Statistical Intervals

William Q. Meeker is Professor of Statistics and Distinguished Professor of Liberal Arts and Sciences at Iowa State University. He is the co-author of Statistical Methods for Reliability Data, 2nd Edition (Wiley, 2021) and of numerous publications in the engineering and statistical literature and has won many awards for his research.Gerald J. Hahn served for 46 years as applied statistician and manager of an 18-person statistics group supporting General Electric and has co-authored four books. His accomplishments have been recognized by GE's prestigious Coolidge Fellowship and 19 professional society awards.Luis A. Escobar is Professor of Statistics at Louisiana State University. He is the co-author of Statistical Methods for Reliability Data, 2nd Edition (Wiley, 2021) and several book chapters. His publications have appeared in the engineering and statistical literature and he has won several research and teaching awards.

• Preface to Second Edition iii Preface to First Edition viiAcknowledgments x1 Introduction, Basic Concepts, and Assumptions 11.1 Statistical Inference 21.2 Different Types of Statistical Intervals: An Overview 21.3 The Assumption of Sample Data 31.4 The Central Role of Practical Assumptions Concerning Representative Data 41.5 Enumerative Versus Analytic Studies 51.6 Basic Assumptions for Enumerative Studies 71.7 Considerations in the Conduct of Analytic Studies 101.8 Convenience and Judgment Samples 111.9 Sampling People 121.10 Infinite Population Assumptions 131.11 Practical Assumptions: Overview 141.12 Practical Assumptions: Further Example 141.13 Planning the Study 171.14 The Role of Statistical Distributions 171.15 The Interpretation of Statistical Intervals 181.16 Statistical Intervals and Big Data 191.17 Comment Concerning Subsequent Discussion 192 Overview of Different Types of Statistical Intervals 212.1 Choice of a Statistical Interval 212.2 Confidence Intervals 232.3 Prediction Intervals 242.4 Statistical Tolerance Intervals 262.5 Which Statistical Interval Do I Use? 272.6 Choosing a Confidence Level 282.7 Two-Sided Statistical Intervals Versus One-Sided Statistical Bounds 292.8 The Advantage of Using Confidence Intervals Instead of Significance Tests 302.9 Simultaneous Statistical Intervals 313 Constructing Statistical Intervals Assuming a Normal Distribution Using Simple Tabulations 333.1 Introduction 343.2 Circuit Pack Voltage Output Example 353.3 Two-Sided Statistical Intervals 363.4 One-Sided Statistical Bounds 384 Methods for Calculating Statistical Intervals for a Normal Distribution 434.1 Notation 444.2 Confidence Interval for the Mean of a Normal Distribution 454.3 Confidence Interval for the Standard Deviation of a Normal Distribution 454.4 Confidence Interval for a Normal Distribution Quantile 464.5 Confidence Interval for the Distribution Proportion Less (Greater) Than a Specified Value 474.6 Statistical Tolerance Intervals 484.7 Prediction Interval to Contain a Single Future Observation or the Mean of m Future Observations 504.8 Prediction Interval to Contain at least k of m Future Observations 514.9 Prediction Interval to Contain the Standard Deviation of m Future Observations 524.10 The Assumption of a Normal Distribution 534.11 Assessing Distribution Normality and Dealing with Nonnormality 544.12 Data Transformations and Inferences from Transformed Data 574.13 Statistical Intervals for Linear Regression Analysis 604.14 Statistical Intervals for Comparing Populations and Processes 625 Distribution-Free Statistical Intervals 655.1 Introduction 665.2 Distribution-Free Confidence Intervals and One-Sided Confidence Bounds for a Quantile 685.3 Distribution-Free Tolerance Intervals and Bounds to Contain a Specified Proportion of a Distribution 785.4 Prediction Intervals to Contain a Specified Ordered Observation in a Future Sample 815.5 Distribution-Free Prediction Intervals and Bounds to Contain at Least k of m Future Observations 846 Statistical Intervals for a Binomial Distribution 896.1 Introduction to Binomial Distribution Statistical Intervals 906.2 Confidence Intervals for the Actual Proportion Nonconforming in the Sampled Distribution 926.3 Confidence Interval for the Proportion of Nonconforming Units in a Finite Population 1026.4 Confidence Intervals for the Probability that the Number of Nonconforming Units in a Sample is Less than or Equal to (or Greater than) a Specified Number 1046.5 Confidence Intervals for the Quantile of the Distribution of the Number of Nonconforming Units 1056.6 Tolerance Intervals and One-Sided Tolerance Bounds for the Distribution of the Number of Nonconforming Units 1076.7 Prediction Intervals for the Number Nonconforming in a Future Sample 1087 Statistical Intervals for a Poisson Distribution 1157.1 Introduction 1167.2 Confidence Intervals for the Event-Occurrence Rate of a Poisson Distribution 1177.3 Confidence Intervals for the Probability that the Number of Events in a Specified Amount of Exposure is Less than or Equal to (or Greater than) a Specified Number 1247.4 Confidence Intervals for the Quantile of the Distribution of the Number of Events in a Specified Amount of Exposure 1257.5 Tolerance Intervals and One-Sided Tolerance Bounds for the Distribution of the Number of Events in a Specified Amount of Exposure 1277.6 Prediction Intervals for the Number of Events in a Future Amount of Exposure 1288 Sample Size Requirements for Confidence Intervals on Distribution Parameters 1358.1 Basic Requirements for Sample Size Determination 1368.2 Sample Size for a Confidence Interval for a Normal Distribution Mean 1378.3 Sample Size to Estimate a Normal Distribution Standard Deviation 1418.4 Sample Size to Estimate a Normal Distribution Quantile 1438.5 Sample Size to Estimate a Binomial Proportion 1438.6 Sample Size to Estimate a Poisson Occurrence Rate 1449 Sample Size Requirements for Tolerance Intervals, Tolerance Bounds, and Related Demonstration Tests 1489.1 Sample Size for Normal Distribution Tolerance Intervals and One-Sided Tolerance Bounds1489.2 Sample Size to Pass a One-Sided Demonstration Test Based on Normally Distributed Measurements 1509.3 Minimum Sample Size For Distribution-Free Two-Sided Tolerance Intervals and One-Sided Tolerance Bounds 1529.4 Sample Size for Controlling the Precision of Two-Sided Distribution-Free Tolerance In-tervals and One-Sided Distribution-Free Tolerance Bounds 1539.5 Sample Size to Demonstrate that a Binomial Proportion Exceeds (is Exceeded by) a Specified Value 15410 Sample Size Requirements for Prediction Intervals 16410.1 Prediction Interval Width: The Basic Idea 16410.2 Sample Size for a Normal Distribution Prediction Interval 16510.3 Sample Size for Distribution-Free Prediction Intervals for k of m Future Observations 17011 Basic Case Studies 17211.1 Demonstration that the Operating Temperature of Most Manufactured Devices will not Exceed a Specified Value 17311.2 Forecasting Future Demand for Spare Parts 17711.3 Estimating the Probability of Passing an Environmental Emissions Test 18011.4 Planning a Demonstration Test to Verify that a Radar System has a Satisfactory Prob-ability of Detection 18211.5 Estimating the Probability of Exceeding a Regulatory Limit 18411.6 Estimating the Reliability of a Circuit Board 18911.7 Using Sample Results to Estimate the Probability that a Demonstration Test will be Successful 19111.8 Estimating the Proportion within Specifications for a Two-Variable Problem 19411.9 Determining the Minimum Sample Size for a Demonstration Test 19512 Likelihood-Based Statistical Intervals 19712.1 Introduction to Likelihood-Based Inference 19812.2 Likelihood Function and Maximum Likelihood Estimation 20012.3 Likelihood-Based Confidence Intervals for Single-Parameter Distributions 20312.4 Likelihood-Based Estimation Methods for Location-Scale and Log-Location-Scale Distri-butions 20612.5 Likelihood-Based Confidence Intervals for Parameters and Scalar Functions of Parameters21212.6 Wald-Approximation Confidence Intervals 21612.7 Some Other Likelihood-Based Statistical Intervals 22413 Nonparametric Bootstrap Statistical Intervals 22613.1 Introduction 22713.2 Nonparametric Methods for Generating Bootstrap Samples and Obtaining Bootstrap Estimates 22713.3 Bootstrap Operational Considerations 23113.4 Nonparametric Bootstrap Confidence Interval Methods 23314 Parametric Bootstrap and Other Simulation-Based Statistical Intervals 24514.1 Introduction 24614.2 Parametric Bootstrap Samples and Bootstrap Estimates 24714.3 Bootstrap Confidence Intervals Based on Pivotal Quantities 25014.4 Generalized Pivotal Quantities 25314.5 Simulation-Based Tolerance Intervals for Location-Scale or Log-Location-Scale Distribu-tions 25814.6 Simulation-Based Prediction Intervals and One-Sided Prediction Bounds for k of m Fu-ture Observations from Location-Scale or Log-Location-Scale Distributions 26014.7 Other Simulation and Bootstrap Methods and Application to Other Distributions and Models 26315 Introduction to Bayesian Statistical Intervals 27015.1 Bayesian Inference: Overview 27115.2 Bayesian Inference: an Illustrative Example 27415.3 More About Specification of a Prior Distribution 28315.4 Implementing Bayesian Analyses Using Markov Chain Monte Carlo Simulation 28615.5 Bayesian Tolerance and Prediction Intervals 29116 Bayesian Statistical Intervals for the Binomial, Poisson and Normal Distributions 29716.1 Bayesian Intervals for the Binomial Distribution 29816.2 Bayesian Intervals for the Poisson Distribution 30616.3 Bayesian Intervals for the Normal Distribution 31117 Statistical Intervals for Bayesian Hierarchical Models 32117.1 Bayesian Hierarchical Models and Random Effects 32217.2 Normal Distribution Hierarchical Models 32317.3 Binomial Distribution Hierarchical Models 32517.4 Poisson Distribution Hierarchical Models 32817.5 Longitudinal Repeated Measures Models 32918 Advanced Case Studies 33518.1 Confidence Interval for the Proportion of Defective Integrated Circuits 33618.2 Confidence Intervals for Components of Variance in a Measurement Process 33918.3 Tolerance Interval to Characterize the Distribution of Process Output in the Presence of Measurement Error 34418.4 Confidence Interval for the Proportion of Product Conforming to a Two-Sided Specification34518.5 Confidence Interval for the Treatment Effect in a Marketing Campaign 34818.6 Confidence Interval for the Probability of Detection with Limited Hit-Miss Data 34918.7 Using Prior Information to Estimate the Service-Life Distribution of a Rocket Motor 353Epilogue 357A Notation and Acronyms 360B Generic Definition of Statistical Intervals and Formulas for Computing Coverage Probabilities 367B.1 Introduction 367B.2 Two-sided Confidence Intervals and One-sided Confidence Bounds for Distribution Pa-rameters or a Function of Parameters 368B.3 Two Sided Control-the-Center Tolerance Intervals to Contain at Least a Specified Pro-portion of a Distribution 371B.4 Two Sided Tolerance Intervals to Control Both Tails of a Distribution 374B.5 One-Sided Tolerance Bounds 377B.6 Two-sided Prediction Intervals and One-Sided Prediction Bounds for Future Observations378B.7 Two-Sided Simultaneous Prediction Intervals and One-Sided Simultaneous Prediction Bounds 381B.8 Calibration of Statistical Intervals 383C Useful Probability Distributions 384C.1 Probability Distribution and R Computations 384C.2 Important Characteristics of Random Variables 385C.3 Continuous Distributions 388C.4 Discrete Distributions 398D General Results from Statistical Theory and Some Methods Used to Construct Sta-tistical Intervals 404D.1 cdfs and pdfs of Functions of Random Variables 405D.2 Statistical Error Propagation—The Delta Method 409D.3 Likelihood and Fisher Information Matrices 410D.4 Convergence in Distribution 413D.5 Outline of General ML Theory 415D.6 The CDF pivotal method for constructing confidence intervals 419D.7 Bonferroni approximate statistical intervals 424E Pivotal Methods for Constructing Parametric Statistical Intervals 427E.1 General definition and examples of pivotal quantities 428E.2 Pivotal Quantities for the Normal Distribution 428E.3 Confidence intervals for a Normal Distribution Based on Pivotal Quantities 429E.4 Confidence Intervals for Two Normal Distributions Based on Pivotal Quantities 432E.5 Tolerance Intervals for a Normal Distribution Based on Pivotal Quantities 432E.6 Normal Distribution Prediction Intervals Based on Pivotal Quantities 434E.7 Pivotal Quantities for Log-Location-Scale Distributions 436F Generalized Pivotal Quantities 440F.1 Definition of Generalized Pivotal Quantities 440F.2 A Substitution Method to Obtain GPQs 441F.3 Examples of GPQs for Functions of Location-Scale Distribution Parameters 441F.4 Conditions for Exact Intervals Derived from GPQs 443G Distribution-Free Intervals Based on Order Statistics 446G.1 Basic Statistical Results Used in this Appendix 446G.2 Distribution-Free Confidence Intervals and Bounds for a Distribution Quantile 447G.3 Distribution-Free Tolerance Intervals to Contain a Given Proportion of a Distribution 448G.4 Distribution-Free Prediction Interval to Contain a Specified Ordered Observation From a Future Sample 449G.5 Distribution-Free Prediction Intervals and Bounds to Contain at Least k of m Future Observations From a Future Sample 451H Basic Results from Bayesian Inference Models 455H.1 Basic Statistical Results Used in this Appendix 455H.2 Bayes’ Theorem 456H.3 Conjugate Prior Distributions 456H.4 Jeffreys Prior Distributions 459H.5 Posterior Predictive Distributions 463H.6 Posterior Predictive Distributions Based on Jeffreys Prior Distributions 465I Probability of Successful Demonstration 468I.1 Demonstration Tests Based on a Normal Distribution Assumption 468I.2 Distribution-Free Demonstration Tests 469J Tables 471References 508Subject Index 525