K.S. KUSHWAHA, RAJESH KUMAR
K.S. KUSHWAHA
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
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
Preface
CHAPTER-1
Preliminaries on Sample Survey Theory
1.1 Introduction,
1.2 Use of Sampling Techniques and its Limitations,
1.3 Sample Surveys and Complete Enumeration,
1.3.1 Sample Survey,
1.3.2 Complete Enumeration (Census Survey),
1.4 Advantage of Sample Survey Over Complete Enumeration,
1.5 Principles of Sampling Theory,
1.5.1 Principle of Validity,
1.5.2 Principle of Statistical Regularity,
1.5.3 Principle of Optimisation,
1.6 Principal Stages in Planning and Execution of a Sample Survey Work,
1.6.1 Statement of Objectives,
1.6.2 Specification of Data Required,
1.6.3 Survey Reference and Reporting Periods,
1.6.4 Determination of Sampling Units and Preparation of Sampling Frame,
1.6.5 Choice of Sampling Design,
1.6.6 Method of Data Collection,
1.6.7 Organisation of Field Work and Training of Personal,
1.6.8 Processing of Survey Data,
1.6.9 Preparation of Reports,
1.6.10 Information Gained for Future Surveys,
1.7 Role of Sampling Theory,
1.8 Some Definitions with Preliminaries Based on Them,
1.8.1 Population (Universe),
1.8.2 Sampling Units,
1.8.3 Sampling Frame,
1.8.4 Population Parameter (Population constant),
1.8.5 Sample,
1.8.6 Sample Statistic,
1.8.7 Sample Estimator and Estimates,
1.8.8 Population Variance and Standard Deviation,
1.8.9 Population Mean Square and Sample Mean Square,
1.8.10 Sampling Variance of an Estimator,
1.8.11 Standard Error (S.E) of an Estimator,
1.9 Probability and Non-Probability Sampling,
1.9.1 Sampling Design,
1.9.2 Quota Sampling,
1.10 Mathematical Expectation,
1.10.1 Mathematical Expectation of an Estimator,
1.10.2 Unbiased and Biased Estimator,
1.10.3 Effect of Bias on Accuracy of an Estimate,
1.10.4 Mean Square Error (M.S.E) of an Estimate,
1.10.5 Consistent Estimator,
1.10.6 Accuracy and Precision of an Estimate,
1.11 Sampling and Non-Sampling Error,
1.11.1 Sampling Error,
1.11.2 Non-Sampling Error,
1.12 Questions and Problems for Exercise,
CHAPTER-2
Methods of Simple Random Sampling
2.1 Simple Random Sampling,
2.1.1 Definition,
2.1.2 Definition,
2.1.3 Simple Random Sampling without Replacement (SRSWOR),
2.1.4 Simple Random Sampling with Replacement (SRSWR),
2.1.5 Inclusion Probability,
2.2 Procedures of Selecting A Random Sample,
2.2.1 Lottery Method,
2.2.2 Random Number Table Method,
2.2.3 Reminder Method,
2.3 Notations and Definitions,
2.3.1 Estimations of Population Parameters,
2.3.2 Sampling Variance and Standard Error of Estimators,
2.4 Merits and Demerits of Simple Random Sampling,
2.5 Determination of Sample Size for A Specified Precision,
2.6 Confidence Limits and Confidence Coefficient,
2.7 Numerical Illustration,
2.8 Sampling for Proportion and Percentage of Attributes,
2.8.1 Introduction,
2.8.2 Notations and Terminology,
2.8.3 Sample Estimator and Its Variance,
2.9 Numerical Illustration,
2.10 Questions and Problems for Exercises,
CHAPTER-3
Stratified Random Sampling
3.1 Introduction,
3.2 Definition,
3.3 Principles of Stratification,
3.4 Advantages of Stratification,
3.5 Notations,
3.6 Estimation of Population Mean,
3.7 Properties and Variance of the Estimator,
3.8 Allocation of Sample Sizes in Different Strata,
3.8.1 Equal Allocation,
3.8.2 Proportional Allocation,
3.8.3 Neyman Allocation,
3.8.4 Optimum Allocation,
3.9 Sampling Variance image in Different Allocations,
3.10 Presentation of Heterogeneous Population,
3.11 Comparison of Stratified Random Sampling with Simple Random Sampling,
3.11.1 Proportional Allocation Verses S.R.S. Scheme,
3.11.2 Neyman Allocation Verses S.R.S. Scheme,
3.11.3 Neyman Allocation Verses Proportional Allocation,
3.11.4 Arbitrary Allocation Verses S.R.S. Scheme,
3.12 Practical Difficulties in Adopting Neyman Allocation,
3.13 Questions and Problems for Exercises,
CHAPTER-4
Ratio, Product and RegressionMethods of Estimation
4.1 Introduction,
4.2 Definition and Notations: Let us denote,
4.3 Ratio Method of Estimation,
4.4 Bias and Mean Square Error (M.S.E) of Ratio Estimators,
4.4.1 Bias of image
4.4.2 Condition for Bias to be Zero,
4.4.3 Upper Limits of Bias,
4.4.4 Variance (M.S.E) of image
4.4.5 Efficiency of the Estimators,
4.5 Unbiased Ratio-Type Estimator,
4.6 Product Method of Estimation,
4.6.1 Bias of image
4.6.2 Variance of image
4.6.3 Efficiency of Product Estimator image over Sample Mean image
4.7 Numerical Illustrations,
4.7.1 Questions and Problems for Exercises,
4.8 Regression Method of Estimation,
4.8.1 Difference Estimator,
4.8.2 Bias of Regression Estimator,
4.8.3 Sampling Variance of Regression Estimator,
4.8.4 Efficiency of Regression Estimator over Sample Estimator,
4.8.5 Efficiency of Regression Estimator over Ratio Estimator,
4.8.6 Choice of Estimators,
4.9 Numerical Illustrations,
4.10 Questions and Problems for Exercises,
CHAPTER-5
Cluster Sampling
5.1 Introduction,
5.2 Merits of Cluster Sampling,
5.3 Demerits of Cluster Sampling,
5.4 Selection of Clusters,
5.5 Notations (For Cluster Sizes Equal),
5.6 Numerical Illustration,
5.6.1 Relative Efficiency of Cluster Sampling,
5.7 Unequal Clusters,
5.7.1 Estimation of Population Mean,
5.7.2 Sampling Variance and their Estimation,
5.8 Numerical Illustration,
5.9 Questions and Problems for Exercises,
CHAPTER-6
Systematic Random Sampling
6.1 Introduction,
6.2 Situations Suitable for Applying Systematic Sampling,
6.3 Sample Selection Procedure,
6.3.1 Linear Systematic Sampling,
6.3.2 Circular Systematic Sampling,
6.4 Sample Mean and Its Sampling Variance,
6.5 Numerical Illustrations,
6.6 Populations of Different Nature,
6.6.1 Population in Random Order,
6.6.2 Population with Linear Trend,
6.6.3 Population with Periodic Variation,
6.6.4 Natural Population,
6.6.5 Auto Correlated Population,
6.7 Questions and Problems for Exercises,
CHAPTER-7
Multistage Sampling
7.1 Introduction,
7.2 Definition,
7.3 Advantages,
7.4 Two Stage Sampling with Equal First Stage Units,
7.4.1 Sampling Variance,
7.4.2 Estimation of Variance of Sample Mean image
7.5 Numerical Illustrations,
7.6 Two Stage Sampling with Unequal First and Second Stage Units,
7.7 Biased Estimator for Population Mean,
7.7.1 Sampling Variance and its Estimation,
7.7.2 Unbiased Estimator for Population Mean,
7.7.3 Sampling Variance and its Estimation,
7.8 Numerical Illustration,
7.9 Three Stage Sampling with Equal First and Second Stage Units,
7.10 Multiphase Sampling (or Double Sampling),
7.10.1 Definition,
7.10.2 Advantage
7.10.3 Difference Between Multiphase and Multistage Sample,
7.11 Questions and Problems for Exercises,
CHAPTER-8
Statistical Decision(Point and Internal Estimation Theory)
8.1 Introduction,
8.2 Definition (Statistical Decision),
8.3 Problem of Estimation,
8.4 Point Estimate,
8.5 Properties of an Estimator,
8.6 Unbiased Estimator,
8.7 Efficient Estimator,
8.8 Best Linear Unbiased Estimator (BLUE),
8.9 Assumption of Best Linear Unbiased Estimates,
8.10 Numerical Illustration on Unbiased and Efficient Estimators,
8.11 Interval Estimation,
8.12 Confidence Interval for Population Parameter in Case of N (µ,σ2),
8.13 Confidence Interval Estimates for Means,
8.14 Numerical Illustration on Confidence Interval Estimates for Population Means,
8.15 Confidence Intervals for Proportions,
8.16 Numerical Illustrations on Confidence Interval Estimates for Proportions,
8.17 Confidence Intervals for Differences and Sums of Populations Parameters,
8.18 Numerical Illustrations,
8.19 Confidence Interval for Standard Deviation,
8.20 Probable Error,
8.21 Numerical Illustrations,
8.22 Questions and Problems for Exercise,
CHAPTER-9
Test of Hypothesis and its Significance (Preliminaries)
9.1 Introduction,
9.2 Significant and Non-Significant Events,
9.3 Statistical Hypothesis,
9.4 Null Hypothesis,
9.5 Alternative or Complementary Hypothesis,
9.6 Problem of Testing of Hypothesis,
9.7 Tests of Hypothesis and their Significance,
9.8 Parametric and Non-Parametric Tests,
9.9 Simple and Composite Hypothesis,
9.10 Types of Decision Taken,
9.10.1 Types of Error,
9.10.2 Type One Error,
9.10.3 Type Two Error,
9.11 Level of Significance,
9.12 Critical (Rejection) and Acceptance Regions,
9.13 One Tailed and Two Tailed Tests,
9.14 Critical Values or Significant Values,
9.15 Questions and Problems for Exercises,
CHAPTER-10
Normal Distribution and Test Based on it(Large Sample Test or Normal Test or Z Test)
10.1 Normal Distribution,
10.2 Definition,
10.3 Standard Normal Variable,
10.4 Importance of Normal Distribution in Hypothesis Testing Problem,
10.5 Test of Significance for Large Samples,
10.6 An Important Assumption,
10.7 Test of Significance for Single Mean,
10.8 Test of Significance for Difference of Means (Two Means Test),
10.9 Test of Significance for Proportion,
10.10 Test for Single Proportion,
10.11 Test of Significance for Difference of Two Proportions,
10.12 Test of Significance for the Difference of Standard Deviations,
10.13 Questions and Problems for Exercises,
CHAPTER-11
Exact Sampling Distribution andRelated Small Sample Tests (F, t)
11.1 Introduction,
11.2 F Statistic and its Distribution,
11.2.1 Assumptions of F Statistic,
11.3 An Application of F Distribution,
11.3.1 Test of Equality of Two Population Variances,
11.4 Numerical Illustrations,
11.5 Student’s t Statistics and its Applications,
11.5.1 Fisher’s (Statistic),
11.5.2 Assumptions for Student’s t Test,
11.5.3 Test for Single Mean with Example,
11.5.4 Test for Significance of Two Means with Example,
11.5.5 Paired t Test with Example,
11.6 Questions and Problems for Exercises,
CHAPTER-12
Chi-square Distribution and its Application
12.1 Chi-Square Variate,
12.2 Chi-Square Distribution,
12.3 Additive Property of χ2 Variates,
12.4 Condition for the Validity of χ2 Test,
12.5 Characteristics of Critical Values of χ2 Statistic,
12.6 Application of Chi-Square Distribution,
12.7 Degree of Freedom (d.f.),
12.8 Chi-Square Test for Population Variance,
12.9 Chi-Square Test of Goodness of Fit,
12.10 A rxc Contingency Table (Independence of Two Attributes),
12.11 A 2x2 Contingency Table,
12.12 Yate’s Correction,
12.13 Coefficient of Contingency,
12.14 Correlation of Attributes,
12.15 Questions and Problems for Exercises,
CHAPTER-13
Miscellaneous Tests of Significance
13.1 Introduction,
13.2 t Test for Testing the Significance of an Observed Sample Correlation Coefficient ‘r’xy,
13.3 Test of Significance for an Observed Simple Regression Coefficient byx,
13.4 Test for Significance of an Observed Partial Correlation Coefficient R,
13.5 F Test for Significance of an Observed Multiple Correlation Coefficient,
13.6 Fisher’s z Transformation,
13.7 Test of Significance for an Observed Value of ‘r’ from A Hypothetical Value ρ,
13.8 Test of Significance for the Difference Between Two Independent Sample Correlation Coefficients,
13.9 χ2 Test of Homogeneity of Correlation Coefficients,
13.10 Bartlett’s Test for Homogeneity of Several Independent Estimates of the Same Population Variance,
13.11 Serial Correlation in Residuals (Error) and Its Test of Significance,
13.11.1 Durbin-Watson Test for Serial Correlation,
13.11.2 Precautions to be Taken While Using d statistic,
13.12 Questions and Problems for Exercises,
Bibliography
Appendix
Start Pages
Preliminaries on Sample Survey Theory
Methods of Simple Random Sampling
Stratified Random Sampling
Ratio, Product and Regression Methods of Estimation
Cluster Sampling
Systematic Random Sampling
Multistage Sampling
Statistical Decision (Point and Internal Estimation Theory)
Test of Hypothesis and its Significance (Preliminaries)
Normal Distribution and Tests Based on it (Large Sample Test or Normal Test or Z Test)
Exact Sampling Distributions and Related Small Sample Tests (F, t)
Chi-Square Distribution and Its Applications (Or Chi-Square Statistic)
Miscellaneous Tests of Significance
End Pages