9780130272942

This user-friendly introduction to the mathematics of probability and statistics (for readers with a background in calculus) uses numerous applications--drawn from biology, education, economics, engineering, environmental studies, exercise science, health science, manufacturing, opinion polls, psychology, sociology, and sports--to help explain and motivate the concepts. A review of selected mathematical techniques is included, and an accompanying CD-ROM contains many of the figures (many animated), and the data included in the examples and exercises (stored in both Minitab compatible format and ASCII). Empirical and Probability Distributions. Probability. Discrete Distributions. Continuous Distributions. Multivariable Distributions. Sampling Distribution Theory. Importance of Understanding Variability. Estimation. Tests of Statistical Hypotheses. Theory of Statistical Inference. Quality Improvement Through Statistical Methods. For anyone interested in the Mathematics of Probability and Statistics.

Preface

vii

Prologue

xi

Empirical and Probability Distributions

1

(65)

Basic Concepts

1

(9)

The Mean, Variance, and Standard Deviation

10

(6)

Continuous-Type Data

16

(14)

Exploratory Data Analysis

30

(11)

Graphical Comparisons of Data Sets

41

(8)

Time Sequences

49

(9)

Probability Density and Mass Functions

58

(8)

Probability

66

(42)

Properties of Probability

66

(9)

Methods of Enumeration

75

(11)

Conditional Probability

86

(10)

Independent Events

96

(7)

Bayes' Theorem

103

(5)

Discrete Distributions

108

(57)

Random Variables of the Discrete Type

108

(10)

Mathematical Expectation

118

(13)

Bernoulli Trials and the Binomial Distribution

131

(11)

The Moment-Generating Function

142

(11)

The Poisson Distribution

153

(12)

Continuous Distributions

165

(57)

Random Variables of the Continuous Type

165

(11)

The Uniform and Exponential Distributions

176

(10)

The Gamma and Chi-Square Distributions

186

(7)

The Normal Distribution

193

(12)

Distributions of Functions of a Random Variable

205

(11)

Mixed Distributions and Censoring

216

(6)

Multivariable Distributions

222

(61)

Distributions of Two Random Variables

222

(13)

The Correlation Coefficient

235

(9)

Conditional Distributions

244

(10)

The Bivariate Normal Distribution

254

(7)

Transformations of Random Variables

261

(12)

Order Statistics

273

(10)

Sampling Distribution Theory

283

(58)

Independent Random Variables

283

(7)

Distributions of Sums of Independent Random Variables

290

(9)

Random Functions Associated with Normal Distributions

299

(8)

The Central Limit Theorem

307

(8)

Approximations for Discrete Distributions

315

(5)

The t and F Distributions

320

(5)

Limiting Moment-Generating Functions

325

(6)

Chebyshev's Inequality and Convergence in Probability

331

(10)

Centerpiece: Importance of Understanding Variability

336

(5)

Estimation

341

(85)

Point Estimation

341

(13)

Confidence Intervals for Means

354

(10)

Confidence Intervals for Difference of Two Means

364

(9)

Confidence Intervals for Variances

373

(6)

Confidence Intervals for Proportions

379

(7)

Sample Size

386

(8)

Distribution-Free Confidence Intervals for Percentiles

394

(8)

A Simple Regression Problem

402

(14)

More Regression

416

(10)

Tests of Statistical Hypotheses

426

(130)

Tests About Proportions

426

(10)

Tests About One Mean and One Variance

436

(17)

Tests of the Equality of Two Normal Distributions

453

(14)

Chi-Square Goodness of Fit Tests

467

(11)

Contingency Tables

478

(13)

Tests of the Equality of Several Means

491

(11)

Two-Factor Analysis of Variance

502

(13)

Tests Concerning Regression and Correlation

515

(7)

The Wilcoxon Tests

522

(12)

Kolmogorov-Smirnov Goodness of Fit Test

534

(7)

Resampling Methods

541

(6)

Run Test and Test for Randomness

547

(9)

Theory of Statistical Inference

556

(39)

Sufficient Statistics

556

(5)

Power of a Statistical Test

561

(9)

Best Critical Regions

570

(7)

Likelihood Ratio Tests

577

(6)

Bayesian Estimation

583

(6)

Asymptotic Distributions of Maximum Likelihood Estimators

589

(6)

Quality Improvement Through Statistical Methods

595

(28)

Statistical Quality Control

595

(13)

General Factorial and 2k Factorial Designs

608

(7)

More on Design of Experiments

615

(8)

Epilogue

621

(2)

A REVIEW OF SELECTED MATHEMATICAL TECHNIQUES

623

(20)

A.1 Algebra of Sets

623

(4)

A.2 Mathematical Tools for the Hypergeometric Distribution

627

(3)

A.3 Limits

630

(2)

A.4 Infinite Series

632

(4)

A.5 Integration

636

(2)

A.6 Multivariate Calculus

638

(5)

B REFERENCES

643

(2)

C TABLES

645

(28)

D ANSWERS TO ODD-NUMBERED EXERCISES

673

(26)

Index

699

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Preface We are pleased with the reception that was given to the first five editions of Probability and Statistical Inference.The sixth edition is still designed for use in a course having from three to six semester hours of credit. No previous study of statistics is assumed, and a standard course in calculus provides an adequate mathematical background. Certain sections have been starred and are not needed in subsequent sections. This, however, does not mean that these starred sections are unimportant, and we hope many of you will study them. We still view this book as the basis of a junior or senior level course in the mathematics of probability and statistics that is taught by many departments of mathematics and/or statistics. We have tried to make it more "user friendly"; yet we do want to reinforce certain basic concepts of mathematics, particularly calculus. To help the student with methods of algebra of sets and calculus, we include a Review of Selected Mathematical Techniquesin Appendix A. This review includes a method that makes integration by parts easier. Also we derive the important Rule of 72that provides an approximation to the number of years necessary for money to double. MAJOR CHANGES IN THIS EDITION Chapter 1 still provides an excellent introduction to good descriptive statistics and exploratory data analysis and the corresponding empirical distributions. However, probability models are also introduced in Chapter 1 so that the student recognizes from the beginning that the characteristics of the empirical distributions are estimates of those of probability distributions. Hopefully, this creates some interest among students in checking to see if a probability model is appropriate for the situation under consideration throughout the text. Chapters 2-4 provide concepts in probability and basic distributions. These have been simplified somewhat from the previous edition by introducing a few of the easiest distributions through examples. Also the probability generating function has been dropped in this edition, although we note that the moment-generating function can serve in that capacity. Of course, the latter can also be used to compute the moments of a distribution. By request of many statisticians, we have introduced multivariate distributions much earlier, but in such a way that only the first section of that chapter is needed for the chapters that follow on techniques used in statistical inference. Hence, if an instructor so desires, he or she can start statistical methods sooner without conditional distributions. While this book is written primarily as a mathematical introduction to probability and statistics, there are a great many examples and exercises concerned with applications. For illustrations, the reader will find applications in the areas of biology, education, economics, engineering, environmental studies, exercise science, health science, manufacturing, opinion polls, psychology, sociology, and sports. That is, there are many exercises in the text, some illustrating the mathematics of probability and statistics but a great number are concerned with applications. We are certainly more concerned with model checking in this edition than in the previous editions. In addition, there is major effort to emphasize confidence intervals more than previously, and we clearly spell out the relationship between confidence intervals and tests of hypotheses. In that regard, we have increased the emphasis on one-sided confidence intervals somewhat because often a practitioner wants a lower or upper bound for the parameter in question, and these have a natural relationship with one-sided tests of hypotheses. The chapter on confidence intervals is so organized that the instructor can introduce early basic concepts of regression and distribution-free techniques, if he or she chooses to do so. That is

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Probability and Statistical Inference

Probability and Statistical Inference

0130272949

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