3 Joint Distributions 71

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1 2.2.3 The Normal Distribution The Beta Density Functions of a Random Variable Concluding Remarks Problems 64 3 Joint Distributions Introduction Discrete Random Variables Continuous Random Variables Independent Random Variables Conditional Distributions The Discrete Case The Continuous Case Functions of Jointly Distributed Random Variables Sums and Quotients The General Case Extrema and Order Statistics Problems Expected Values The Expected Value of a Random Variable Expectations of Functions of Random Variables Expectations of Linear Combinations of Random Variables Variance and Standard Deviation A Model for Measurement Error Covariance and Correlation Conditional Expectation and Prediction Definitions and Examples Prediction The Moment-Generating Function Approximate Methods Problems 166

2 vi 5 Limit Theorems Introduction The Law of Large Numbers Convergence in Distribution and the Central Limit Theorem Problems Distributions Derived from the Normal Distribution Introduction x 2, f, and F Distributions The Sample Mean and the Sample Variance Problems Survey Sampling Introduction Population Parameters Simple Random Sampling The Expectation and Variance of the Sample Mean Estimation of the Population Variance The Normal Approximation to the Sampling Distribution of X Estimation of a Ratio Stratified Random Sampling Introduction and Notation Properties of Stratified Estimates Methods of Allocation Concluding Remarks Problems Estimation of Parameters and Fitting of Probability Distributions Introduction Fitting the Poisson Distribution to Emissions of Alpha Particles Parameter Estimation The Method of Moments The Method of Maximum Likelihood 267

3 vii Maximum Likelihood Estimates of Multinomial Cell Probabilities Large Sample Theory for Maximum Likelihood Estimates Confidence Intervals from Maximum Likelihood Estimates The Bayesian Approach to Parameter Estimation Further Remarks on Priors Large Sample Normal Approximation to the Posterior Computational Aspects Efficiency and the Cramér-Rao Lower Bound An Example: The Negative Binomial Distribution Sufficiency A Factorization Theorem The Rao-Blackwell Theorem Concluding Remarks Problems Testing Hypotheses and Assessing Goodness of Fit Introduction The Neyman-Pearson Paradigm Specification of the Significance Level and the Concept of a p-value The Null Hypothesis Uniformly Most Powerful Tests The Duality of Confidence Intervals and Hypothesis Tests Generalized Likelihood Ratio Tests Likelihood Ratio Tests for the Multinomial Distribution The Poisson Dispersion Test Hanging Rootograms Probability Plots Tests for Normality Concluding Remarks Problems Summarizing Data Introduction Methods Based on the Cumulative Distribution Function 378

4 viii The Empirical Cumulative Distribution Function The Survival Function Quantile-Quantile Plots Histograms, Density Curves, and Stem-and-Leaf Plots Measures of Location The Arithmetic Mean The Median The Trimmed Mean M Estimates Comparison of Location Estimates Estimating Variability of Location Estimates by the Bootstrap Measures of Dispersion Boxplots Exploring Relationships with Scatterplots Concluding Remarks Problems Comparing Two Samples Introduction Comparing Two Independent Samples Methods Based on the Normal Distribution Power A Nonparametric Method The Mann-Whitney Test Bayesian Approach Comparing Paired Samples Methods Based on the Normal Distribution A Nonparametric Method The Signed Rank Test An Example Measuring Mercury Levels in Fish Experimental Design Mammary Artery Ligation The Placebo Effect The Lanarkshire Milk Experiment The Portacaval Shunt FD&CRedNo Further Remarks on Randomization 456

5 ix Observational Studies, Confounding, and Bias in Graduate Admissions Fishing Expeditions Concluding Remarks Problems The Analysis of Variance Introduction The One-Way Layout Normal Theory; the F Test The Problem of Multiple Comparisons A Nonparametric Method The Kruskal-Wallis Test The Two-Way Layout Additive Parametrization Normal Theory for the Two-Way Layout Randomized Block Designs A Nonparametric Method Friedman's Test Concluding Remarks Problems The Analysis of Categorical Data Introduction Fisher's Exact Test The Chi-Square Test of Homogeneity The Chi-Square Test of Independence Matched-Pairs Designs Odds Ratios Concluding Remarks Problems Linear Least Squares Introduction Simple Linear Regression Statistical Properties of the Estimated Slope and Intercept 547

6 Assessing the Fit Correlation and Regression The Matrix Approach to Linear Least Squares Statistical Properties of Least Squares Estimates Vector-Valued Random Variables Mean and Covariance of Least Squares Estimates Estimation of a Residuals and Standardized Residuals Inference about ß Multiple Linear Regression An Example Conditional Inference, Unconditional Inference, and the Bootstrap Local Linear Smoothing Concluding Remarks Problems 591 Appendix A Common Distributions Al Appendix B Tables A4 Bibliography A25 Answers to Selected Problems A32 Author Index A48 Applications Index A51 Subject Index A54

Contents. Acknowledgments. xix

Contents. Acknowledgments. xix Table of Preface Acknowledgments page xv xix 1 Introduction 1 The Role of the Computer in Data Analysis 1 Statistics: Descriptive and Inferential 2 Variables and Constants 3 The Measurement of Variables