Statistical Inference

MICHAEL J. PANIK, PhD, is Professor Emeritus in the Department of Economics at the University of Hartford. He has served as a consultant to the Connecticut Department of Motor Vehicles as well as a variety of healthcare organizations. Dr. Panik has published numerous journal articles in the areas of economics, mathematics, and applied econometrics.

“The book is addressed to courses on probability, mathematical statistics, and statistical inference at the upper-undergraduate and graduate levels. It also serves as a valuable reference for researchers and practitioners who would like to develop further insights into essential statistical tools.”  (Zentralblatt Math, 1 August 2013)“If an undergraduate student seeks a guide that will introduce the basic ideas of statistics, or a lecturer wants interesting life examples and a source of valid intuitions to improve his teaching skills, then this book is a great place to start. . . This book, with its explanations of basic intuitions, its many examples, the easy language, and a minimal requirement for mathematical training, is a good self-contained starting point to prepare one for the jump into those heavier works.”  (Computing Reviews, 30 September 2013)

• Preface xv1 The Nature of Statistics 11.1 Statistics Defined 11.2 The Population and the Sample 21.3 Selecting a Sample from a Population 31.4 Measurement Scales 41.5 Let us Add 6Exercises 72 Analyzing Quantitative Data 92.1 Imposing Order 92.2 Tabular and Graphical Techniques: Ungrouped Data 92.3 Tabular and Graphical Techniques: Grouped Data 11Exercises 16Appendix 2.A Histograms with Classes of Different Lengths 183 Descriptive Characteristics of Quantitative Data 223.1 The Search for Summary Characteristics 223.2 The Arithmetic Mean 233.3 The Median 263.4 The Mode 273.5 The Range 273.6 The Standard Deviation 283.7 Relative Variation 333.8 Skewness 343.9 Quantiles 363.10 Kurtosis 383.11 Detection of Outliers 393.12 So What Do We Do with All This Stuff? 41Exercises 47Appendix 3.A Descriptive Characteristics of Grouped Data 513.A.1 The Arithmetic Mean 523.A.2 The Median 533.A.3 The Mode 553.A.4 The Standard Deviation 573.A.5 Quantiles (Quartiles, Deciles, and Percentiles) 584 Essentials of Probability 614.1 Set Notation 614.2 Events within the Sample Space 634.3 Basic Probability Calculations 644.4 Joint, Marginal, and Conditional Probability 684.5 Sources of Probabilities 73Exercises 755 Discrete Probability Distributions and Their Properties 815.1 The Discrete Probability Distribution 815.2 The Mean, Variance, and Standard Deviation of a Discrete Random Variable 855.3 The Binomial Probability Distribution 895.3.1 Counting Issues 895.3.2 The Bernoulli Probability Distribution 915.3.3 The Binomial Probability Distribution 91Exercises 966 The Normal Distribution 1016.1 The Continuous Probability Distribution 1016.2 The Normal Distribution 1026.3 Probability as an Area Under the Normal Curve 1046.4 Percentiles of the Standard Normal Distribution and Percentiles of the Random Variable X 114Exercises 116Appendix 6.A The Normal Approximation to Binomial Probabilities 1207 Simple Random Sampling and the Sampling Distribution of the Mean 1227.1 Simple Random Sampling 1227.2 The Sampling Distribution of the Mean 1237.3 Comments on the Sampling Distribution of the Mean 1277.4 A Central Limit Theorem 130Exercises 132Appendix 7.A Using a Table of Random Numbers 133Appendix 7.B Assessing Normality via the Normal Probability Plot 136Appendix 7.C Randomness, Risk, and Uncertainty 1397.C.1 Introduction to Randomness 1397.C.2 Types of Randomness 1427.C.2.1 Type I Randomness 1427.C.2.2 Type II Randomness 1437.C.2.3 Type III Randomness 1437.C.3 Pseudo-Random Numbers 1447.C.4 Chaotic Behavior 1457.C.5 Risk and Uncertainty 1468 Confidence Interval Estimation of m 1528.1 The Error Bound on X as an Estimator of m 1528.2 A Confidence Interval for the Population Mean m (s Known) 1548.3 A Sample Size Requirements Formula 1598.4 A Confidence Interval for the Population Mean m (s Unknown) 160Exercises 165Appendix 8.A A Confidence Interval for the Population Median MED 1679 The Sampling Distribution of a Proportion and its Confidence Interval Estimation 1709.1 The Sampling Distribution of a Proportion 1709.2 The Error Bound on ^p as an Estimator for p 1739.3 A Confidence Interval for the Population Proportion (of Successes) p 1749.4 A Sample Size Requirements Formula 176Exercises 177Appendix 9.A Ratio Estimation 17910 Testing Statistical Hypotheses 18410.1 What is a Statistical Hypothesis? 18410.2 Errors in Testing 18510.3 The Contextual Framework of Hypothesis Testing 18610.3.1 Types of Errors in a Legal Context 18810.3.2 Types of Errors in a Medical Context 18810.3.3 Types of Errors in a Processing or Control Context 18910.3.4 Types of Errors in a Sports Context 18910.4 Selecting a Test Statistic 19010.5 The Classical Approach to Hypothesis Testing 19010.6 Types of Hypothesis Tests 19110.7 Hypothesis Tests for m (s Known) 19410.8 Hypothesis Tests for m (s Unknown and n Small) 19510.9 Reporting the Results of Statistical Hypothesis Tests 19810.10 Hypothesis Tests for the Population Proportion (of Successes) p 201Exercises 204Appendix 10.A Assessing the Randomness of a Sample 208Appendix 10.B Wilcoxon Signed Rank Test (of a Median) 210Appendix 10.C Lilliefors Goodness-of-Fit Test for Normality 21311 Comparing Two Population Means and Two Population Proportions 21711.1 Confidence Intervals for the Difference of Means when Sampling from Two Independent Normal Populations 21711.1.1 Sampling from Two Independent Normal Populations with Equal and Known Variances 21711.1.2 Sampling from Two Independent Normal Populations with Unequal but Known Variances 21811.1.3 Sampling from Two Independent Normal Populations with Equal but Unknown Variances 21811.1.4 Sampling from Two Independent Normal Populations with Unequal and Unknown Variances 21911.2 Confidence Intervals for the Difference of Means when Sampling from Two Dependent Populations: Paired Comparisons 22411.3 Confidence Intervals for the Difference of Proportions when Sampling from Two Independent Binomial Populations 22711.4 Statistical Hypothesis Tests for the Difference of Means when Sampling from Two Independent Normal Populations 22811.4.1 Population Variances Equal and Known 22911.4.2 Population Variances Unequal but Known 22911.4.3 Population Variances Equal and Unknown 22911.4.4 Population Variances Unequal and Unknown (an Approximate Test) 23011.5 Hypothesis Tests for the Difference of Means when Sampling from Two Dependent Populations: Paired Comparisons 23411.6 Hypothesis Tests for the Difference of Proportions when Sampling from Two Independent Binomial Populations 236Exercises 239Appendix 11.A Runs Test for Two Independent Samples 243Appendix 11.B Mann–Whitney (Rank Sum) Test for Two Independent Populations 245Appendix 11.C Wilcoxon Signed Rank Test when Sampling from Two Dependent Populations: Paired Comparisons 24912 Bivariate Regression and Correlation 25312.1 Introducing an Additional Dimension to our Statistical Analysis 25312.2 Linear Relationships 25412.2.1 Exact Linear Relationships 25412.3 Estimating the Slope and Intercept of the Population Regression Line 25712.4 Decomposition of the Sample Variation in Y 26212.5 Mean, Variance, and Sampling Distribution of the Least Squares Estimators ^b0 and ^b1 26412.6 Confidence Intervals for b0 and b1 26612.7 Testing Hypotheses about b0 and b1 26712.8 Predicting the Average Value of Y given X 26912.9 The Prediction of a Particular Value of Y given X 27012.10 Correlation Analysis 27212.10.1 Case A: X and Y Random Variables 27212.10.1.1 Estimating the Population Correlation Coefficient r 27412.10.1.2 Inferences about the Population Correlation Coefficient r 27512.10.2 Case B: X Values Fixed, Y a Random Variable 277Exercises 278Appendix 12.A Assessing Normality (Appendix 7.B Continued) 280Appendix 12.B On Making Causal Inferences 28112.B.1 Introduction 28112.B.2 Rudiments of Experimental Design 28212.B.3 Truth Sets, Propositions, and Logical Implications 28312.B.4 Necessary and Sufficient Conditions 28512.B.5 Causality Proper 28612.B.6 Logical Implications and Causality 28712.B.7 Correlation and Causality 28812.B.8 Causality from Counterfactuals 28912.B.9 Testing Causality 29212.B.10 Suggestions for Further Reading 29413 An Assortment of Additional Statistical Tests 29513.1 Distributional Hypotheses 29513.2 The Multinomial Chi-Square Statistic 29513.3 The Chi-Square Distribution 29813.4 Testing Goodness of Fit 29913.5 Testing Independence 30413.6 Testing k Proportions 30913.7 A Measure of Strength of Association in a Contingency Table 31113.8 A Confidence Interval for s2 under Random Sampling from a Normal Population 31213.9 The F Distribution 31413.10 Applications of the F Statistic to Regression Analysis 31613.10.1 Testing the Significance of the Regression Relationship Between X and Y 31613.10.2 A Joint Test of the Regression Intercept and Slope 317Exercises 318Appendix A 323Table A.1 Standard Normal Areas [Z is N(0,1)] 323Table A.2 Quantiles of the t Distribution (T is tv) 325Table A.3 Quantiles of the Chi-Square Distribution (X is w2v) 327Table A.4 Quantiles of the F Distribution (F is Fv1;v2 ) 329Table A.5 Binomial Probabilities P(X;n,p) 334Table A.6 Cumulative Binomial Probabilities 338Table A.7 Quantiles of Lilliefors’ Test for Normality 342Solutions to Exercises 343References 369Index 373