Statistics and Probability with Applications for Engineers and Scientists

BHISHAM C. GUPTA, PhD, is Professor in the Department of Mathematics and Statistics at the University of Southern Maine. Dr. Gupta has written four books and more than thirty articles. IRWIN GUTTMAN, PhD, is Professor Emeritus of Statistics in the Department of Mathematics at the State University of New York at Buffalo and Department of Statistics at the University of Toronto, Canada. Dr. Guttman has written five books and over 140 articles.

Preface xvii Chapter 1 | Introduction 1 1.1 Designed Experiment 2 1.1.1 Motivation for the Study 2 1.1.2 Investigation 2 1.1.3 Changing Criteria 2 1.1.4 A Summary of the Various Phases of the Investigation 3 1.2 A Survey 5 1.3 An Observational Study 6 1.4 A Set of Historical Data 6 1.5 A Brief Description of What is Covered in This Book 6 PART I Chapter 2 | Describing Data Graphically and Numerically 11 2.1 Getting Started with Statistics 12 2.1.1 What Is Statistics? 12 2.1.2 Population and Sample in a Statistical Study 12 2.2 Classification of Various Types of Data 15 2.2.1 Nominal Data 15 2.2.2 Ordinal Data 16 2.2.3 Interval Data 16 2.2.4 Ratio Data 16 2.3 Frequency Distribution Tables for Qualitative and Quantitative Data 17 2.3.1 Qualitative Data 17 2.3.2 Quantitative Data 20 2.4 Graphical Description of Qualitative and Quantitative Data 25 2.4.1 Dot Plot 25 2.4.2 Pie Chart 25 2.4.3 Bar Chart 27 2.4.4 Histograms 30 2.4.5 Line Graph 35 2.4.6 Stem-and-Leaf Plot 37 2.5 Numerical Measures of Quantitative Data 41 2.5.1 Measures of Centrality 42 2.5.2 Measures of Dispersion 46 2.6 Numerical Measures of Grouped Data 55 2.6.1 Mean of a Grouped Data 56 2.6.2 Median of a Grouped Data 56 2.6.3 Mode of a Grouped Data 57 2.6.4 Variance of a Grouped Data 57 2.7 Measures of Relative Position 59 2.7.1 Percentiles 59 2.7.2 Quartiles 60 2.7.3 Interquartile Range 60 2.7.4 Coefficient of Variation 61 2.8 Box-Whisker Plot 62 2.8.1 Construction of a Box Plot 62 2.8.2 How to Use the Box Plot 63 2.9 Measures of Association 68 2.10 Case Studies 71 2.11 Using JMP1 73 Review Practice Problems 73 Chapter 3 | Elements of Probability 83 3.1 Introduction 84 3.2 Random Experiments, Sample Spaces, and Events 84 3.2.1 Random Experiments and Sample Spaces 84 3.2.2 Events 85 3.3 Concepts of Probability 88 3.4 Techniques of Counting Sample Points 93 3.4.1 Tree Diagram 93 3.4.2 Permutations 94 3.4.3 Combinations 95 3.4.4 Arrangements of n Objects Involving Several Kinds of Objects 96 3.5 Conditional Probability 98 3.6 Bayes s Theorem 100 3.7 Introducing Random Variables 104 Review Practice Problems 105 Chapter 4 | Discrete Random Variables and Some Important Discrete Probability Distributions 111 4.1 Graphical Descriptions of Discrete Distributions 112 4.2 Mean and Variance of a Discrete Random Variable 113 4.2.1 Expected Value of Discrete Random Variables and Their Functions 113 4.2.2 The Moment-Generating Function Expected Value of a Special Function of X 115 4.3 The Discrete Uniform Distribution 117 4.4 The Hypergeometric Distribution 119 4.5 The Bernoulli Distribution 122 4.6 The Binomial Distribution 123 4.7 The Multinomial Distribution 126 4.8 The Poisson Distribution 128 4.8.1 Definition and Properties of the Poisson Distribution 128 4.8.2 Poisson Process 128 4.8.3 Poisson Distribution as a Limiting Form of the Binomial 128 4.9 The Negative Binomial Distribution 132 4.10 Some Derivations and Proofs (Optional) 135 4.11 A Case Study 135 4.12 Using JMP 135 Review Practice Problems 136 Chapter 5 | Continuous Random Variables and Some Important Continuous Probability Distributions 143 5.1 Continuous Random Variables 144 5.2 Mean and Variance of Continuous Random Variables 146 5.2.1 Expected Value of Continuous Random Variables and Their Function 146 5.2.2 The Moment-Generating Function Expected Value of a Special Function of X 149 5.3 Chebychev s Inequality 151 5.4 The Uniform Distribution 152 5.4.1 Definition and Properties 152 5.4.2 Mean and Standard Deviation of the Uniform Distribution 155 5.5 The Normal Distribution 157 5.5.1 Definition and Properties 157 5.5.2 The Standard Normal Distribution 158 5.5.3 The Moment-Generating Function of the Normal Distribution 164 5.6 Distribution of Linear Combination of Independent Normal Variables 165 5.7 Approximation of the Binomial and Poisson Distribution by the Normal Distribution 169 5.7.1 Approximation of the Binomial Distribution by the Normal Distribution 169 5.7.2 Approximation of the Poisson Distribution by the Normal Distribution 171 5.8 A Test of Normality 171 5.9 Probability Models Commonly Used in Reliability Theory 175 5.9.1 The Lognormal Distribution 176 5.9.2 The Exponential Distribution 180 5.9.3 The Gamma Distribution 184 5.9.4 The Weibull Distribution 187 5.10 A Case Study 191 5.11 Using JMP 192 Review Practice Problems 192 Chapter 6 | Distribution of Functions of Random Variables 199 6.1 Introduction 200 6.2 Distribution Functions of Two Random Variables 200 6.2.1 Case of Two Discrete Random Variables 200 6.2.2 Case of Two Continuous Random Variables 202 6.2.3 The Mean Value and Variance of Functions of Two Random Variables 204 6.2.4 Conditional Distributions 206 6.2.5 Correlation between Two Random Variables 208 6.2.6 Bivariate Normal Distribution 211 6.3 Extension to Several Random Variables 214 6.4 The Moment-Generating Function Revisited 214 Review Practice Problems 218 Chapter 7 | Sampling Distributions 223 7.1 Random Sampling 224 7.1.1 Random Sampling from an Infinite Population 224 7.1.2 Random Sampling from a Finite Population 225 7.2 The Sampling Distribution of the Mean 228 7.2.1 Normal Sampled Population 228 7.2.2 Nonnormal Sampled Population 228 7.2.3 The Central Limit Theorem 228 7.3 Sampling from a Normal Population 234 7.3.1 The Chi-Square Distribution 234 7.3.2 The Student t-Distribution 240 7.3.3 Snedecor s F-Distribution 244 7.4 Order Statistics 247 7.5 Using JMP 247 Review Practice Problems 247 Chapter 8 | Estimation of Population Parameters 251 8.1 Introduction 252 8.2 Point Estimators for the Population Mean and Variance 252 8.2.1 Properties of Point Estimators 253 8.2.2 Methods of Finding Point Estimators 256 8.3 Interval Estimators for the Mean m of a Normal Population 262 8.3.1 s2 Known 262 8.3.2 s2 Unknown 264 8.3.3 Sample Size Is Large 266 8.4 Interval Estimators for the Difference of Means of Two Normal Populations 272 8.4.1 Variances Are Known 272 8.4.2 Variances Are Unknown 273 8.5 Interval Estimators for the Variance of a Normal Population 280 8.6 Interval Estimator for the Ratio of Variances of Two Normal Populations 284 8.7 Point and Interval Estimators for the Parameters of Binomial Populations 288 8.7.1 One Binomial Population 288 8.7.2 Two Binomial Populations 290 8.8 Determination of Sample Size 294 8.8.1 One Population Mean 294 8.8.2 Difference of Two Population Means 295 8.8.3 One Population Proportion 296 8.8.4 Difference of Two Population Proportions 296 8.9 Some Supplemental Information 298 8.10 A Case Study 298 8.11 Using JMP 299 Review Practice Problems 299 Chapter 9 | Hypothesis Testing 307 9.1 Introduction 308 9.2 Basic Concepts of Testing a Statistical Hypothesis 308 9.2.1 Hypothesis Formulation 308 9.2.2 Risk Assessment 310 9.3 Tests Concerning the Mean of a Normal Population Having Known Variance 312 9.3.1 Case of a One-Tail (Left-Sided) Test 312 9.3.2 Case of a One-Tail (Right-Sided) Test 316 9.3.3 Case of a Two-Tail Test 317 9.4 Tests Concerning the Mean of a Normal Population Having Unknown Variance 324 9.4.1 Case of a Left-Tail Test 324 9.4.2 Case of a Right-Tail Test 326 9.4.3 The Two-Tail Case 326 9.5 Large Sample Theory 330 9.6 Tests Concerning the Difference of Means of Two Populations Having Distributions with Known Variances 332 9.6.1 The Left-Tail Test 332 9.6.2 The Right-Tail Test 333 9.6.3 The Two-Tail Test 334 9.7 Tests Concerning the Difference of Means of Two Populations Having Normal Distributions with Unknown Variances 339 9.7.1 Two Population Variances Are Equal 339 9.7.2 Two Population Variances Are Unequal 342 9.7.3 The Paired t-Test 344 9.8 Testing Population Proportions 349 9.8.1 Test Concerning One Population Proportion 349 9.8.2 Test Concerning the Difference between Two Population Proportions 351 9.9 Tests Concerning the Variance of a Normal Population 355 9.10 Tests Concerning the Ratio of Variances of Two Normal Populations 358 9.11 Testing of Statistical Hypotheses Using Confidence Intervals 362 9.12 Sequential Tests of Hypotheses 367 9.12.1 A One-Tail Sequential Testing Procedure 367 9.12.2 A Two-Tail Sequential Testing Procedure 371 9.13 Case Studies 374 9.14 Using JMP 375 Review Practice Problems 375 PART II Chapter 10 | Elements of Reliability Theory 389 10.1 The Reliability Function 390 10.1.1 The Hazard Rate Function 391 10.1.2 Employing the Hazard Function 398 10.2 Estimation: Exponential Distribution 399 10.3 Hypothesis Testing: Exponential Distribution 406 10.4 Estimation: Weibull Distribution 407 10.5 Case Studies 414 10.6 Using JMP 416 Review Practice Problems 416 Chapter 11 | Statistical Quality Control Phase I Control Charts 419 11.1 Basic Concepts of Quality and Its Benefits 420 11.2 What a Process Is and Some Valuable Tools 420 11.2.1 Check Sheet 422 11.2.2 Pareto Chart 422 11.2.3 Cause-and-Effect (Fishbone or Ishikawa) Diagram 425 11.2.4 Defect Concentration Diagram 427 11.3 Common and Assignable Causes 427 11.3.1 Process Evaluation 427 11.3.2 Action on the Process 428 11.3.3 Action on Output 428 11.3.4 Variation 428 11.4 Control Charts 429 11.4.1 Preparation for Use of Control Charts 430 11.4.2 Benefits of a Control Chart 431 11.4.3 Control Limits Versus Specification Limits 433 11.5 Control Charts for Variables 434 11.5.1 Shewhart X and R Control Charts 434 11.5.2 Shewhart X and R Control Charts When Process Mean m and Process Standard Deviation s Are Known 440 11.5.3 Shewhart X and S Control Charts 441 11.6 Control Charts for Attributes 448 11.6.1 The p Chart: Control Chart for the Fraction of Nonconforming Units 449 11.6.2 The p Chart: Control Chart for the Fraction Nonconforming with Variable Sample Sizes 454 11.6.3 The np Control Chart: Control Chart for the Number of Nonconforming Units 456 11.6.4 The c Control Chart 458 11.6.5 The u Control Chart 461 11.7 Process Capability 468 11.8 Case Studies 470 11.9 Using JMP 472 Review Practice Problems 472 Chapter 12 | Statistical Quality Control Phase II Control Charts 479 12.1 Introduction 480 12.2 Basic Concepts of CUSUM Control Chart 480 12.3 Designing a CUSUM Control Chart 483 12.3.1 Two-Sided CUSUM Control Chart Using a Numerical Procedure 484 12.3.2 The Fast Initial Response (FIR) Feature for CUSUM Control Chart 489 12.3.3 The Combined Shewhart CUSUM Control Chart 492 12.3.4 The CUSUM Control Chart for Controlling Process Variability 493 12.4 The Moving Average (MA) Control Chart 495 12.5 The Exponentially Weighted Moving Average (EWMA) Control Chart 499 12.6 Case Studies 504 12.7 Using JMP 505 Review Practice Problems 506 Chapter 13 | Analysis of Categorical Data 509 13.1 Introduction 509 13.2 The Chi-Square Goodness-of-Fit Test 510 13.3 Contingency Tables 517 13.3.1 The 2 2 Case Parameters Known 517 13.3.2 The 2 2 Case with Unknown Parameters 519 13.3.3 The r s Contingency Table 521 13.4 Chi-Square Test for Homogeneity 525 13.5 Comments on the Distribution of the Lack-of-Fit Statistics 528 13.6 Case Studies 529 Review Practice Problems 531 Chapter 14 | Nonparametric Tests 537 14.1 Introduction 537 14.2 The Sign Test 538 14.2.1 One-Sample Test 538 14.2.2 The Wilcoxon Signed-Rank Test 541 14.2.3 Two-Sample Test 543 14.3 Mann Whitney (Wilcoxon) W Test for Two Samples 548 14.4 Runs Test 551 14.4.1 Runs Above and Below the Median 551 14.4.2 The Wald Wolfowitz Run Test 553 14.5 Spearman Rank Correlation 556 14.6 Using JMP 559 Review Practice Problems 559 Chapter 15 | Simple Linear Regression Analysis 565 15.1 Introduction 566 15.2 Fitting the Simple Linear Regression Model 567 15.2.1 Simple Linear Regression Model 567 15.2.2 Fitting a Straight Line by Least Squares 569 15.2.3 Sampling Distribution of the Estimators of Regression Coefficients 573 15.3 Unbiased Estimator of s2 578 15.4 Further Inferences Concerning Regression Coefficients (b0, b1), E(Y), and Y 580 15.4.1 Confidence Interval for b1 with Confidence Coefficient (1 a) 580 15.4.2 Confidence Interval for b0 with Confidence Coefficient (1a) 581 15.4.3 Confidence Interval for E(YjX) with Confidence Coefficient (1 a) 582 15.4.4 Prediction Interval for a Future Observation Y with Confidence Coefficient (1 a) 585 15.5 Tests of Hypotheses for b0 and b1 590 15.5.1 Test of Hypotheses for b1 590 15.5.2 Test of Hypotheses for b0 590 15.6 Analysis of Variance Approach to Simple Linear Regression Analysis 596 15.7 Residual Analysis 601 15.8 Transformations 609 15.9 Inference About r 615 15.10 A Case Study 618 15.11 Using JMP 619 Review Practice Problems 619 Chapter 16 | Multiple Linear Regression Analysis 627 16.1 Introduction 628 16.2 Multiple Linear Regression Models 628 16.3 Estimation of Regression Coefficients 632 16.3.1 Estimation of Regression Coefficients Using Matrix Notation 633 16.3.2 Properties of the Least-Squares Estimators 635 16.3.3 The Analysis of Variance Table 636 16.3.4 More Inferences about Regression Coefficients 639 16.4 Multiple Linear Regression Model Using Quantitative and Qualitative Predictor Variables 646 16.4.1 Single Qualitative Variable with Two Categories 646 16.4.2 Single Qualitative Variable with Three or More Categories 647 16.5 Standardized Regression Coefficients 658 16.5.1 Multicollinearity 660 16.5.2 Consequences of Multicollinearity 661 16.6 Building Regression Type Prediction Models 662 16.6.1 First Variable to Enter into the Model 662 16.7 Residual Analysis and Certain Criteria for Model Selection 665 16.7.1 Residual Analysis 665 16.7.2 Certain Criteria for Model Selection 667 16.8 Logistic Regression 672 16.9 Case Studies 676 16.10 Using JMP 677 Review Practice Problems 678 Chapter 17 | Analysis of Variance 685 17.1 Introduction 686 17.2 The Design Models 686 17.2.1 Estimable Parameters 686 17.2.2 Estimable Functions 688 17.3 One-Way Experimental Layouts 689 17.3.1 The Model and Its Analysis 689 17.3.2 Confidence Intervals for Treatment Means 695 17.3.3 Multiple Comparisons 700 17.3.4 Determination of Sample Size 706 17.3.5 The Kruskal Wallis Test for One-Way Layouts (Nonparametric Method) 707 17.4 Randomized Complete Block Designs 710 17.4.1 The Friedman Fr-Test for Randomized Complete Block Design (Nonparametric Method) 718 17.4.2 Experiments with One Missing Observation in an RCB-Design Experiment 719 17.4.3 Experiments with Several Missing Observations in an RCB-Design Experiment 719 17.5 Two-Way Experimental Layouts 722 17.5.1 Two-Way Experimental Layouts with One Observation per Cell 724 17.5.2 Two-Way Experimental Layouts with r>1 Observations per Cell 725 17.5.3 Blocking in Two-Way Experimental Layouts 734 17.5.4 Extending Two-Way Experimental Designs to n-Way Experimental Layouts 734 17.6 Latin Square Designs 736 17.7 Random-Effects and Mixed-Effects Models 742 17.7.1 Random-Effects Model 742 17.7.2 Mixed-Effects Model 744 17.7.3 Nested (Hierarchical) Designs 746 17.8 A Case Study 752 17.9 Using JMP 753 Review Practice Problems 753 Chapter 18 | The 2k Factorial Designs 765 18.1 Introduction 766 18.2 The Factorial Designs 766 18.3 The 2k Factorial Design 768 18.4 Unreplicated 2k Factorial Designs 776 18.5 Blocking in the 2k Factorial Design 782 18.5.1 Confounding in the 2k Factorial Design 783 18.5.2 Yates s Algorithm for the 2k Factorial Designs 788 18.6 The 2k Fractional Factorial Designs 790 18.6.1 One-half Replicate of a 2k Factorial Design 790 18.6.2 One-quarter Replicate of a 2k Factorial Design 795 18.7 Case Studies 799 18.8 Using JMP 801 Review Practice Problems 801 Chapter 19 | Response Surfaces This chapter is not included in text, but is available for download via the book s website: www.wiley.com/go/statsforengineers Appendices 807 Appendix A | Statistical Tables 809 Appendix B | Answers to Selected Problems 845 Appendix C | Bibliography 863 Index 867