Introduction to Probability and Statistics for Engineers and Scientists (eBook)

Front Cover 1
Introduction to Probability and Statistics for Engineers and Scientists 4
Copyright Page 5
Table of Contents 8
Preface 14
Chapter 1. Introduction to Statistics 18
1.1 Introduction 18
1.2 Data Collection and Descriptive Statistics 18
1.3 Inferential Statistics and Probability Models 19
1.4 Populations and Samples 20
1.5 A Brief History of Statistics 20
Problems 24
Chapter 2. Descriptive Statistics 26
2.1 Introduction 26
2.2 Describing Data Sets 26
2.2.1 Frequency Tables and Graphs 27
2.2.2 Relative Frequency Tables and Graphs 27
2.2.3 Grouped Data, Histograms, Ogives, and Stem and Leaf Plots 31
2.3 Summarizing Data Sets 34
2.3.1 Sample Mean, Sample Median, and Sample Mode 34
2.3.2 Sample Variance and Sample Standard Deviation 39
2.3.3 Sample Percentiles and Box Plots 41
2.4 Chebyshev’s Inequality 44
2.5 Normal Data Sets 48
2.6 Paired Data Sets and the Sample Correlation Coefficient 50
Problems 58
Chapter 3. Elements of Probability 72
3.1 Introduction 72
3.2 Sample Space and Events 73
3.3 Venn Diagrams and the Algebra of Events 75
3.4 Axioms of Probability 76
3.5 Sample Spaces Having Equally Likely Outcomes 78
3.6 Conditional Probability 84
3.7 Bayes’ Formula 87
3.8 Independent Events 93
Problems 97
Chapter 4. Random Variables and Expectation 106
4.1 Random Variables 106
4.2 Types of Random Variables 109
4.3 Jointly Distributed Random Variables 112
4.3.1 Independent Random Variables 118
*4.3.2 Conditional Distributions 122
4.4 Expectation 124
4.5 Properties of the Expected Value 128
4.5.1 Expected Value of Sums of Random Variables 132
4.6 Variance 135
4.7 Covariance and Variance of Sums of Random Variables 138
4.8 Moment Generating Functions 142
4.9 Chebyshev’s Inequality and the Weak Law of Large Numbers 144
Problems 147
Chapter 5. Special Random Variables 158
5.1 The Bernoulli and Binomial Random Variables 158
5.1.1 Computing the Binomial Distribution Function 164
5.2 The Poisson Random Variable 165
5.2.1 Computing the Poisson Distribution Function 172
5.3 The Hypergeometric Random Variable 173
5.4 The Uniform Random Variable 177
5.5 Normal Random Variables 185
5.6 Exponential Random Variables 193
*5.6.1 The Poisson Process 197
*5.7 The Gamma Distribution 200
5.8 Distributions Arising from the Normal 203
5.8.1 The Chi-Square Distribution 203
5.8.2 The t-Distribution 207
5.8.3 The F-Distribution 209
*5.9 The Logistics Distribution 210
Problems 212
Chapter 6. Distributions of Sampling Statistics 220
6.1 Introduction 220
6.2 The Sample Mean 221
6.3 The Central Limit Theorem 223
6.3.1 Approximate Distribution of the Sample Mean 229
6.3.2 How Large a Sample Is Needed? 231
6.4 The Sample Variance 232
6.5 Sampling Distributions from a Normal Population 233
6.5.1 Distribution of the Sample Mean 234
6.5.2 Joint Distribution of X and S2 234
6.6 Sampling from a Finite Population 236
Problems 240
Chapter 7. Parameter Estimation 248
7.1 Introduction 248
7.2 Maximum Likelihood Estimators 249
*7.2.1 Estimating Life Distributions 257
7.3 Interval Estimates 259
7.3.1 Confidence Interval for a Normal Mean When the Variance Is Unknown 265
7.3.2 Confidence Intervals for the Variance of a Normal Distribution 270
7.4 Estimating the Difference in Means of Two Normal Populations 272
7.5 Approximate Confidence Interval for the Mean of a Bernoulli Random Variable 279
*7.6 Confidence Interval of the Mean of the Exponential Distribution 284
*7.7 Evaluating a Point Estimator 285
*7.8 The Bayes Estimator 291
Problems 296
Chapter 8. Hypothesis Testing 310
8.1 Introduction 310
8.2 Significance Levels 311
8.3 Tests Concerning the Mean of a Normal Population 312
8.3.1 Case of Known Variance 312
8.3.2 Case of Unknown Variance: The t-Test 324
8.4 Testing the Equality of Means of Two Normal Populations 331
8.4.1 Case of Known Variances 331
8.4.2 Case of Unknown Variances 333
8.4.3 Case of Unknown and Unequal Variances 337
8.4.4 The Paired t-Test 338
8.5 Hypothesis Tests Concerning the Variance of a Normal Population 340
8.5.1 Testing for the Equality of Variances of Two Normal Populations 341
8.6 Hypothesis Tests in Bernoulli Populations 342
8.6.1 Testing the Equality of Parameters in Two Bernoulli Populations 346
8.7 Tests Concerning the Mean of a Poisson Distribution 349
8.7.1 Testing the Relationship Between Two Poisson Parameters 350
Problems 353
Chapter 9. Regression 370
9.1 Introduction 370
9.2 Least Squares Estimators of the Regression Parameters 372
9.3 Distribution of the Estimators 374
9.4 Statistical Inferences about the Regression Parameters 380
9.4.1 Inferences Concerning ß 381
9.4.2 Inferences Concerning a 389
9.4.3 Inferences Concerning the Mean Response a+ßx0 390
9.4.4 Prediction Interval of a Future Response 392
9.4.5 Summary of Distributional Results 394
9.5 The Coefficient of Determination and the Sample Correlation Coefficient 395
9.6 Analysis of Residuals: Assessing the Model 397
9.7 Transforming to Linearity 400
9.8 Weighted Least Squares 403
9.9 Polynomial Regression 410
*9.10 Multiple Linear Regression 413
9.10.1 Predicting Future Responses 424
9.11 Logistic Regression Models for Binary Output Data 429
Problems 432
Chapter 10. Analysis of Variance 458
10.1 Introduction 458
10.2 An Overview 459
10.3 One-Way Analysis of Variance 461
10.3.1 Multiple Comparisons of Sample Means 469
10.3.2 One-Way Analysis of Variance with Unequal Sample Sizes 471
10.4 Two-Factor Analysis of Variance:Introduction and Parameter Estimation 473
10.5 Two-Factor Analysis of Variance:Testing Hypotheses 477
10.6 Two-Way Analysis of Variance with Interaction 482
Problems 490
Chapter 11. Goodness of Fit Tests and Categorical Data Analysis 502
11.1 Introduction 502
11.2 Goodness of Fit Tests When All Parameters are Specified 503
11.2.1 Determining the Critical Region by Simulation 509
11.3 Goodness of Fit Tests When Some Parameters are Unspecified 512
11.4 Tests of Independence in Contingency Tables 514
11.5 Tests of Independence in Contingency Tables Having Fixed Marginal Totals 518
*11.6 The Kolmogorov–Smirnov Goodness of Fit Test for Continuous Data 523
Problems 527
Chapter 12. Nonparametric Hypothesis Tests 534
12.1 Introduction 534
12.2 The Sign Test 534
12.3 The Signed Rank Test 538
12.4 The Two-Sample Problem 544
*12.4.1 The Classical Approximation and Simulation 548
12.5 The Runs Test for Randomness 552
Problems 556
Chapter 13. Quality Control 564
13.1 Introduction 564
13.2 Control Charts for Average Values:The X-Control Chart 565
13.2.1 Case of Unknown µ and s 568
13.3 S-Control Charts 573
13.4 Control Charts for the Fraction Defective 576
13.5 Control Charts for Number of Defects 578
13.6 Other Control Charts for Detecting Changes in the Population Mean 582
13.6.1 Moving-Average Control Charts 582
13.6.2 Exponentially Weighted Moving-Average Control Charts 584
13.6.3 Cumulative Sum Control Charts 590
Problems 592
Chapter 14*. Life Testing 600
14.1 Introduction 600
14.2 Hazard Rate Functions 600
14.3 The Exponential Distribution in Life Testing 603
14.3.1 Simultaneous Testing — Stopping at the rth Failure 603
14.3.2 Sequential Testing 609
14.3.3 Simultaneous Testing — Stopping by a Fixed Time 613
14.3.4 The Bayesian Approach 615
14.4 A Two-Sample Problem 617
14.5 The Weibull Distribution in Life Testing 619
14.5.1 Parameter Estimation by Least Squares 621
Problems 623
Chapter 15. Simulation, Bootstrap Statistical Methods, and Permutation Tests 630
15.1 Introduction 630
15.2 Random Numbers 631
15.2.1 The Monte Carlo Simulation Approach 633
15.3 The Bootstrap Method 634
15.4 Permutation Tests 641
15.4.1 Normal Approximations in Permutation Tests 644
15.4.2 Two-Sample Permutation Tests 648
15.5 Generating Discrete Random Variables 649
15.6 Generating Continuous Random Variables 651
15.6.1 Generating a Normal Random Variable 653
15.7 Determining the Number of Simulation Runs in a Monte Carlo Study 654
Problems 655
Appendix of Tables 658
Index 664