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# Advanced Statistics from an Elementary Point of View

**Michael Panik**

ISBN **0120884941**

Pages **625**

Description

The highly readable text captures the flavor of a course in mathematical statistics without imposing too much rigor; students can concentrate on the statistical strategies without getting lost in the theory.

Students who use this book will be well on their way to thinking like a statistician. Practicing statisticians will find this book useful in that it is replete with statistical test procedures (both parametric and non-parametric) as well as numerous detailed examples.

Contents

1 Introduction 1.1 Statistics Defined 1.2 Types of Statistics 1.3 Levels of Discourse: Sample vs. Population 1.4 Levels of Discourse: Target vs. Sampled Population 1.5 Measurement Scales 1.6 Sampling and Sampling Errors 1.7 Exercises 2 Elementary Descriptive Statistical Techniques 2.1 Summarizing Sets of Data Measured on a Ratio or Interval Scale 2.2 Tabular Methods 2.3 Quantitative Summary Characteristics 2.3.1 Measures of Central Location 2.3.2 Measures of Dispersion 2.3.3 Standardized Variables 2.3.4 Moments 2.3.5 Skewness and Kurtosis 2.3.6 Relative Variation 2.3.7 Comparison of the Mean, Median, and Mode 2.3.8 The Sample Variance and Standard Deviation 2.4 Correlation between Variables X and Y 2.5 Rank Correlation between Variables X and Y 2.6 Exercises 3 Probability Theory 3.1 Mathematical Foundations: Sets, Set Relations, and Functions 3.2 The Random Experiment, Events, Sample Space, and the Random Variable 3.3 Axiomatic Development of Probability Theory 3.4 The Occurrence and Probability of an Event 3.5 General Addition Rule for Probabilities 3.6 Joint, Marginal, and Conditional Probability 3.7 Classification of Events 3.8 Sources of Probabilities 3.9 Bayes???Rule 3.10 Exercises 4 Random Variables and Probability Distributions 4.1 Random Variables 4.2 Discrete Probability Distributions 4.3 Continuous Probability Distributions 4.4 Mean and Variance of a Random Variable 4.5 Chebyshev’s Theorem for Random Variables 4.6 Moments of a Random Variable 4.7 Quantiles of a Probability Distribution 4.8 Moment-Generating Function 4.9 Probability-Generating Function 4.10 Exercises 5 Bivariate Probability Distributions 5.1 Bivariate Random Variables 5.2 Discrete Bivariate Probability Distributions 5.3 Continuous Bivariate Probability Distributions 5.4 Expectations and Moments of Bivariate Probability Distributions 5.5 Chebyshev’s Theorem for Bivariate Probability Distributions 5.6 Joint Moment–Generating Function 5.7 Exercises 6 Discrete Parametric Probability Distributions 6.1 Introduction 6.2 Counting Rules 6.3 Discrete Uniform Distribution 6.4 The Bernoulli Distribution 6.5 The Binomial Distribution 6.6 The Multinomial Distribution 6.7 The Geometric Distribution 6.8 The Negative Binomial Distribution 6.9 The Poisson Distribution 6.10 The Hypergeometric Distribution 6.11 The Generalized Hypergeometric Distribution 6.12 Exercises 7 Continuous Parametric Probability Distributions 7.1 Introduction 7.2 The Uniform Distribution 7.3 The Normal Distribution 7.3.1 Introduction to Normality 7.3.2 The Z Transformation 7.3.3 Moments, Quantiles, and Percentage Points 7.3.4 The Normal Curve of Error 7.4 The Normal Approximation to Binomial Probabilities 7.5 The Normal Approximation to Poisson Probabilities 7.6 The Exponential Distribution 7.6.1 Source of the Exponential Distribution 7.6.2 Features/Uses of the Exponential Distribution 7.7 Gamma and Beta Functions 7.8 The Gamma Distribution 7.9 The Beta Distribution 7.10 Other Useful Continuous Distributions 7.10.1 The Lognormal Distribution 7.10.2 The Logistic Distribution 7.11 Exercises 8 Sampling and the Sampling Distribution of a Statistic 8.1 The Purpose of Random Sampling 8.2 Sampling Scenarios 8.2.1 Data Generating Process or Infinite Population 8.2.2 Drawings from a Finite Population 8.3 The Arithmetic of Random Sampling 8.4 The Sampling Distribution of a Statistic 8.5 The Sampling Distribution of the Mean 8.5.1 Sampling from an Infinite Population 8.5.2 Sampling from a Finite Population 8.6 A Weak Law of Large Numbers 8.7 Convergence Concepts 8.8 A Central Limit Theorem 8.9 The Sampling Distribution of a Proportion 8.10 The Sampling Distribution of the Variance 8.11 A Note on Sample Moments 8.12 Exercises 9 The Chi-Square, Student’s t, and Snedecor’s F Distributions 9.1 Derived Continuous Parametric Distributions 9.2 The Chi-Square Distribution 9.3 The Sampling Distribution of the Variance When sampling from a Normal Population 9.4 Student’s t Distribution 9.5 Snedecor’s F Distribution 9.6 Exercises 10 Point Estimation and Properties of Point Estimators 10.1 Statistics as Point Estimators 10.2 Desirable Properties of Estimators as Statistical Properties 10.3 Small Sample Properties of Point Estimators 10.3.1 Unbiased, Minimum Variance, and Minimum MSE Estimators 10.3.2 Efficient Estimators 10.3.3 Most Efficient Estimators 10.3.4 Sufficient Statistics 10.3.5 Minimal Sufficient Statistics 10.3.6 On the Use of Sufficient Statistics 10.3.7 Completeness 10.3.8 Best Linear Unbiased Estimators 10.3.9 Jointly Sufficient Statistics 10.4 Large Sample Properties of Point Estimators 10.4.1 Asymptotic or Limiting Properties 10.4.2 Asymptotic Mean and Variance 10.4.3 Consistency 10.4.4 Asymptotic Efficiency 10.4.5 Asymptotic Normality 10.5 Techniques for Finding Good Point Estimators 10.5.1 Method of Maximum Likelihood 10.5.2 Method of Least Squares 10.6 Exercises 11 Interval Estimation and Confidence Interval Estimates 11.1 Interval Estimators 11.2 Central Confidence Intervals 11.3 The Pivotal Quantity Method 11.4 A Confidence Interval for µ Under Random Sampling from a Normal Population with Known Variance 11.5 A Confidence Interval for µ Under Random Sampling from a Normal Population with Unknown Variance 11.6 A Confidence Interval for ??? Under Random Sampling from a Normal Population with Unknown Mean 11.7 A Confidence Interval for p Under Random Sampling from a Binomial Population 11.8 Joint Estimation of a Family of Population Parameters 11.9 Confidence Intervals for the Difference of Means When Sampling from Two Independent Normal Populations 11.9.1 Population Variances Known 11.9.2 Population Variances Unknown But Equal 11.9.3 Population Variances Unknown and Unequal 11.10 Confidence Intervals for the Difference of Means When Sampling from Two Dependent Populations: Paired Comparisons 11.11 Confidence Intervals for the Difference of Proportions When Sampling from Two Independent Binomial Populations 11.12 Confidence Interval for the Ratio of Two Variances When Sampling from Two Independent Normal Populations 11.13 Exercises 12 Tests of Parametric Statistical Hypotheses 12.1 Statistical Inference Revisited 12.2 Fundamental Concepts for Testing Statistical Hypotheses 12.3 What Is the Research Question? 12.4 Decision Outcomes 12.5 Devising a Test for a Statistical Hypothesis 12.6 The Classical Approach to Statistical Hypothesis Testing 12.7 Types of Tests or Critical Regions 12.8 The Essentials of Conducting a Hypothesis Test 12.9 Hypothesis Test for µ Under Random Sampling from a Normal Population with Known Variance 12.10 Reporting Hypothesis Test Results 12.11 Determining the Probability of a Type II Error ???12.12 Hypothesis Tests for µ Under Random Sampling from a Normal Population with Unknown Variance 12.13 Hypothesis Tests for p Under Random Sampling from a Binomial Population 12.14 Hypothesis Tests for ??? Under Random Sampling from a Normal Population 12.15 The Operating Characteristic and Power Functions of a Test 12.16 Determining the Best Test for a Statistical Hypothesis 12.17 Generalized Likelihood Ratio Tests 12.18 Hypothesis Tests for the Difference of Means When Sampling from Two Independent Normal Populations 12.18.1 Population Variances Equal and Known 12.18.2 Population Variances Equal But Known 12.18.3 Population Variances Equal But Unknown 12.18.4 Population Variances Unequal and Unknown 12.19 Hypothesis Tests for the Difference of Means When Sampling from Two Dependent Populations: Paired Comparisons 12.20 Hypothesis Tests for the Difference of Proportions When Sampling from Two Independent Binomial Populations 12.21 Hypothesis Tests for the Difference of Variances When Sampling from Two Independent Normal Populations 12.22 Hypothesis Tests for Spearman’s Rank Correlation Coefficient •S 12.23 Exercises 13 Nonparametric Statistical Techniques 13.1 Parametric vs. Nonparametric Methods 13.2 Tests for the Randomness of a Single Sample 13.3 Single-Sample Sign Test Under Random Sampling 13.4 Wilcoxon Signed Rank Test of a Median 13.5 Runs Test for Two Independent Samples 13.6 Mann-Whitney (Rank-Sum) Test for Two Independent Samples 13.7 The Sign Test When Sampling from Two Dependent Populations: Paired Comparisons 13.8 Wilcoxon Signed-Rank Test When Sampling from Two Dependent Populations: Paired Comparisons 13.9 Exercises 14 Testing Goodness of Fit 14.1 Distributional Hypotheses 14.2 The Multinomial Chi-Square Statistic: Complete Specification of H0 14.3 The Multinomial Chi-Square Statistic: Incomplete Specification of H0 14.4 The Kolmogorov-Smirnov Test for Goodness of Fit 14.5 The Lilliefors Goodness-of-Fit Test for Normality 14.6 The Shapiro-Wilk Goodness-of-Fit Test for Normality 14.7 The Kolmogorov-Smirnov Test for Goodness of Fit: Two Independent Samples 14.8 Assessing Normality via Sample Moments 14.9 Exercises 15 Testing Goodness of Fit: Contingency Tables 15.1 An Extension of the Multinomial Chi-Square Statistic 15.2 Testing Independence 15.3 Testing k Proportions 15.4 Testing for Homogeneity 15.5 Measuring Strength of Association in Contingency Tables 15.6 Testing Goodness of Fit with Nominal-Scale Data: Paired Samples 15.7 Exercises 16 Bivariate Linear Regression and Correlation 16.1 The Regression Model 16.2 The Strong Classical Linear Regression Model 16.3 Estimating the Slope and Intercept of the Population Regression Line 16.4 Mean, Variance, and Sampling Distribution of the Least-Squares Estimators ˆ ??? and ˆ ??? 16.5 Precision of the Least Squares Estimators ˆ ???, ˆ ???: Confidence Intervals 16.6 Testing Hypotheses Concerning ???, ??? 16.7 The Precision of the Entire Least Squares Regression Equation: A Confidence Band 16.8 The Prediction of a Particular Value of Y Given X 16.9 Decomposition of the Sample Variation of Y 16.10 The Correlation Model 16.11 Estimating the Population Correlation Coefficient ???16.12 Inferences about the Population Correlation Coefficient ???16.13 Exercises Appendix A Table A.1 Standard Normal Areas Table A.2 Cumulative Distribution Function Values for the Standard Normal Distribution Table A.3 Quantiles of Student’s t Distribution Table A.4 Quantiles of the Chi-Square Distribution Table A.5 Quantiles of Snedecor’s F Distribution Table A.6 Binomial Probabilities Table A.7 Cumulative Distribution Function Values for the Binomial Distribution Table A.8 Poisson Probabilities Table A.9 Fisher’s ˆ ???= r) to ???Transformation Table A.10 R Distribution for the Runs Test of Randomness Table A.11 W+ Distribution for the Wilcoxon Signed-Rank Test Table A.12 R1 Distribution for the Mann-Whitney Rank-Sum Test Table A.13 Quantiles of the Lilliefors Test Statistic ˆ Dn Table A.14 Quantiles of the Kolmogorov-Smirnov Test Statistic Dn Table A.15 Quantiles of the Kolmogorov-Smirnov Test Statistic Dn,m When n = m Table A.16 Quantiles of the Kolmogorov-Smirnov Test Statistic Dn,m When n = m Table A.17 Quantiles of the Shapiro-Wilk Test Statistic W Table A.18 Coefficients for the Shapiro-Wilk Test Table A.19 Durbin-Watson DW Statistic Table A.20 D Distribution of the Von Neumann Ratio of the Mean-Square Successive Difference to the Variance Solutions to Selected Exercises References and Suggested Reading Index