Preface xiii

**1 Introduction to Statistical Science 1**

1.1 The Scientic Method: A Process for Learning 3

1.2 The Role of Statistics in the Scientic Method 5

1.3 Main Approaches to Statistics 5

1.4 Purpose and Organization of This Text 8

**2 Scientic Data Gathering 13**

2.1 Sampling from a Real Population 14

2.2 Observational Studies and Designed Experiments 17

Monte Carlo Exercises 23

**3 Displaying and Summarizing Data 31**

3.1 Graphically Displaying a Single Variable 32

3.2 Graphically Comparing Two Samples 39

3.3 Measures of Location 41

3.4 Measures of Spread 44

3.5 Displaying Relationships Between Two or More Variables 46

3.6 Measures of Association for Two or More Variables 49

Exercises 52

**4 Logic, Probability, and Uncertainty 59**

4.1 Deductive Logic and Plausible Reasoning 60

4.2 Probability 62

4.3 Axioms of Probability 64

4.4 Joint Probability and Independent Events 65

4.5 Conditional Probability 66

4.6 Bayes' Theorem 68

4.7 Assigning Probabilities 74

4.8 Odds and Bayes Factor 75

4.9 Beat the Dealer 76

Exercises 80

**5 Discrete Random Variables 83**

5.1 Discrete Random Variables 84

5.2 Probability Distribution of a Discrete Random Variable 86

5.3 Binomial Distribution 90

5.4 Hypergeometric Distribution 92

5.5 Poisson Distribution 93

5.6 Joint Random Variables 96

5.7 Conditional Probability for Joint Random Variables 100

Exercises 104

**6 Bayesian Inference for Discrete Random Variables 109**

6.1 Two Equivalent Ways of Using Bayes' Theorem 114

6.2 Bayes' Theorem for Binomial with Discrete Prior 116

6.3 Important Consequences of Bayes' Theorem 119

6.4 Bayes' Theorem for Poisson with Discrete Prior 120

Exercises 122

Computer Exercises 126

**7 Continuous Random Variables 129**

7.1 Probability Density Function 131

7.2 Some Continuous Distributions 135

7.3 Joint Continuous Random Variables 143

7.4 Joint Continuous and Discrete Random Variables 144

Exercises 147

**8 Bayesian Inference for Binomial Proportion 149**

8.1 Using a Uniform Prior 150

8.2 Using a Beta Prior 151

8.3 Choosing Your Prior 154

8.4 Summarizing the Posterior Distribution 158

8.5 Estimating the Proportion 161

8.6 Bayesian Credible Interval 162

Exercises 164

Computer Exercises 167

**9 Comparing Bayesian and Frequentist Inferences for Proportion 169**

9.1 Frequentist Interpretation of Probability and Parameters 170

9.2 Point Estimation 171

9.3 Comparing Estimators for Proportion 174

9.4 Interval Estimation 175

9.5 Hypothesis Testing 178

9.6 Testing a One-Sided Hypothesis 179

9.7 Testing a Two-Sided Hypothesis 182

Exercises 187

Monte Carlo Exercises 190

**10 Bayesian Inference for Poisson 193**

10.1 Some Prior Distributions for Poisson 194

10.2 Inference for Poisson Parameter 200

Exercises 207

Computer Exercises 208

**11 Bayesian Inference for Normal Mean 211**

11.1 Bayes' Theorem for Normal Mean with a Discrete Prior 211

11.2 Bayes' Theorem for Normal Mean with a Continuous Prior 218

11.3 Choosing Your Normal Prior 222

11.4 Bayesian Credible Interval for Normal Mean 224

11.5 Predictive Density for Next Observation 227

Exercises 230

Computer Exercises 232

**12 Comparing Bayesian and Frequentist Inferences for Mean 237**

12.1 Comparing Frequentist and Bayesian Point Estimators 238

12.2 Comparing Condence and Credible Intervals for Mean 241

12.3 Testing a One-Sided Hypothesis about a Normal Mean 243

12.4 Testing a Two-Sided Hypothesis about a Normal Mean 247

Exercises 251

**13 Bayesian Inference for Di erence Between Means 255**

13.1 Independent Random Samples from Two Normal Distributions 256

13.2 Case 1: Equal Variances 257

13.3 Case 2: Unequal Variances 262

13.4 Bayesian Inference for Dierence Between Two Proportions Using Normal Approximation 265

13.5 Normal Random Samples from Paired Experiments 266

Exercises 272

**14 Bayesian Inference for Simple Linear Regression 283**

14.1 Least Squares Regression 284

14.2 Exponential Growth Model 288

14.3 Simple Linear Regression Assumptions 290

14.4 Bayes' Theorem for the Regression Model 292

14.5 Predictive Distribution for Future Observation 298

Exercises 303

Computer Exercises 312

**15 Bayesian Inference for Standard Deviation 315**

15.1 Bayes' Theorem for Normal Variance with a Continuous Prior 316

15.2 Some Specic Prior Distributions and the Resulting Posteriors 318

15.3 Bayesian Inference for Normal Standard Deviation 326

Exercises 332

Computer Exercises 335

**16 Robust Bayesian Methods 337**

16.1 Eect of Misspecied Prior 338

16.2 Bayes' Theorem with Mixture Priors 340

Exercises 349

Computer Exercises 351

**17 Bayesian Inference for Normal with Unknown Mean and Variance 355**

17.1 The Joint Likelihood Function 358

17.2 Finding the Posterior when Independent Jeffreys' Priors for μ and σ2 Are Used 359

17.3 Finding the Posterior when a Joint Conjugate Prior for μ and σ2 Is Used 361

17.4 Difference Between Normal Means with Equal Unknown Variance 367

17.5 Difference Between Normal Means with Unequal Unknown Variances 377

Computer Exercises 383

Appendix: Proof that the Exact Marginal Posterior Distribution of μ is *Student's t* 385

**18 Bayesian Inference for Multivariate Normal Mean Vector 393**

18.1 Bivariate Normal Density 394

18.2 Multivariate Normal Distribution 397

18.3 The Posterior Distribution of the Multivariate Normal Mean Vector when Covariance Matrix Is Known 398

18.4 Credible Region for Multivariate Normal Mean Vector when Covariance Matrix Is Known 400

18.5 Multivariate Normal Distribution with Unknown Covariance Matrix 402

Computer Exercises 406

**19 Bayesian Inference for the Multiple Linear Regression Model 411**

19.1 Least Squares Regression for Multiple Linear Regression Model 412

19.2 Assumptions of Normal Multiple Linear Regression Model 414

19.3 Bayes' Theorem for Normal Multiple Linear Regression Model 415

19.4 Inference in the Multivariate Normal Linear Regression Model 419

19.5 The Predictive Distribution for a Future Observation 425

Computer Exercises 428

**20 Computational Bayesian Statistics Including Markov Chain Monte Carlo 431**

20.1 Direct Methods for Sampling from the Posterior 436

20.2 Sampling - Importance - Resampling 450

20.3 Markov Chain Monte Carlo Methods 454

20.4 Slice Sampling 470

20.5 Inference from a Posterior Random Sample 473

20.6 Where to Next? 475

A Introduction to Calculus 477

B Use of Statistical Tables 497

C Using the Included Minitab Macros 523

D Using the Included R Functions 543

E Answers to Selected Exercises 565

References 591

Index 595