clusters and the second for selecting the units (elements) within the selected clusters where sampling units are the elements. The process of sampling can be done at more than two stages. In this case the first stage unit will be larger than the second stage units and the second stage units will be larger than the third stage units and so on. Due to process of sampling in stages, such sampling is called “Multistage sampling.”

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8 Statistical Decision (Point and Internal Estimation Theory)

8.1 Introduction We have already discussed in details how different sampling schemes can be employed to draw the representative samples suitable to many kinds of population whose sampling units differ in nature. From practical view points, however, it is often more important to be able to make decision about certain unknown characteristics of the population on the basis of informations available in the representative samples which are drawn from the parent populations. Such decisions taken, are known as the “Statistical decisions”. For example, we may wish to decide on the basis of sample data whether (i) One educational system is better than the other or not, (ii) A given six faced die is unbiased (balanced) or not, (iii) A new insecticide is really effective to control the attack of insects or not? etc. Such problems are dealt with in the theory of statistical decisions which uses the principles of sampling theory.

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9 Test of Hypothesis and its Significance (Preliminaries)

9.1 Introduction In the preceeding chapter, we have studied the estimation part of statistical decision (inference) infered about the parameters of the population on the basis of information available in the sample observations drawn from the parent population under study. In the present chapter we discuss the problems related with the tests of statistical hypothesis and its significance about the population parameters under study. For understanding of the subject matter considered in this chapter, very clearly and easily, one must be familier with the statistical terminologies used in the subject matter. Hence we discuss the various notable terminologies with some suitable examples in a simple way for better understanding.

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10 Normal Distribution and Tests Based on it (Large Sample Test or Normal Test or Z Test)

10.1 Normal Distribution The normal distribution is the most widely used distribution in statistics. Almost all biological data including the agricultural data eg. crop yield, plant height, seed weight, seed size, protein percent in pulse, oil content in an oil seed etc are assumed to be normally distributed. In practice their distribution is rarely verified. In distribution theory, for large values of n, the number of trial (or sample size), almost all the distributions e.g. Binomial, poission, Negative binomial, chi square, student’s t, and Snedenor’s F distributions etc. are very closely approximated by normal distribution. It is a continuous distribution with mean and variance as its two parameters.

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11 Exact Sampling Distributions and Related Small Sample Tests (F, t)

11.1 Introduction In the preceding chapter 10, we have discussed in detail, the test procedures for which it has been assumed that the sample size n is large, (i.e n >,30). Those tests are also known as approximate tests or z tests. But in case when sample size n is small i.e. n<30, the approximate test fails to deal with such problem and we apply the theory of exact sample, for small sample tests. The exact sample tests however can be applied to large samples also though the converse is not plausible. In all the exact sample tests, the basic assumption is that the parent population is normally distributed. In this part we will discusse in details the exact sample tests (based on exact sampling distributions namely snedekor’s F, student’s t and chi square (χ2) distributions. These distributions will be simply introduced in short because main emphasis has to be given only on the test procedures based on them.

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12 Chi-Square Distribution and Its Applications (Or Chi-Square Statistic)

12.1 Chi-Square Variate The square of a standard normal variate is known as a chi – square variate or chi square statistic with one degree of freedom.

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13 Miscellaneous Tests of Significance

13.1 Introduction In the previous chapters, we have considered different tests of significance based on normal distribution, F distribution, student’s t distribution and Chi-square (χ2) distribution which are frequently applied by the users of different disciplines of applied sciences and social sciences. In this chapter some more tests of significance are considered which have not been covered in those chapters. These tests are also based on t, F, Fisher’s z transformation, χ2 and some other test statistics are also discussed one by one in the following sections.

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14 End Pages

Bibliography 1.Agarwal, B.L. (1999): Basic Statistics, 2nd edition, Wiley Eastern Ltd, New Delhi. 2.Alexander M. Mood, Franklin A. Gryabill and Duane C. Boes: Introduction to the theory of statistics, McGraw -Hill 3.Bowley, A.L. (1926): Measurement of precision attained in sampling. Bull. Inter. Statist. Instt. 22, 1-62. 4.Cochran, W.G. (1946): Relative accuracy of systematic and stratified random sampling for a certain class of population. Ann. Math. Statist, 17, 164-177. 5.Cochram, W.G. (1977): Sampling Techniques, 3rd edition, Wiley Eastern, Ltd, New Delhi 6.Drapper, N.R. and Smith, H (1966): Applied Regression Analysis Wiley series in probability and mathematical statistics, 2nd edition, 615-16. 7.Fisher, R.A. and F. Yates. (1963): Statistical table for biological, agricultural and medical research. 8.Ganguli, M. (1941): A note on nested sampling. Sankhya, 5, 449-452. 9.Gupta, S.C. and Kapoor, V.K. (1983): Fundamentals of mathematical statistics, Sultan Chand and Sons, New Delhi. 10.Hanson. M.H., Hurwitz, W.N. and Madow, W.G. (1953): Sample survey methods and theory. John Wiley and Sons, New York.

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