Elements of Survey Sampling

Kluwer Texts in the Mathematical Sciences VOLUME 15

A Graduate-Level Book Series

The titles published in this series are listed at the end 0/ this volume.

Elements of Survey Sampling by

Ravindra Singh and

Naurang Singh Mangat Punjab Agricultural University, Ludhiana, Punjab, India

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-4703-8 ISBN 978-94-017-1404-4 (eBook) DOI 10.1007/978-94-017-1404-4

Printed on acid-free paper

All Rights Reserved

e 1996 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1996 No part of the material protectec1 by this copyright notice may be reprodUCeci or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

TO THE MEMORY OF OUR PARENTS

Gayatri Devi and Narendra Singh

RS

Gurdial Kaur and Hernam Singh NSM

Table of Contents PREFACE

1.

COLLECTION OF SURVEY DATA Need for statistical data 1.2 Types of data 1.3 Methods of collecting primary data 1.4 Framing of questionnaire/schedule 1.5 Some technical terms 1.6 Need for a sample 1.7 Sampling procedures 1.8 With and without replacement sampling 1.9 Planning and execution of sample surveys Let us do

xiii

1.1

2.

11

ELEMENTARY CONCEPTS

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3.

2 3 4 5 6 7 9

Introduction Statistical preliminaries Estimator and its sampling distribution Unbiased estimator Measures of error Confidence intervals Sample size determination Sampling and nonsampling errors Let us do

14 14 16 19 21 23 26 27 28

SIMPLE RANDOM SAMPLING

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

What is simple random sampling? How to draw a simple random sample? Estimation of population mean/total Estimation of mean/total using distinct units Determining sample size for estimating population mean/total Estimation of population proportion Sample size for estimation of proportion Estimation of proportion using inverse sampling

30 30 33 48 50 53 55 56

viii

CONTENTS

3.9 Estimation over subpopulations 3.10 Some further remarks Let us do

58 62 63

4. SAMPLING WITH VARYING PROBABILITIES 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13

Introduction Methods of selecting a PPS sample Estimation in PPSWR sampling Relative efficiency of PPSWR estimator Determining sample size for estimating population mean/total Sampling with PPS without replacement Des Raj's ordered estimator Murthy's unordered estimator Horvitz-Thompson estimator Sen-Midzuno method Random group method Relative efficiency of RHC estimator Some further remarks Let us do

67 67 70 72

76 77

78 82 84 86 91 95 97 98

5. STRATIFIED SAMPLING 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

Introduction Notations Estimation of mean and total using simple random sampling Allocation of sample size Relative efficiency of stratified estimator Estimation of population proportion Construction of strata Poststratification Some further remarks Let us do

102 104 104 108 123 129 132 136 138 140

6. SYSTEMATIC SAMPLING 6.1 6.2 6.3 6.4 6.5 6.6

Linear systematic sampling Circular systematic sampling Estimating mean/total Estimating mean/total through interpenetrating subsamples Sample .size determination for estimating mean/total Estimation of proportion

145 148 149 153 156 158

CONTENTS

6.7

ix

Some further remarks Let us do

160 161

7. RATIO AND PRODUCT METHODS OF ESTIMATION 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 8.

8.3 8.4 8.5

8.6

175 178 184 185 189 191

Introduction Estimation of mean/total using difference estimator Estimation of mean/total using estimated regression coefficient Sample size determination for estimating mean/total Separate and combined regression estimators Some further remarks Let us do

197 197 201 204 206 216 217

TWO-PHASE SAMPLING

9.1 9.2 9.3 9.4 9.5 9.6 10.

165 166 169

REGRESSION METHOD OF ESTIMATION

8.1 8.2

9.

Need for ratio estimation Estimation of population ratio Ratio estimator for population mean/total Determining the sample size for estimation of ratio, mean, and total Separate and combined ratio estimators Some further remarks Product method for estimating mean/total Determination of sample size for product estimator Let us do

Need for two-phase sampling Two-phase sampling in ratio, product, and regression methods of estimation Sample size determination for ratio, product, and regression estimators Two-phase PPS sampling Sampling on two occasions Some further remarks Let us do

221 222 231 233 237 242 243

CLUSTER SAMPLING

10.1 Introduction 10.2 Notations

248 249

x

CONTENTS

10.3 10.4 10.5 10.6

Estimation of mean using simple random sampling Estimation of total using simple random sampling Relative efficiency of cluster sampling Determining the sample size for estimating mean/total 10.7 Estimation of proportion 10.8 Sample size required for estimation of proportion 10.9 Selection of clusters with unequal probabilities 10.10 Some further remarks Let us do

250 258 264 266 269 273 275 278 278

11. MULTISTAGE SAMPLING ILl Introduction 11.2 Notations 11.3 Estimation of mean/total in two-stage sampling using SRSWOR at both the stages 11.4 Estimation of proportion 11.5 Estimation of mean/total using PPSWR and SRSWOR 11.6 Some further remarks Let us do 12.

283 284 284 296 304 307 308

SAMPLING FROM MOBILE POPULATIONS

12.1 Introduction 12.2 Estimation of population size using direct sampling 12.3 Estimation of population size using inverse sampling 12.4 Determining the sample sizes 12.5 Some further remarks Let us do

314 315 320 324 328 329

13. NONRESPONSE ERRORS 13.1 13.2 13.3 13.4 13.5 13.6 13.7

Introduction Hansen and Hurwitz technique Bias reduction without call-backs Warner's randomized response model Mangat and Singh's two-stage model Unrelated question model Estimation of mean for quantitative characters Let us do

331 332 337 340 342 345 350 358

CONTENTS

xi

APPENDIXES A. B. C.

D. E.

Standard normal probability distribution Random numbers Number of tractors, tube wells, and net irrigated area (in hectares) for 69 villages of Doraha development block of Punjab, India Fifty WOR simple random samples Explanation of certain local terms used

364 365

369 371 372

REFERENCES

373

AUTHOR INDEX

383

SUBJEtTINDEX

385

Preface Modern statistics consists of methods which help in drawing inferences about the population under consideration. These populations may actually exist, or could be generated by repeated· experimentation. The medium of drawing inferences about the population is the sample, which is a subset of measurements selected from the population. Each measurement in the sample is used for making inferences about the population. The populations and also the methods of sample selection differ from one field of science to the other. Social scientists use surveys tocollectthe sample information, whereas the physical scientists employ the method of experimentation for obtaining this information. This is because in social sciences the factors that cause variation in the measurements on the study variable for the population units can not be controlled, whereas in physical sciences these factors can be controlled, at least to some extent, through proper experimental design. Several excellent books on sampling theory are available in the market. These books discuss the theory of sample surveys in great depth and detail, and are suited to the postgraduate students majoring in statistics. Research workers in the field of sampling methodology can also make use of these books. However, not many suitable books are available, which can be used by the students and researchers in the fields of economics, social sciences, extension education, agriculture, medical sciences, business management, etc. These students and workers usually conduct sample surveys during their research projects. Since they do not have statistics/ mathematics as their major subject, their level of understanding in these fields is not adequate enough. They, therefore, cannot make use of the sampling theory books available in the market, profitably. The present volume is meant to serve these students and research workers. Besides, the book will also serve as a guide for those professionals, who have to design and conduct surveys as a routine for private and state organizations. The information collected through such surveys may not be used directly for research purposes, but could be used in framing the policies for the welfare of the people. This book is oriented to discuss various situations that arise in the fields of social sciences, agriculture, business and natural resource management, etc. Suitable methods of sample selection and the estimation procedures for such cases have been considered. This helps in minimizing the survey cost for a given level of estimator's accuracy or in maximizing precision for the budget at disposal. The book uses only elementary algebraic symbols and will be easily understood by the students and research workers. Each chapter begins with an introduction of the topic, which gives an overview ofthe content and scope ofthe chapter besides its linkage with the other chapters. Formulas appropriate to different sampling strategies have been presented without any proofs. It is expected that the readers will try to understand the basic concepts,

xiv

PREFACE

and will not be overwhelmed by the formulas. Important definitions and algebraic expressions have been put in boxes so as to enable the reader to have a quick review of the material presented. The pervasive feature that characterizes the present volume is a good number of solved examples. It will help the reader in better understanding of the various steps involved in the calculations of estimator values and their sampling errors from the sample data. Sufficient exercises presenting the applications of methods under discussion are included at the end of each chapter. The reader can use them for further practice. This practice will be of great help in fixing and clarifying different calculation steps involved in any particular case. The data presented in some of the illustrations and exercises are from actual surveys, while in other cases these may not be real but relate to and reflect real life situations. In certain examples and exercises, the data are given for all the population units. This has been done with a view to clarify certain concepts involved. There could be readers, who may be interested in having an idea about some other developments that have taken place on the topics considered in the chapter. They may find the section "Some Further Remarks" useful. The references cited, after the appendixes in the book, are intended to arouse readers' curiosity for further learning, and help them in acquiring more detailed accounts of the topics considered. The material in the book has been arranged into 13 chapters. The first and second chapters are intended to provide the base for subsequent chapters. Chapter 1 discusses briefly the need for sample surveys, and various steps involved in their planning and execution. Important elementary concepts and definitions used in sample surveys are presented in chapter 2. This will help in better understanding of the bulk of the material presented in subsequent chapters. The basic sampling schemes such as simple random sampling, unequal probability sampling, stratified sampling, and systematic sampling are covered in chapters 3 to 6. Chapters 7 and 8 deal with ratio, product, and regression estimators. Other sampling schemes (multiphase, cluster, and multistage sampling) are discussed in chapters 9 through 11. Chapter 12 is devoted to the estimation of size of mobile populations, where a little deviant approach is followed to examine the units. In the last chapter, in addition to the techniques dealing with nonsampling errors, survey methods used for stigmatized and/or sensitive characters are considered. The book in the end also gives author index, subject index, and certain statistical tables. The explanation to certain Indian terms used in the text is given in appendix E. This will facilitate the readers to understand the material presented.' The help we have obtained from the available statistical literature is obvious. The excellent works of W.G. Cochran, M.N. Murthy, P.V. Sukhatme, B.V. Sukhatme, Ms. Shashikala Sukhatme, C. Asok, Des Raj, M.H. Hansen, W.N. Hurwitz, W.G. Madow, F. Yates, S.S. Zarkovich, R.L. Scheaffer, W. Mendenhall, L. Ott, W.E. Deming, L. Kish, I.M. Chakravarti, R.G. Laha, J. Roy, G.A.F. Seber, etc. have been useful in formulating the pattern of development that is followed in the book. All these are gratefully acknowledged. We have also derived some material from the research papers which are listed after the appendixes. Attempts have been made to refer to the original source, but still it is possible that in some instances we might not have recorded proper credit to the authors. We offer them our regrets and apologies.

PREFACE

xv

The American Fisheries Society, Washington, D.C. has our warmest thanks for permission to reprint table 12.1 from their joumal Transactions of the American Fisheries Society. The permission granted by Indian Statistical Institute, Calcutta, for reproducing the standard normal probability distribution table and the table of random numbers from the book Formulae and Tablesfor Statistical Work, is also gratefully acknowledged. We record our sincere appreciation to Professor S.K. Mehta for his most perceptive and valuable comments on all the chapters of the book. The authors also express their thanks to S.E.H. Rizvi, M. Bhargava, Inderjit S. Grewal, and M. Javed, doctoral students of the Department of Mathematics and Statistics, for verifying the numerical calculations in the examples and for helping the authors in going through the proofs of the book. The help rendered by Ms. Santosh and Ms. Loveleen in preparing the manuscript is gratefully acknowledged. Thanks are also due to Professor Pritpal Singh Phul, Head, Department of Plant Breeding, for constant encouragement for accomplishment of the project. Many friends, too numerous to mention by name, had made suggestions that are reflected in the final draft of the book. The authors express their thanks to P.L. Printers, Ludhiana, for preparing an excellent camera-ready version of the manuscript. Ms. Mamta, working with P.L. Printers, has worked hard in typesetting the material in correct and attractive form. Her sincerity of effort is highly appreciated. We shall be failing in our duty if we do not appreciate the personal sacrifices of our family members: Urrnila Singh, Shivendra, Ina, Shailendra, and Shalini; Paramjit Kaur, Parvinder, Gurvinder, and Jaswinder. All these thanks are, however, only a fraction of what is due to the Almighty who granted us an opportunity and mettle to make successful accomplishment of our project. In spite of the care taken, some errors might have escaped our notice. We shall appreciate if such errors are brought to our attention. Suggestions for improving the present volume will be gratefully received. Ravindra Singh Naurang Singh Mangat