# Applied Statistics and Probability for Engineers

Applied Statistics and Probability for Engineers Third Edition (822 Pages)

By Douglas. C. Montgomery, George. C. Runger

Contents of Applied Statistics and Probability for Engineers

CHAPTER 1 The Role of

Statistics in Engineering 1

1-1 The Engineering Method and

Statistical Thinking 2

1-2 Collecting Engineering Data 5

1-2.1 Basic Principles 5

1-2.2 Retrospective Study 5

1-2.3 Observational Study 6

1-2.4 Designed Experiments 6

1-2.5 A Factorial Experiment for

the Connector Pull-Off Force

Problem (CD Only) 8

1-2.6 Observing Processes Over Time 8

1-3 Mechanistic and Empirical Models 11

1-4 Probability and Probability Models 14

CHAPTER 2 Probability 16

2-1 Sample Spaces and Events 17

2-1.1 Random Experiments 17

2-1.2 Sample Spaces 18

2-1.3 Events 22

2-1.4 Counting Techniques

(CD Only) 25

2-2 Interpretations of Probability 27

2-2.1 Introduction 27

2-2.2 Axioms of Probability 30

2-3 Addition Rules 33

2-4 Conditional Probability 37

2-5 Multiplication and Total Probability

Rules 42

2-5.1 Multiplication Rule 42

2-5.2 Total Probability Rule 43

2-6 Independence 46

2-7 Bayes’ Theorem 51

2-8 Random Variables 53

CHAPTER 3 Discrete Random

Variables and Probability

Distributions 59

3-1 Discrete Random Variables 60

3-2 Probability Distributions and

Probability Mass Functions 61

3-3 Cumulative Distribution

Functions 63

3-4 Mean and Variance of a Discrete

Random Variable 66

3-5 Discrete Uniform Distribution 70

3-6 Binomial Distribution 72

3-7 Geometric and Negative Binomial

Distributions 78

3-7.1 Geometric Distribution 78

3-7.2 Negative Binomial

Distribution 80

3-8 Hypergeometric Distribution 84

3-9 Poisson Distribution 89

CHAPTER 4 Continuous Random

Variables and Probability

Distributions 97

4-1 Continuous Random

Variables 98

4-2 Probability Distributions

and Probability Density

Functions 98

4-3 Cumulative Distribution

Functions 102

4-4 Mean and Variance of a

Continuous Random Variable 105

4-5 Continuous Uniform

Distribution 107

4-6 Normal Distribution 109

4-7 Normal Approximation to the

Binomial and Poisson

Distributions 118

4-8 Continuity Corrections to

Improve the Approximation

(CD Only) 122

4-9 Exponential Distribution 122

4-10 Erlang and Gamma

Distribution 128

4-10.1 Erlang Distribution 128

4-10.2 Gamma Distribution 130

4-11 Weibull Distribution 133

4-12 Lognormal Distribution 135

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CHAPTER 5 Joint Probability

Distributions 141

5-1 Two Discrete Random Variables 142

5-1.1 Joint Probability

Distributions 142

5-1.2 Marginal Probability

Distributions 144

5-1.3 Conditional Probability

Distributions 146

5-1.4 Independence 148

5-2 Multiple Discrete Random

Variables 151

5-2.1 Joint Probability

Distributions 151

5-2.2 Multinomial Probability

Distribution 154

5-3 Two Continuous Random

Variables 157

5-3.1 Joint Probability

Distributions 157

5-3.2 Marginal Probability

Distributions 159

5-3.3 Conditional Probability

Distributions 162

5-3.4 Independence 164

5-4 Multiple Continuous Random

Variables 167

5-5 Covariance and Correlation 171

5-6 Bivariate Normal Distribution 177

5-7 Linear Combinations of Random

Variables 180

5-8 Functions of Random Variables

(CD Only) 185

5-9 Moment Generating Functions

(CD Only) 185

5-10 Chebyshev’s Inequality

(CD Only) 185

CHAPTER 6 Random Sampling

and Data Description 189

6-1 Data Summary and Display 190

6-2 Random Sampling 195

6-3 Stem-and-Leaf Diagrams 197

6-4 Frequency Distributions and

Histograms 203

6-5 Box Plots 207

6-6 Time Sequence Plots 209

6-7 Probability Plots 212

6-8 More About Probability Plotting

(CD Only) 216

CHAPTER 7 Point Estimation of

Parameters 220

7-1 Introduction 221

7-2 General Concepts of Point

Estimation 222

7-2.1 Unbiased Estimators 222

7-2.2 Proof that S is a Biased Estimator

of (CD Only) 224

7-2.3 Variance of a Point Estimator 224

7-2.4 Standard Error: Reporting a Point

Estimator 225

7-2.5 Bootstrap Estimate of the Standard

Error (CD Only) 226

7-2.6 Mean Square Error of an

Estimator 226

7-3 Methods of Point Estimation 229

7-3.1 Method of Moments 229

7-3.2 Method of Maximum

Likelihood 230

7-3.3 Bayesian Estimation of Parameters

(CD Only) 237

7-4 Sampling Distributions 238

7-5 Sampling Distribution of

Means 239

CHAPTER 8 Statistical Intervals

for a Single Sample 247

8-1 Introduction 248

8-2 Confidence Interval on the Mean of

a Normal Distribution, Variance

Known 249

8-2.1 Development of the Confidence

Interval and Its Basic

Properties 249

8-2.2 Choice of Sample Size 252

8-2.3 One-sided Confidence

Bounds 253

8-2.4 General method to Derive a

Confidence Interval 253

8-2.5 A Large-Sample Confidence

Interval for 254

8-2.6 Bootstrap Confidence Intervals

(CD Only) 256

8-3 Confidence Interval on the Mean of a

Normal Distribution, Variance

Unknown 257

8-3.1 The t Distribution 258

8-3.2 Development of the t Distribution

(CD Only) 259

8-3.3 The t Confidence Interval

on 259

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8-4 Confidence Interval on the Variance

and Standard Deviation of a Normal

Distribution 261

8-5 A Large-Sample Confidence Interval

for a Population Proportion 265

8-6 A Prediction Interval for a Future

Observation 268

8-7 Tolerance Intervals for a Normal

Distribution 270

CHAPTER 9 Tests of Hypotheses

for a Single Sample 277

9-1 Hypothesis Testing 278

9-1.1 Statistical Hypotheses 278

9-1.2 Tests of Statistical

Hypotheses 280

9-1.3 One-Sided and Two-Sided

Hypotheses 286

9-1.4 General Procedure for Hypothesis

Testing 287

9-2 Tests on the Mean of a

Normal Distribution, Variance

Known 289

9-2.1 Hypothesis Tests on the Mean 289

9-2.2 P-Values in Hypothesis

Tests 292

9-2.3 Connection Between Hypothesis

Tests and Confidence

Intervals 293

9-2.4 Type II Error and Choice of Sample

Size 293

9-2.5 Large Sample Test 297

9-2.6 Some Practical Comments on

Hypothesis Tests 298

9-3 Tests on the Mean of a Normal

Distribution, Variance

Unknown 300

9-3.1 Hypothesis Tests on the

Mean 300

9-3.2 P-Value for a t-Test 303

9-3.3 Choice of Sample Size 304

9-3.4 Likelihood Ratio Approach to

Development of Test Procedures

(CD Only) 305

9-4 Tests on the Variance and

Standard Deviation of a Normal

Distribution 307

9-4.1 The Hypothesis Testing

Procedures 307

9-4.2 -Error and Choice of

Sample Size 309

9-5 Tests on a Population

Proportion 310

9-5.1 Large-Sample Tests on a

Proportion 310

9-5.2 Small-Sample Tests on a

Proportion (CD Only) 312

9-5.3 Type II Error and Choice of Sample

Size 312

9-6 Summary of Inference Procedures for

a Single Sample 315

9-7 Testing for Goodness of Fit 315

9-8 Contingency Table Tests 320

CHAPTER 10 Statistical Inference

for Two Samples 327

10-1 Introduction 328

10-2 Inference For a Difference in Means

of Two Normal Distributions,

Variances Known 328

10-2.1 Hypothesis Tests for a

Difference in Means, Variances

Known 329

10-2.2 Choice of Sample Size 331

10-2.3 Identifying Cause and Effect 333

10-2.4 Confidence Interval on a

Difference in Means, Variances

Known 334

10-3 Inference For a Difference in Means

of Two Normal Distributions,

Variances Unknown 337

10-3.1 Hypothesis Tests for a

Difference in Means, Variances

Unknown 337

10-3.2 More About the Equal Variance

Assumption (CD Only) 344

10-3.3 Choice of Sample Size 344

10-3.4 Confidence Interval on a

Difference in Means, Variances

Unknown 345

10-4 Paired t-Test 349

10-5 Inference on the Variances of Two

Normal Distributions 355

10-5.1 The F Distribution 355

10-5.2 Development of the F

Distribution (CD Only) 357

10-5.3 Hypothesis Tests on the Ratio of

Two Variances 357

10-5.4 -Error and Choice of Sample

Size 359

10-5.5 Confidence Interval on the Ratio

of Two Variances 359

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10-6 Inference on Two Population

Proportions 361

10-6.1 Large-Sample Test for

H0 : p1 p2 361

10-6.2 Small Sample Test for

H0 : p1 p2 (CD Only) 364

10-6.3 -Error and Choice of

Sample Size 364

10-6.4 Confidence Interval for

P1 P2 365

10-7 Summary Table for Inference

Procedures for Two Samples 367

CHAPTER 11 Simple Linear

Regression and Correlation 372

11-1 Empirical Models 373

11-2 Simple Linear Regression 375

11-3 Properties of the Least Squares

Estimators 383

11-4 Some Comments on Uses of

Regression (CD Only) 384

11-5 Hypothesis Tests in Simple Linear

Regression 384

11-5.1 Use of t-Tests 384

11-5.2 Analysis of Variance Approach

to Test Significance of

Regression 387

11-6 Confidence Intervals 389

11-6.1 Confidence Intervals on the

Slope and Intercept 389

11-6.2 Confidence Interval on the

Mean Response 390

11-7 Prediction of New Observations 392

11-8 Adequacy of the Regression

Model 395

11-8.1 Residual Analysis 395

11-8.2 Coefficient of Determination

(R2

) 397

11-8.3 Lack-of-Fit Test

(CD Only) 398

11-9 Transformations to a Straight

Line 400

11-10 More About Transformations

(CD Only) 400

11-11 Correlation 400

CHAPTER 12 Multiple Linear

Regression 410

12-1 Multiple Linear Regression

Model 411

12-1.1 Introduction 411

12-1.2 Least Squares Estimation of the

Parameters 414

12-1.3 Matrix Approach to Multiple

Linear Regression 417

12-1.4 Properties of the Least Squares

Estimators 421

12-2 Hypothesis Tests in Multiple Linear

Regression 428

12-2.1 Test for Significance of

Regression 428

12-2.2 Tests on Individual Regression

Coefficients and Subsets of

Coefficients 432

12-2.3 More About the Extra Sum of

Squares Method (CD Only) 435

12-3 Confidence Intervals in Multiple

Linear Regression 437

12-3.1 Confidence Intervals on Individual

Regression Coefficients 437

12-3.2 Confidence Interval on the Mean

Response 438

12-4 Prediction of New Observations 439

12-5 Model Adequacy Checking 441

12-5.1 Residual Analysis 441

12-5.2 Influential Observations 444

12-6 Aspects of Multiple Regression

Modeling 447

12-6.1 Polynomial Regression

Models 447

12-6.2 Categorical Regressors and

Indicator Variables 450

12-6.3 Selection of Variables and Model

Building 452

12-6.4 Multicollinearity 460

12-6.5 Ridge Regression

(CD Only) 461

12-6.6 Nonlinear Regression Models

(CD Only) 461

CHAPTER 13 Design and

Analysis of Single-Factor

Experiments: The Analysis

of Variance 468

13-1 Designing Engineering

Experiments 469

13-2 The Completely Randomized

Single-Factor Experiment 470

13-2.1 An Example 470

13-2.2 The Analysis of Variance 472

13-2.3 Multiple Comparisons Following

the ANOVA 479

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13-2.4 More About Multiple

Comparisons (CD Only) 481

13-2.5 Residual Analysis and Model

Checking 481

13-2.6 Determining Sample Size 482

13-2.7 Technical Details about the

Analysis of Variance

(CD Only) 485

13-3 The Random Effects Model 487

13-3.1 Fixed Versus Random

Factors 487

13-3.2 ANOVA and Variance

Components 487

13-3.3 Determining Sample Size in

the Random Model

(CD Only) 490

13-4 Randomized Complete Block

Design 491

13-4.1 Design and Statistical

Analysis 491

13-4.2 Multiple Comparisons 497

13-4.3 Residual Analysis and Model

Checking 498

13-4.4 Randomized Complete Block

Design with Random Factors

(CD Only) 498

CHAPTER 14 Design of

Experiments with Several

Factors 505

14-1 Introduction 506

14-2 Some Applications of Designed

Experiments (CD Only) 506

14-3 Factorial Experiments 506

14-4 Two-Factor Factorial

Experiments 510

14-4.1 Statistical Analysis of the FixedEffects

Model 511

14-4.2 Model Adequacy Checking 517

14-4.3 One Observation Per Cell 517

14-4.4 Factorial Experiments with

Random Factors: Overview 518

14-5 General Factorial Experiments 520

14-6 Factorial Experiments with Random

Factors (CD Only) 523

14-7 2k Factorial Designs 523

14-7.1 22 Design 524

14-7.2 2k Design for k 3 Factors 529

14-7.3 Single Replicate of the 2k

Design 537

14-7.4 Addition of Center Points to a 2k

Design (CD Only) 541

14-8 Blocking and Confounding in the 2k

Design 543

14-9 Fractional Replication of the 2k

Design 549

14-9.1 One Half Fraction of the

2k Design 549

14-9.2 Smaller Fractions: The 2kp

Fractional Factorial 555

14-10 Response Surface Methods and

Designs (CD Only) 564

CHAPTER 15 Nonparametric

Statistics 571

15-1 Introduction 572

15-2 Sign Test 572

15-2.1 Description of the Test 572

15-2.2 Sign Test for Paired Samples 576

15-2.3 Type II Error for the Sign

Test 578

15-2.4 Comparison to the t-Test 579

15-3 Wilcoxon Signed-Rank Test 581

15-3.1 Description of the Test 581

15-3.2 Large-Sample

Approximation 583

15-3.3 Paired Observations 583

15-3.4 Comparison to the t-Test 584

15-4 Wilcoxon Rank-Sum Test 585

15-4.1 Description of the Test 585

15-4.2 Large-Sample

Approximation 587

15-4.3 Comparison to the t-Test 588

15-5 Nonparametric Methods in the

Analysis of Variance 589

15-5.1 Kruskal-Wallis Test 589

15-5.2 Rank Transformation 591

CHAPTER 16 Statistical Quality

Control 595

16-1 Quality Improvement and

Statistics 596

16-2 Statistical Quality Control 597

16-3 Statistical Process Control 597

16-4 Introduction to Control Charts 598

16-4.1 Basic Principles 598

16-4.2 Design of a Control Chart 602

16-4.3 Rational Subgroups 603

16-4.4 Analysis of Patterns on Control

Charts 604

16-5 and R or S Control Chart 607

16-6 Control Charts for Individual

Measurements 615

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16-7 Process Capability 619

16-8 Attribute Control Charts 625

16-8.1 P Chart (Control Chart for

Proportion) 625

16-8.2 U Chart (Control Chart for

Defects per Unit) 627

16-9 Control Chart Performance 630

16-10 Cumulative Sum Control

Chart 632

16-11 Other SPC Problem-Solving

Tools 639

16-12 Implementing SPC 641

APPENDICES 649

APPENDIX A: Statistical Tables

and Charts 651

Table I Summary of Common Probability

Distributions 652

Table II Cumulative Standard Normal

Distribution 653

Table III Percentage Points 2

,

of the ChiSquared

Distribution 655

Table IV Percentage Points t ,

of the

t-distribution 656

Table V Percentage Points f ,v1,v2 of the

F-distribution 657

Chart VI Operating Characteristic

Curves 662

Table VII Critical Values for the Sign

Test 671

Table VIII Critical Values for the Wilcoxon

Signed-Rank Test 671

Table IX Critical Values for the Wilcoxon

Rank-Sum Test 672

Table X Factors for Constructing Variables

Control Charts 673

Table XI Factors for Tolerance

Intervals 674

APPENDIX B: Bibliography 677

APPENDIX C: Answers to

Selected Exercises

679

GLOSSARY 689

INDEX 703

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