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Statistical Intervals, 2nd Edition

Book Description

 Describes statistical intervals to quantify sampling uncertainty,focusing on key application needs and recently developed methodology in an easy-to-apply format

Statistical intervals provide invaluable tools for quantifying sampling uncertainty. The widely hailed first edition, published in 1991, described the use and construction of the most important statistical intervals. Particular emphasis was given to intervals—such as prediction intervals, tolerance intervals and confidence intervals on distribution quantiles—frequently needed in practice, but often neglected in introductory courses.

Vastly improved computer capabilities over the past 25 years have resulted in an explosion of the tools readily available to analysts. This second edition—more than double the size of the first—adds these new methods in an easy-to-apply format. In addition to extensive updating of the original chapters, the second edition includes new chapters on:

• Likelihood-based statistical intervals

• Nonparametric bootstrap intervals

• Parametric bootstrap and other simulation-based intervals

• An introduction to Bayesian intervals

• Bayesian intervals for the popular binomial, Poisson and normal distributions

• Statistical intervals for Bayesian hierarchical models

• Advanced case studies, further illustrating the use of the newly described methods

New technical appendices provide justification of the methods and pathways to extensions and further applications. A webpage directs readers to current readily accessible computer software and other useful information.

Statistical Intervals: A Guide for Practitioners and Researchers, Second Edition is an up-to-date working guide and reference for all who analyze data, allowing them to quantify the uncertainty in their results using statistical intervals.

William Q. Meeker is Professor of Statistics and Distinguished Professor of Liberal Arts and Sciences at Iowa State University. He is co-author of Statistical Methods for Reliability Data (Wiley, 1998) 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 co-author of Statistical Methods for Reliability Data (Wiley, 1998) and several book chapters. His publications have appeared in the engineering and statistical literature and he has won several research and teaching awards.

Table of Contents

  1. Preface to Second Edition
  2. Preface to First Edition
  3. Acknowledgments
  4. About the Companion Website
  5. Chapter 1: Introduction, Basic Concepts, and Assumptions
    1. Objectives and Overview
    2. 1.1 Statistical Inference
    3. 1.2 Different Types of Statistical Intervals: An Overview
    4. 1.3 The Assumption of Sample Data
    5. 1.4 The Central Role of Practical Assumptions Concerning Representative Data
    6. 1.5 Enumerative Versus Analytic Studies
    7. 1.6 Basic Assumptions for Inferences from Enumerative Studies
    8. 1.7 Considerations in the Conduct of Analytic Studies
    9. 1.8 Convenience and Judgment Samples
    10. 1.9 Sampling People
    11. 1.10 Infinite Population Assumption
    12. 1.11 Practical Assumptions: Overview
    13. 1.12 Practical Assumptions: Further Example
    14. 1.13 Planning the Study
    15. 1.14 The Role of Statistical Distributions
    16. 1.15 The Interpretation of Statistical Intervals
    17. 1.16 Statistical Intervals and Big Data
    18. 1.17 Comment Concerning Subsequent Discussion
    19. Bibliographic Notes
  6. Chapter 2: Overview of Different Types of Statistical Intervals
    1. Objectives and Overview
    2. 2.1 Choice of a Statistical Interval
    3. 2.2 Confidence Intervals
    4. 2.3 Prediction Intervals
    5. 2.4 Statistical Tolerance Intervals
    6. 2.5 Which Statistical Interval do I Use?
    7. 2.6 Choosing a Confidence Level
    8. 2.7 Two-Sided Statistical Intervals Versus One-Sided Statistical Bounds
    9. 2.8 The Advantage of Using Confidence Intervals Instead of Significance Tests
    10. 2.9 Simultaneous Statistical Intervals
    11. Bibliographic Notes
  7. Chapter 3: Constructing Statistical Intervals Assuming a Normal Distribution Using Simple Tabulations
    1. Objectives and Overview
    2. 3.1 Introduction
    3. 3.2 Circuit Pack Voltage Output Example
    4. 3.3 Two-Sided Statistical Intervals
    5. 3.4 One-Sided Statistical Bounds
  8. Chapter 4: Methods for Calculating Statistical Intervals for a Normal Distribution
    1. Objectives and Overview
    2. 4.1 Notation
    3. 4.2 Confidence Interval for the Mean of A Normal Distribution
    4. 4.3 Confidence Interval for The Standard Deviation of a Normal Distribution
    5. 4.4 Confidence Interval for a Normal Distribution Quantile
    6. 4.5 Confidence Interval for the Distribution Proportion Less (Greater) than a Specified Value
    7. 4.6 Statistical Tolerance Intervals
    8. 4.7 Prediction Interval to Contain a Single Future Observation or the Mean of m Future Observations
    9. 4.8 Prediction Interval to Contain at Least k of m Future Observations
    10. 4.9 Prediction Interval to Contain the Standard Deviation of m Future Observations
    11. 4.10 The Assumption of a Normal Distribution
    12. 4.11 Assessing Distribution Normality and Dealing with Nonnormality
    13. 4.12 Data Transformations and Inferences from Transformed Data
    14. 4.13 Statistical Intervals for Linear Regression Analysis
    15. 4.14 Statistical Intervals for Comparing Populations and Processes
    16. Bibliographic Notes
  9. Chapter 5: Distribution-Free Statistical Intervals
    1. Objectives and Overview
    2. 5.1 Introduction
    3. 5.2 Distribution-Free Confidence Intervals and One-Sided Confidence Bounds for a Quantile
    4. 5.3 Distribution-Free Tolerance Intervals and Bounds to Contain a Specified Proportion of a Distribution
    5. 5.4 Prediction Intervals and Bounds to Contain a Specified Ordered Observation in a Future Sample
    6. 5.5 Distribution-Free Prediction Intervals and Bounds to Contain at Least k of m Future Observations
    7. Bibliographic Notes
  10. Chapter 6: Statistical Intervals for a Binomial Distribution
    1. Objectives and Overview
    2. 6.1 Introduction
    3. 6.2 Confidence Intervals for the Actual Proportion Nonconforming in the Sampled Distribution
    4. 6.3 Confidence Interval for the Proportion of Nonconforming Units in a Finite Population
    5. 6.4 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
    6. 6.5 Confidence Intervals for the Quantile of the Distribution of the Number of Nonconforming Units
    7. 6.6 Tolerance Intervals and One-Sided Tolerance Bounds for the Distribution of the Number of Nonconforming Units
    8. 6.7 Prediction Intervals for the Number Nonconforming in a Future Sample
    9. Bibliographic Notes
  11. Chapter 7: Statistical Intervals for a Poisson Distribution
    1. Objectives and Overview
    2. 7.1 Introduction
    3. 7.2 Confidence Intervals for the Event-Occurrence Rate of a Poisson Distribution
    4. 7.3 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
    5. 7.4 Confidence Intervals for the Quantile of the Distribution of the Number of Events in a Specified Amount of Exposure
    6. 7.5 Tolerance Intervals and One-Sided Tolerance Bounds for the Distribution of the Number of Events in a Specified Amount of Exposure
    7. 7.6 Prediction Intervals for the Number of Events in a Future Amount of Exposure
    8. Bibliographic Notes
  12. Chapter 8: Sample Size Requirements for Confidence Intervals on Distribution Parameters
    1. Objectives and Overview
    2. 8.1 Basic Requirements for Sample Size Determination
    3. 8.2 Sample Size for a Confidence Interval for a Normal Distribution Mean
    4. 8.3 Sample Size to Estimate a Normal Distribution Standard Deviation
    5. 8.4 Sample Size to Estimate a Normal Distribution Quantile
    6. 8.5 Sample Size to Estimate a Binomial Proportion
    7. 8.6 Sample Size to Estimate a Poisson Occurrence Rate
    8. Bibliographic Notes
  13. Chapter 9: Sample Size Requirements for Tolerance Intervals, Tolerance Bounds, and Related Demonstration Tests
    1. Objectives and Overview
    2. 9.1 Sample Size for Normal Distribution Tolerance Intervals and One-Sided Tolerance Bounds
    3. 9.2 Sample Size to Pass a One-Sided Demonstration Test Based on Normally Distributed Measurements
    4. 9.3 Minimum Sample Size for Distribution-Free Two-Sided Tolerance Intervals and One-Sided Tolerance Bounds
    5. 9.4 Sample Size for Controlling The Precision of Two-Sided Distribution-Free Tolerance Intervals and One-Sided Distribution-Free Tolerance Bounds
    6. 9.5 Sample Size to Demonstrate that a Binomial Proportion Exceeds (Is Exceeded By) a Specified Value
    7. Bibliographic Notes
  14. Chapter 10: Sample Size Requirements for Prediction Intervals
    1. Objectives and Overview
    2. 10.1 Prediction Interval Width: The Basic Idea
    3. 10.2 Sample Size for a Normal Distribution Prediction Interval
    4. 10.3 Sample Size for Distribution-Free Prediction Intervals for at least k of m Future Observations
    5. Bibliographic Notes
  15. Chapter 11: Basic Case Studies
    1. Objectives and Overview
    2. 11.1 Demonstration That the Operating Temperature of Most Manufactured Devices will not Exceed a Specified Value
    3. 11.2 Forecasting Future Demand for Spare Parts
    4. 11.3 Estimating the Probability of Passing an Environmental Emissions Test
    5. 11.4 Planning A Demonstration Test to Verify that a Radar System has a Satisfactory Probability of Detection
    6. 11.5 Estimating the Probability of Exceeding a Regulatory Limit
    7. 11.6 Estimating the Reliability of a Circuit Board
    8. 11.7 Using Sample Results to Estimate the Probability that a Demonstration Test will be Successful
    9. 11.8 Estimating the Proportion within Specifications for a Two-Variable Problem
    10. 11.9 Determining the Minimum Sample Size for a Demonstration Test
  16. Chapter 12: Likelihood-Based Statistical Intervals
    1. Objectives and Overview
    2. 12.1 Introduction to Likelihood-Based Inference
    3. 12.2 Likelihood Function and Maximum Likelihood Estimation
    4. 12.3 Likelihood-Based Confidence Intervals for Single-Parameter Distributions
    5. 12.4 Likelihood-Based Estimation Methods for Location-Scale and Log-Location-Scale Distributions
    6. 12.5 Likelihood-Based Confidence Intervals for Parameters and Scalar Functions of Parameters
    7. 12.6 Wald-Approximation Confidence Intervals
    8. 12.7 Some Other Likelihood-Based Statistical Intervals
    9. Bibliographic Notes
  17. Chapter 13: Nonparametric Bootstrap Statistical Intervals
    1. Objectives and Overview
    2. 13.1 Introduction
    3. 13.2 Nonparametric Methods for Generating Bootstrap Samples and Obtaining Bootstrap Estimates
    4. 13.3 Bootstrap Operational Considerations
    5. 13.4 Nonparametric Bootstrap Confidence Interval Methods
    6. Bibliographic Notes
  18. Chapter 14: Parametric Bootstrap and Other Simulation-Based Statistical Intervals
    1. Objectives and Overview
    2. 14.1 Introduction
    3. 14.2 Parametric Bootstrap Samples and Bootstrap Estimates
    4. 14.3 Bootstrap Confidence Intervals Based on Pivotal Quantities
    5. 14.4 Generalized Pivotal Quantities
    6. 14.5 Simulation-Based Tolerance Intervals for Location-Scale or Log-Location-Scale Distributions
    7. 14.6 Simulation-Based Prediction Intervals and One-Sided Prediction Bounds for at least k of m Future Observations from Location-Scale or Log-Location-Scale Distributions
    8. 14.7 Other Simulation and Bootstrap Methods and Application to Other Distributions and Models
    9. Bibliographic Notes
  19. Chapter 15: Introduction to Bayesian Statistical Intervals
    1. Objectives and Overview
    2. 15.1 Bayesian Inference: Overview
    3. 15.2 Bayesian Inference: An Illustrative Example
    4. 15.3 More About Specification of A Prior Distribution
    5. 15.4 Implementing Bayesian Analyses using Markov Chain Monte Carlo Simulation
    6. 15.5 Bayesian Tolerance and Prediction Intervals
    7. Bibliographic Notes
  20. Chapter 16: Bayesian Statistical Intervals for the Binomial, Poisson, and Normal Distributions
    1. Objectives and Overview
    2. 16.1 Bayesian Intervals for the Binomial Distribution
    3. 16.2 Bayesian Intervals for the Poisson Distribution
    4. 16.3 Bayesian Intervals for the Normal Distribution
    5. Bibliographic Notes
  21. Chapter 17: Statistical Intervals for Bayesian Hierarchical Models
    1. Objectives and Overview
    2. 17.1 Bayesian Hierarchical Models and Random Effects
    3. 17.2 Normal Distribution Hierarchical Models
    4. 17.3 Binomial Distribution Hierarchical Models
    5. 17.4 Poisson Distribution Hierarchical Models
    6. 17.5 Longitudinal Repeated Measures Models
    7. Bibliographic Notes
  22. Chapter 18: Advanced Case Studies
    1. Objectives and Overview
    2. 18.1 Confidence Interval for the Proportion of Defective Integrated Circuits
    3. 18.2 Confidence Intervals for Components of Variance in a Measurement Process
    4. 18.3 Tolerance Interval to Characterize the Distribution of Process Output in the Presence of Measurement Error
    5. 18.4 Confidence Interval for the Proportion of Product Conforming to a Two-Sided Specification
    6. 18.5 Confidence Interval for the Treatment Effect in a Marketing Campaign
    7. 18.6 Confidence Interval for the Probability of Detection with Limited Hit/Miss Data
    8. 18.7 Using Prior Information to Estimate the Service-Life Distribution of a Rocket Motor
    9. Bibliographic Notes
  23. Epilogue
  24. Appendix A: Notation and Acronyms
  25. Appendix B: Generic Definition of Statistical Intervals and Formulas for Computing Coverage Probabilities
    1. B.1 Introduction
    2. B.2 Two-Sided Confidence Intervals and One-Sided Confidence Bounds for Distribution Parameters or a Function of Parameters
    3. B.3 Two-Sided Control-the-Center Tolerance Intervals to Contain at Least a Specified Proportion of a Distribution
    4. B.4 Two-Sided Tolerance Intervals to Control Both Tails of a Distribution
    5. B.5 One-Sided Tolerance Bounds
    6. B.6 Two-Sided Prediction Intervals and One-Sided Prediction Bounds for Future Observations
    7. B.7 Two-Sided Simultaneous Prediction Intervals and One-Sided Simultaneous Prediction Bounds
    8. B.8 Calibration of Statistical Intervals
  26. Appendix C: Useful Probability Distributions
    1. Introduction
    2. C.1 Probability Distributions and R Computations
    3. C.2 Important Characteristics of Random Variables
    4. C.3 Continuous Distributions
    5. C.4 Discrete Distributions
  27. Appendix D: General Results from Statistical Theory and Some Methods Used to Construct Statistical Intervals
    1. Introduction
    2. D.1 The Cdfs and Pdfs of Functions of Random Variables
    3. D.2 Statistical Error Propagation—The Delta Method
    4. D.3 Likelihood and Fisher Information Matrices
    5. D.4 Convergence in Distribution
    6. D.5 Outline of General Maximum Likelihood Theory
    7. D.6 The Cdf Pivotal Method for Obtaining Confidence Intervals
    8. D.7 Bonferroni Approximate Statistical Intervals
  28. Appendix E: Pivotal Methods for Constructing Parametric Statistical Intervals
    1. Introduction
    2. E.1 General Definition and Examples of Pivotal Quantities
    3. E.2 Pivotal Quantities for the Normal Distribution
    4. E.3 Confidence Intervals for a Normal Distribution Based on Pivotal Quantities
    5. E.4 Confidence Intervals for two Normal Distributions Based on Pivotal Quantities
    6. E.5 Tolerance Intervals for a Normal Distribution Based on Pivotal Quantities
    7. E.6 Normal Distribution Prediction Intervals Based on Pivotal Quantities
    8. E.7 Pivotal Quantities for Log-Location-Scale Distributions
  29. Appendix F: Generalized Pivotal Quantities
    1. Introduction
    2. F.1 Definition of a Generalized Pivotal Quantity
    3. F.2 A Substitution Method to Obtain Generalized Pivotal Quantities
    4. F.3 Examples of Generalized Pivotal Quantities for Functions of Location-Scale Distribution Parameters
    5. F.4 Conditions for Exact Confidence Intervals Derived from Generalized Pivotal Quantities
  30. Appendix G: Distribution-Free Intervals Based on Order Statistics
    1. Introduction
    2. G.1 Basic Statistical Results Used in this Appendix
    3. G.2 Distribution-Free Confidence Intervals and Bounds for a Distribution Quantile
    4. G.3 Distribution-Free Tolerance Intervals to Contain a Given Proportion of a Distribution
    5. G.4 Distribution-Free Prediction Interval to Contain a Specified Ordered Observation from a Future Sample
    6. G.5 Distribution-Free Prediction Intervals and Bounds to Contain at Least k of m Future Observations from a Future Sample
  31. Appendix H: Basic Results from Bayesian Inference Models
    1. Introduction
    2. H.1 Basic Results Used in this Appendix
    3. H.2 Bayes’ Theorem
    4. H.3 Conjugate Prior Distributions
    5. H.4 Jeffreys Prior Distributions
    6. H.5 Posterior Predictive Distributions
    7. H.6 Posterior Predictive Distributions Based on Jeffreys Prior Distributions
  32. Appendix I: Probability of Successful Demonstration
    1. I.1 Demonstration Tests Based on a Normal Distribution Assumption
    2. I.2 Distribution-Free Demonstration Tests
  33. Appendix J: Tables
  34. References
  35. Index
  36. Wiley Series in Probability and Statistics
  37. EULA