Probability and statistics for engineers and sciencetists

Probability & Statistics for

Engineers & Scientists

NINTH EDITION

GLOBAL EDITION

Ronald E. Walpole

Roanoke College

Raymond H. Myers

Virginia Tech

Sharon L. Myers

Radford University

Keying Ye

University of Texas at San Antonio

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Authorized adaptation from the United States edition, entitled Probability & Statistics for Engineers & Scientists,9t h

Edition MyStatLab Update, ISBN 978-0-13-411585-6, by Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, and

Keying Ye published by Pearson Education c 2017.

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British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library

10 9 8 7 6 5 4 3 2 1

ISBN 10: 1292161361

ISBN 13: 9781292161365

Printed and bound in Italy by LEGO

Typeset by Aptara

This book is dedicated to

Billy and Julie

R.H.M. and S.L.M.

Limin, Carolyn and Emily

K.Y.

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Contents

Preface .......................................................... 13

1 Introduction to Statistics and Data Analysis ........... 21

1.1 Overview: Statistical Inference, Samples, Populations, and the

Role of Probability... ........................................... 21

1.2 Sampling Procedures; Collection of Data . ....................... 27

1.3 Measures of Location: The Sample Mean and Median . .......... 31

Exercises ................................................... 33

1.4 Measures of Variability. ......................................... 34

Exercises ................................................... 37

1.5 Discrete and Continuous Data................................... 37

1.6 Statistical Modeling, Scientific Inspection, and Graphical Diag￾nostics .......................................................... 38

1.7 General Types of Statistical Studies: Designed Experiment,

Observational Study, and Retrospective Study .................. 47

Exercises ................................................... 50

2 Probability .................................................. 55

2.1 Sample Space ................................................... 55

2.2 Events .......................................................... 58

Exercises ................................................... 62

2.3 Counting Sample Points......................................... 64

Exercises ................................................... 71

2.4 Probability of an Event ......................................... 72

2.5 Additive Rules ... ............................................... 76

Exercises ................................................... 79

2.6 Conditional Probability, Independence, and the Product Rule ... 82

Exercises ................................................... 89

2.7 Bayes’ Rule ..................................................... 92

Exercises ................................................... 96

Review Exercises............................................ 97

6 Contents

2.8 Potential Misconceptions and Hazards; Relationship to Material

in Other Chapters............................................... 99

3 Random Variables and Probability Distributions ...... 101

3.1 Concept of a Random Variable .. ................................ 101

3.2 Discrete Probability Distributions . .............................. 104

3.3 Continuous Probability Distributions. ........................... 107

Exercises ................................................... 111

3.4 Joint Probability Distributions ... ............................... 114

Exercises ................................................... 124

Review Exercises............................................ 127

3.5 Potential Misconceptions and Hazards; Relationship to Material

in Other Chapters............................................... 129

4 Mathematical Expectation ................................ 131

4.1 Mean of a Random Variable..................................... 131

Exercises ................................................... 137

4.2 Variance and Covariance of Random Variables................... 139

Exercises ................................................... 147

4.3 Means and Variances of Linear Combinations of Random Variables 148

4.4 Chebyshev’s Theorem ........................................... 155

Exercises ................................................... 157

Review Exercises............................................ 159

4.5 Potential Misconceptions and Hazards; Relationship to Material

in Other Chapters............................................... 162

5 Some Discrete Probability Distributions ................ 163

5.1 Introduction and Motivation .................................... 163

5.2 Binomial and Multinomial Distributions. ........................ 163

Exercises ................................................... 170

5.3 Hypergeometric Distribution .. .................................. 172

Exercises ................................................... 177

5.4 Negative Binomial and Geometric Distributions . ................ 178

5.5 Poisson Distribution and the Poisson Process.................... 181

Exercises ................................................... 184

Review Exercises............................................ 186

5.6 Potential Misconceptions and Hazards; Relationship to Material

in Other Chapters............................................... 189

Contents 7

6 Some Continuous Probability Distributions............. 191

6.1 Continuous Uniform Distribution. ............................... 191

6.2 Normal Distribution ............................................ 192

6.3 Areas under the Normal Curve .................................. 196

6.4 Applications of the Normal Distribution. ........................ 202

Exercises ................................................... 205

6.5 Normal Approximation to the Binomial . ........................ 207

Exercises ................................................... 213

6.6 Gamma and Exponential Distributions . ......................... 214

6.7 Chi-Squared Distribution. ....................................... 220

6.8 Beta Distribution .... ........................................... 221

6.9 Lognormal Distribution . ........................................ 221

6.10 Weibull Distribution (Optional) .... ............................. 223

Exercises ................................................... 226

Review Exercises............................................ 227

6.11 Potential Misconceptions and Hazards; Relationship to Material

in Other Chapters............................................... 229

7 Functions of Random Variables (Optional).............. 231

7.1 Introduction .................................................... 231

7.2 Transformations of Variables .... ................................ 231

7.3 Moments and Moment-Generating Functions . ................... 238

Exercises ................................................... 242

8 Fundamental Sampling Distributions and

Data Descriptions ........................................ 245

8.1 Random Sampling .............................................. 245

8.2 Some Important Statistics. ...................................... 247

Exercises ................................................... 250

8.3 Sampling Distributions. ......................................... 252

8.4 Sampling Distribution of Means and the Central Limit Theorem. 253

Exercises ................................................... 261

8.5 Sampling Distribution of S2 ..................................... 263

8.6 t-Distribution . .................................................. 266

8.7 F-Distribution .. ................................................ 271

8.8 Quantile and Probability Plots . ................................. 274

Exercises ................................................... 279

Review Exercises............................................ 280

8.9 Potential Misconceptions and Hazards; Relationship to Material

in Other Chapters............................................... 282

8 Contents

9 One- and Two-Sample Estimation Problems ............ 285

9.1 Introduction .................................................... 285

9.2 Statistical Inference .. ........................................... 285

9.3 Classical Methods of Estimation. ................................ 286

9.4 Single Sample: Estimating the Mean . ........................... 289

9.5 Standard Error of a Point Estimate ............................. 296

9.6 Prediction Intervals . ............................................ 297

9.7 Tolerance Limits . ............................................... 300

Exercises ................................................... 302

9.8 Two Samples: Estimating the Difference between Two Means ... 305

9.9 Paired Observations. ............................................ 311

Exercises ................................................... 314

9.10 Single Sample: Estimating a Proportion. ........................ 316

9.11 Two Samples: Estimating the Difference between Two Proportions 320

Exercises ................................................... 322

9.12 Single Sample: Estimating the Variance . ........................ 323

9.13 Two Samples: Estimating the Ratio of Two Variances . .......... 325

Exercises ................................................... 327

9.14 Maximum Likelihood Estimation (Optional). .................... 327

Exercises ................................................... 332

Review Exercises............................................ 333

9.15 Potential Misconceptions and Hazards; Relationship to Material

in Other Chapters............................................... 336

10 One- and Two-Sample Tests of Hypotheses ............. 339

10.1 Statistical Hypotheses: General Concepts ....................... 339

10.2 Testing a Statistical Hypothesis . ................................ 341

10.3 The Use of P-Values for Decision Making in Testing Hypotheses. 351

Exercises ................................................... 354

10.4 Single Sample: Tests Concerning a Single Mean ................. 356

10.5 Two Samples: Tests on Two Means . ............................ 362

10.6 Choice of Sample Size for Testing Means .... .................... 369

10.7 Graphical Methods for Comparing Means ....................... 374

Exercises ................................................... 376

10.8 One Sample: Test on a Single Proportion. ....................... 380

10.9 Two Samples: Tests on Two Proportions ... ..................... 383

Exercises ................................................... 385

10.10 One- and Two-Sample Tests Concerning Variances .............. 386

Exercises ................................................... 389

10.11 Goodness-of-Fit Test. ........................................... 390

10.12 Test for Independence (Categorical Data) ....................... 393

Contents 9

10.13 Test for Homogeneity ........................................... 396

10.14 Two-Sample Case Study ........................................ 399

Exercises ................................................... 402

Review Exercises............................................ 404

10.15 Potential Misconceptions and Hazards; Relationship to Material

in Other Chapters............................................... 406

11 Simple Linear Regression and Correlation .............. 409

11.1 Introduction to Linear Regression .. ............................. 409

11.2 The Simple Linear Regression Model . ........................... 410

11.3 Least Squares and the Fitted Model.... ......................... 414

Exercises ................................................... 418

11.4 Properties of the Least Squares Estimators . ..................... 420

11.5 Inferences Concerning the Regression Coefficients... ............. 423

11.6 Prediction ...................................................... 428

Exercises ................................................... 431

11.7 Choice of a Regression Model ................................... 434

11.8 Analysis-of-Variance Approach . ................................. 434

11.9 Test for Linearity of Regression: Data with Repeated Observations 436

Exercises ................................................... 441

11.10 Data Plots and Transformations. ................................ 444

11.11 Simple Linear Regression Case Study. ........................... 448

11.12 Correlation . .................................................... 450

Exercises ................................................... 455

Review Exercises............................................ 456

11.13 Potential Misconceptions and Hazards; Relationship to Material

in Other Chapters............................................... 462

12 Multiple Linear Regression and Certain

Nonlinear Regression Models ........................... 463

12.1 Introduction .................................................... 463

12.2 Estimating the Coefficients. ..................................... 464

12.3 Linear Regression Model Using Matrices ........................ 467

Exercises ................................................... 470

12.4 Properties of the Least Squares Estimators . ..................... 473

12.5 Inferences in Multiple Linear Regression......................... 475

Exercises ................................................... 481

12.6 Choice of a Fitted Model through Hypothesis Testing ........... 482

12.7 Special Case of Orthogonality (Optional). ....................... 487

Exercises ................................................... 491

12.8 Categorical or Indicator Variables .... ........................... 492

10 Contents

Exercises ................................................... 496

12.9 Sequential Methods for Model Selection . ........................ 496

12.10 Study of Residuals and Violation of Assumptions (Model Check￾ing)............................................................. 502

12.11 Cross Validation, Cp, and Other Criteria for Model Selection.... 507

Exercises ................................................... 514

12.12 Special Nonlinear Models for Nonideal Conditions .... ........... 516

Exercises ................................................... 520

Review Exercises............................................ 521

12.13 Potential Misconceptions and Hazards; Relationship to Material

in Other Chapters............................................... 526

13 One-Factor Experiments: General........................ 527

13.1 Analysis-of-Variance Technique. ................................. 527

13.2 The Strategy of Experimental Design............................ 528

13.3 One-Way Analysis of Variance: Completely Randomized Design

(One-Way ANOVA). ............................................ 529

13.4 Tests for the Equality of Several Variances ...................... 536

Exercises ................................................... 538

13.5 Single-Degree-of-Freedom Comparisons. ......................... 540

13.6 Multiple Comparisons. .......................................... 543

Exercises ................................................... 549

13.7 Comparing a Set of Treatments in Blocks ....................... 552

13.8 Randomized Complete Block Designs. ........................... 553

13.9 Graphical Methods and Model Checking ........................ 560

13.10 Data Transformations in Analysis of Variance . .................. 563

Exercises ................................................... 565

13.11 Random Effects Models .... ..................................... 567

13.12 Case Study ..................................................... 571

Exercises ................................................... 573

Review Exercises............................................ 575

13.13 Potential Misconceptions and Hazards; Relationship to Material

in Other Chapters............................................... 579

14 Factorial Experiments (Two or More Factors) .......... 581

14.1 Introduction .................................................... 581

14.2 Interaction in the Two-Factor Experiment... .................... 582

14.3 Two-Factor Analysis of Variance .... ............................ 585

Exercises ................................................... 595

14.4 Three-Factor Experiments. ...................................... 599

Exercises ................................................... 606

Contents 11

14.5 Factorial Experiments for Random Effects and Mixed Models.... 608

Exercises ................................................... 612

Review Exercises............................................ 614

14.6 Potential Misconceptions and Hazards; Relationship to Material

in Other Chapters............................................... 616

15 2k Factorial Experiments and Fractions ................. 617

15.1 Introduction .................................................... 617

15.2 The 2k Factorial: Calculation of Effects and Analysis of Variance 618

15.3 Nonreplicated 2k Factorial Experiment .......................... 624

Exercises ................................................... 629

15.4 Factorial Experiments in a Regression Setting . .................. 632

15.5 The Orthogonal Design ......................................... 637

Exercises ................................................... 645

15.6 Fractional Factorial Experiments .... ............................ 646

15.7 Analysis of Fractional Factorial Experiments .................... 652

Exercises ................................................... 654

15.8 Higher Fractions and Screening Designs . ........................ 656

15.9 Construction of Resolution III and IV Designs with 8, 16, and 32

Design Points ................................................... 657

15.10 Other Two-Level Resolution III Designs; The Plackett-Burman

Designs ......................................................... 658

15.11 Introduction to Response Surface Methodology. ................. 659

15.12 Robust Parameter Design . ...................................... 663

Exercises ................................................... 672

Review Exercises............................................ 673

15.13 Potential Misconceptions and Hazards; Relationship to Material

in Other Chapters............................................... 674

16 Nonparametric Statistics .................................. 675

16.1 Nonparametric Tests . ........................................... 675

16.2 Signed-Rank Test ............................................... 680

Exercises ................................................... 683

16.3 Wilcoxon Rank-Sum Test . ...................................... 685

16.4 Kruskal-Wallis Test ............................................. 688

Exercises ................................................... 690

16.5 Runs Test....................................................... 691

16.6 Tolerance Limits . ............................................... 694

16.7 Rank Correlation Coefficient .. .................................. 694

Exercises ................................................... 697

Review Exercises............................................ 699

12 Contents

17 Statistical Quality Control ................................ 701

17.1 Introduction .................................................... 701

17.2 Nature of the Control Limits . ................................... 703

17.3 Purposes of the Control Chart .................................. 703

17.4 Control Charts for Variables .................................... 704

17.5 Control Charts for Attributes ................................... 717

17.6 Cusum Control Charts .......................................... 725

Review Exercises............................................ 726

18 Bayesian Statistics ......................................... 729

18.1 Bayesian Concepts ... ........................................... 729

18.2 Bayesian Inferences ............................................. 730

18.3 Bayes Estimates Using Decision Theory Framework . ............ 737

Exercises ................................................... 738

Bibliography .................................................... 741

Appendix A: Statistical Tables and Proofs.................. 745

Appendix B: Answers to Odd-Numbered Non-Review

Exercises .................................................. 789

Index ........................................................... 805

Preface

General Approach and Mathematical Level

Our emphasis in creating this edition is less on adding new material and more on

providing clarity and deeper understanding. This objective was accomplished in

part by including new end-of-chapter material that adds connective tissue between

chapters. We affectionately call these comments at the end of the chapter “Pot

Holes.” They are very useful to remind students of the big picture and how each

chapter fits into that picture, and they aid the student in learning about limitations

and pitfalls that may result if procedures are misused. A deeper understanding

of real-world use of statistics is made available through class projects, which were

added in several chapters. These projects provide the opportunity for students

alone, or in groups, to gather their own experimental data and draw inferences. In

some cases, the work involves a problem whose solution will illustrate the meaning

of a concept or provide an empirical understanding of an important statistical

result. Some existing examples were expanded and new ones were introduced to

create “case studies,” in which commentary is provided to give the student a clear

understanding of a statistical concept in the context of a practical situation.

In this edition, we continue to emphasize a balance between theory and appli￾cations. Calculus and other types of mathematical support (e.g., linear algebra)

are used at about the same level as in previous editions. The coverage of an￾alytical tools in statistics is enhanced with the use of calculus when discussion

centers on rules and concepts in probability. Probability distributions and sta￾tistical inference are highlighted in Chapters 2 through 10. Linear algebra and

matrices are very lightly applied in Chapters 11 through 15, where linear regres￾sion and analysis of variance are covered. Students using this text should have

had the equivalent of one semester of differential and integral calculus. Linear

algebra is helpful but not necessary so long as the section in Chapter 12 on mul￾tiple linear regression using matrix algebra is not covered by the instructor. As

in previous editions, a large number of exercises that deal with real-life scientific

and engineering applications are available to challenge the student. The many

data sets associated with the exercises are available for download from the website

http://www.pearsonglobaleditions.com/Walpole or in MyStatLab.

Summary of Changes

• We’ve added MyStatLab, a course management systems that delivers proven

results in helping individual students succeed. MyStatLab provides engaging

experiences that personalize, stimulate, and measure learning for each student.

13

14 Preface

To learn more about how MyStatLab combines proven learning applications

with powerful assessment, visit www.mystatlab.com or contact your Pearson

representative.

• Class projects were added in several chapters to provide a deeper understand￾ing of the real-world use of statistics. Students are asked to produce or gather

their own experimental data and draw inferences from these data.

• More case studies were added and others expanded to help students under￾stand the statistical methods being presented in the context of a real-life

situation.

• “Pot Holes” were added at the end of some chapters and expanded in others.

These comments are intended to present each chapter in the context of the big

picture and discuss how the chapters relate to one another. They also provide

cautions about the possible misuse of statistical techniques MSL bullet.

• Chapter 1 has been enhanced to include more on single-number statistics as

well as graphical techniques. New fundamental material on sampling and

experimental design is presented.

• Examples added to Chapter 8 on sampling distributions are intended to moti￾vate P-values and hypothesis testing. This prepares the student for the more

challenging material on these topics that will be presented in Chapter 10.

• Chapter 12 contains additional development regarding the effect of a single re￾gression variable in a model in which collinearity with other variables is severe.

• Chapter 15 now introduces material on the important topic of response surface

methodology (RSM). The use of noise variables in RSM allows the illustration

of mean and variance (dual response surface) modeling.

• The central composite design (CCD) is introduced in Chapter 15.

• More examples are given in Chapter 18, and the discussion of using Bayesian

methods for statistical decision making has been enhanced.

Content and Course Planning

This text is designed for either a one- or a two-semester course. A reasonable plan

for a one-semester course might include Chapters 1 through 10. This would result

in a curriculum that concluded with the fundamentals of both estimation and hy￾pothesis testing. Instructors who desire that students be exposed to simple linear

regression may wish to include a portion of Chapter 11. For instructors who desire

to have analysis of variance included rather than regression, the one-semester course

may include Chapter 13 rather than Chapters 11 and 12. Chapter 13 features one￾factor analysis of variance. Another option is to eliminate portions of Chapters 5

and/or 6 as well as Chapter 7. With this option, one or more of the discrete or con￾tinuous distributions in Chapters 5 and 6 may be eliminated. These distributions

include the negative binomial, geometric, gamma, Weibull, beta, and log normal

distributions. Other features that one might consider removing from a one-semester

curriculum include maximum likelihood estimation, prediction, and/or tolerance

limits in Chapter 9. A one-semester curriculum has built-in flexibility, depending

on the relative interest of the instructor in regression, analysis of variance, ex￾perimental design, and response surface methods (Chapter 15). There are several