# Statistical Intervals: A Guide for Practitioners and Researchers, 2nd Edition

## Books 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.

Preface to Second Edition iii

Preface to First Edition vii

Acknowledgments x

1 Introduction, Basic Concepts, and Assumptions 1

1.1 Statistical Inference 2

1.2 Different Types of Statistical Intervals: An Overview 2

1.3 The Assumption of Sample Data 3

1.4 The Central Role of Practical Assumptions Concerning Representative Data 4

1.5 Enumerative Versus Analytic Studies 5

1.6 Basic Assumptions for Enumerative Studies 7

1.7 Considerations in the Conduct of Analytic Studies 10

1.8 Convenience and Judgment Samples 11

1.9 Sampling People 12

1.10 Infinite Population Assumptions 13

1.11 Practical Assumptions: Overview 14

1.12 Practical Assumptions: Further Example 14

1.13 Planning the Study 17

1.14 The Role of Statistical Distributions 17

1.15 The Interpretation of Statistical Intervals 18

1.16 Statistical Intervals and Big Data 19

1.17 Comment Concerning Subsequent Discussion 19

2 Overview of Different Types of Statistical Intervals 21

2.1 Choice of a Statistical Interval 21

2.2 Confidence Intervals 23

2.3 Prediction Intervals 24

2.4 Statistical Tolerance Intervals 26

2.5 Which Statistical Interval Do I Use? 27

2.6 Choosing a Confidence Level 28

2.7 Two-Sided Statistical Intervals Versus One-Sided Statistical Bounds 29

2.8 The Advantage of Using Confidence Intervals Instead of Significance Tests 30

2.9 Simultaneous Statistical Intervals 31

3 Constructing Statistical Intervals Assuming a Normal Distribution Using Simple Tabulations 33

3.1 Introduction 34

3.2 Circuit Pack Voltage Output Example 35

3.3 Two-Sided Statistical Intervals 36

3.4 One-Sided Statistical Bounds 38

4 Methods for Calculating Statistical Intervals for a Normal Distribution 43

4.1 Notation 44

4.2 Confidence Interval for the Mean of a Normal Distribution 45

4.3 Confidence Interval for the Standard Deviation of a Normal Distribution 45

4.4 Confidence Interval for a Normal Distribution Quantile 46

4.5 Confidence Interval for the Distribution Proportion Less (Greater) Than a Specified Value 47

4.6 Statistical Tolerance Intervals 48

4.7 Prediction Interval to Contain a Single Future Observation or the Mean of m Future Observations 50

4.8 Prediction Interval to Contain at least k of m Future Observations 51

4.9 Prediction Interval to Contain the Standard Deviation of m Future Observations 52

4.10 The Assumption of a Normal Distribution 53

4.11 Assessing Distribution Normality and Dealing with Nonnormality 54

4.12 Data Transformations and Inferences from Transformed Data 57

4.13 Statistical Intervals for Linear Regression Analysis 60

4.14 Statistical Intervals for Comparing Populations and Processes 62

5 Distribution-Free Statistical Intervals 65

5.1 Introduction 66

5.2 Distribution-Free Confidence Intervals and One-Sided Confidence Bounds for a Quantile 68

5.3 Distribution-Free Tolerance Intervals and Bounds to Contain a Specified Proportion of a Distribution 78

5.4 Prediction Intervals to Contain a Specified Ordered Observation in a Future Sample 81

5.5 Distribution-Free Prediction Intervals and Bounds to Contain at Least k of m Future Observations 84

6 Statistical Intervals for a Binomial Distribution 89

6.1 Introduction to Binomial Distribution Statistical Intervals 90

6.2 Confidence Intervals for the Actual Proportion Nonconforming in the Sampled Distribution 92

6.3 Confidence Interval for the Proportion of Nonconforming Units in a Finite Population 102

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 104

6.5 Confidence Intervals for the Quantile of the Distribution of the Number of Nonconforming Units 105

6.6 Tolerance Intervals and One-Sided Tolerance Bounds for the Distribution of the Number of Nonconforming Units 107

6.7 Prediction Intervals for the Number Nonconforming in a Future Sample 108

7 Statistical Intervals for a Poisson Distribution 115

7.1 Introduction 116

7.2 Confidence Intervals for the Event-Occurrence Rate of a Poisson Distribution 117

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 124

7.4 Confidence Intervals for the Quantile of the Distribution of the Number of Events in a Specified Amount of Exposure 125

7.5 Tolerance Intervals and One-Sided Tolerance Bounds for the Distribution of the Number of Events in a Specified Amount of Exposure 127

7.6 Prediction Intervals for the Number of Events in a Future Amount of Exposure 128

8 Sample Size Requirements for Confidence Intervals on Distribution Parameters 135

8.1 Basic Requirements for Sample Size Determination 136

8.2 Sample Size for a Confidence Interval for a Normal Distribution Mean 137

8.3 Sample Size to Estimate a Normal Distribution Standard Deviation 141

8.4 Sample Size to Estimate a Normal Distribution Quantile 143

8.5 Sample Size to Estimate a Binomial Proportion 143

8.6 Sample Size to Estimate a Poisson Occurrence Rate 144

9 Sample Size Requirements for Tolerance Intervals, Tolerance Bounds, and Related Demonstration Tests 148

9.1 Sample Size for Normal Distribution Tolerance Intervals and One-Sided Tolerance Bounds148

9.2 Sample Size to Pass a One-Sided Demonstration Test Based on Normally Distributed Measurements 150

9.3 Minimum Sample Size For Distribution-Free Two-Sided Tolerance Intervals and One-Sided Tolerance Bounds 152

9.4 Sample Size for Controlling the Precision of Two-Sided Distribution-Free Tolerance In-tervals and One-Sided Distribution-Free Tolerance Bounds 153

9.5 Sample Size to Demonstrate that a Binomial Proportion Exceeds (is Exceeded by) a Specified Value 154

10 Sample Size Requirements for Prediction Intervals 164

10.1 Prediction Interval Width: The Basic Idea 164

10.2 Sample Size for a Normal Distribution Prediction Interval 165

10.3 Sample Size for Distribution-Free Prediction Intervals for k of m Future Observations 170

11 Basic Case Studies 172

11.1 Demonstration that the Operating Temperature of Most Manufactured Devices will not Exceed a Specified Value 173

11.2 Forecasting Future Demand for Spare Parts 177

11.3 Estimating the Probability of Passing an Environmental Emissions Test 180

11.4 Planning a Demonstration Test to Verify that a Radar System has a Satisfactory Prob-ability of Detection 182

11.5 Estimating the Probability of Exceeding a Regulatory Limit 184

11.6 Estimating the Reliability of a Circuit Board 189

11.7 Using Sample Results to Estimate the Probability that a Demonstration Test will be Successful 191

11.8 Estimating the Proportion within Specifications for a Two-Variable Problem 194

11.9 Determining the Minimum Sample Size for a Demonstration Test 195

12 Likelihood-Based Statistical Intervals 197

12.1 Introduction to Likelihood-Based Inference 198

12.2 Likelihood Function and Maximum Likelihood Estimation 200

12.3 Likelihood-Based Confidence Intervals for Single-Parameter Distributions 203

12.4 Likelihood-Based Estimation Methods for Location-Scale and Log-Location-Scale Distri-butions 206

12.5 Likelihood-Based Confidence Intervals for Parameters and Scalar Functions of Parameters212

12.6 Wald-Approximation Confidence Intervals 216

12.7 Some Other Likelihood-Based Statistical Intervals 224

13 Nonparametric Bootstrap Statistical Intervals 226

13.1 Introduction 227

13.2 Nonparametric Methods for Generating Bootstrap Samples and Obtaining Bootstrap Estimates 227

13.3 Bootstrap Operational Considerations 231

13.4 Nonparametric Bootstrap Confidence Interval Methods 233

14 Parametric Bootstrap and Other Simulation-Based Statistical Intervals 245

14.1 Introduction 246

14.2 Parametric Bootstrap Samples and Bootstrap Estimates 247

14.3 Bootstrap Confidence Intervals Based on Pivotal Quantities 250

14.4 Generalized Pivotal Quantities 253

14.5 Simulation-Based Tolerance Intervals for Location-Scale or Log-Location-Scale Distribu-tions 258

14.6 Simulation-Based Prediction Intervals and One-Sided Prediction Bounds for k of m Fu-ture Observations from Location-Scale or Log-Location-Scale Distributions 260

14.7 Other Simulation and Bootstrap Methods and Application to Other Distributions and Models 263

15 Introduction to Bayesian Statistical Intervals 270

15.1 Bayesian Inference: Overview 271

15.2 Bayesian Inference: an Illustrative Example 274

15.3 More About Specification of a Prior Distribution 283

15.4 Implementing Bayesian Analyses Using Markov Chain Monte Carlo Simulation 286

15.5 Bayesian Tolerance and Prediction Intervals 291

16 Bayesian Statistical Intervals for the Binomial, Poisson and Normal Distributions 297

16.1 Bayesian Intervals for the Binomial Distribution 298

16.2 Bayesian Intervals for the Poisson Distribution 306

16.3 Bayesian Intervals for the Normal Distribution 311

17 Statistical Intervals for Bayesian Hierarchical Models 321

17.1 Bayesian Hierarchical Models and Random Effects 322

17.2 Normal Distribution Hierarchical Models 323

17.3 Binomial Distribution Hierarchical Models 325

17.4 Poisson Distribution Hierarchical Models 328

17.5 Longitudinal Repeated Measures Models 329

18.1 Confidence Interval for the Proportion of Defective Integrated Circuits 336

18.2 Confidence Intervals for Components of Variance in a Measurement Process 339

18.3 Tolerance Interval to Characterize the Distribution of Process Output in the Presence of Measurement Error 344

18.4 Confidence Interval for the Proportion of Product Conforming to a Two-Sided Specification345

18.5 Confidence Interval for the Treatment Effect in a Marketing Campaign 348

18.6 Confidence Interval for the Probability of Detection with Limited Hit-Miss Data 349

18.7 Using Prior Information to Estimate the Service-Life Distribution of a Rocket Motor 353

Epilogue 357

A Notation and Acronyms 360

B Generic Definition of Statistical Intervals and Formulas for Computing Coverage Probabilities 367

B.1 Introduction 367

B.2 Two-sided Confidence Intervals and One-sided Confidence Bounds for Distribution Pa-rameters or a Function of Parameters 368

B.3 Two Sided Control-the-Center Tolerance Intervals to Contain at Least a Specified Pro-portion of a Distribution 371

B.4 Two Sided Tolerance Intervals to Control Both Tails of a Distribution 374

B.5 One-Sided Tolerance Bounds 377

B.6 Two-sided Prediction Intervals and One-Sided Prediction Bounds for Future Observations378

B.7 Two-Sided Simultaneous Prediction Intervals and One-Sided Simultaneous Prediction Bounds 381

B.8 Calibration of Statistical Intervals 383

C Useful Probability Distributions 384

C.1 Probability Distribution and R Computations 384

C.2 Important Characteristics of Random Variables 385

C.3 Continuous Distributions 388

C.4 Discrete Distributions 398

D General Results from Statistical Theory and Some Methods Used to Construct Sta-tistical Intervals 404

D.1 cdfs and pdfs of Functions of Random Variables 405

D.2 Statistical Error Propagation—The Delta Method 409

D.3 Likelihood and Fisher Information Matrices 410

D.4 Convergence in Distribution 413

D.5 Outline of General ML Theory 415

D.6 The CDF pivotal method for constructing confidence intervals 419

D.7 Bonferroni approximate statistical intervals 424

E Pivotal Methods for Constructing Parametric Statistical Intervals 427

E.1 General definition and examples of pivotal quantities 428

E.2 Pivotal Quantities for the Normal Distribution 428

E.3 Confidence intervals for a Normal Distribution Based on Pivotal Quantities 429

E.4 Confidence Intervals for Two Normal Distributions Based on Pivotal Quantities 432

E.5 Tolerance Intervals for a Normal Distribution Based on Pivotal Quantities 432

E.6 Normal Distribution Prediction Intervals Based on Pivotal Quantities 434

E.7 Pivotal Quantities for Log-Location-Scale Distributions 436

F Generalized Pivotal Quantities 440

F.1 Definition of Generalized Pivotal Quantities 440

F.2 A Substitution Method to Obtain GPQs 441

F.3 Examples of GPQs for Functions of Location-Scale Distribution Parameters 441

F.4 Conditions for Exact Intervals Derived from GPQs 443

G Distribution-Free Intervals Based on Order Statistics 446

G.1 Basic Statistical Results Used in this Appendix 446

G.2 Distribution-Free Confidence Intervals and Bounds for a Distribution Quantile 447

G.3 Distribution-Free Tolerance Intervals to Contain a Given Proportion of a Distribution 448

G.4 Distribution-Free Prediction Interval to Contain a Specified Ordered Observation From a Future Sample 449

G.5 Distribution-Free Prediction Intervals and Bounds to Contain at Least k of m Future Observations From a Future Sample 451

H Basic Results from Bayesian Inference Models 455

H.1 Basic Statistical Results Used in this Appendix 455

H.2 Bayes’ Theorem 456

H.3 Conjugate Prior Distributions 456

H.4 Jeffreys Prior Distributions 459

H.5 Posterior Predictive Distributions 463

H.6 Posterior Predictive Distributions Based on Jeffreys Prior Distributions 465

I Probability of Successful Demonstration 468

I.1 Demonstration Tests Based on a Normal Distribution Assumption 468

I.2 Distribution-Free Demonstration Tests 469

J Tables 471

References 508

Subject Index 525

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