CONTENTS
Chapter 1 Histograms and Empirical Distributions = 1
1.1 Introduction = 1
1.2 Empirical Distributions = 2
1.3 Measures of Central Tendency = 7
1.4 Measures of Variation = 8
1.5 Computation of the Mean and Standard Deviation of the Data from the Frequency Table = 9
Chapter 2 Random Variables and Probability Distributions = 13
2.1 Introduction = 13
2.2 Set of All Possible Outcomes of the Experiment - the Sample Space = 14
2.3 Events = 18
2.4 Random Variables = 19
2.5 Probability = 24
2.6 Cumulative Distribution Functions = 27
2.7 Discrete Probability Distributions = 33
2.8 Continuous Random Variables and Density Functions = 37
2.9 Expectation = 43
2.10 Moments = 47
2.11 Some Properties of Functions of Random Variables = 51
2.12 Bivariate Probability Distributions = 52
2.13 Conditional Probability and Independent Events = 57
2.14 Independent Random Variables and a Random Sample = 60
2.15 Conditional Probability Distributions = 64
Chapter 3 The Normal Distribution = 75
3.1 Definitions = 75
3.2 The Mean and Variance of the Normal Distribution = 76
3.2.1. Evaluation of the Mean and Variance = 78
3.3 Tables of the Normal Integral = 79
3.4 Combinations of Normally Distributed Variables = 82
3.5 The Standardized Normal Random Variable = 84
3.6 The Distribution of the Sample Mean = 84
3.7 Tolerances = 86
3.8 Tolerances in "Complex Items" = 93
3.9 The Central Limit Theorem = 99
Chapter 4 Other Probability Distributions = 106
4.1 Introduction = 106
4.2 The Chi-Square Distribution = 107
4.2.1 The Chi-Square Random Variable = 107
4.2.2 The Addition Theorem = 110
4.2.3 The Distribution of the Sample Variance, S² = 111
4.3 The t Distribution = 114
4.3.1 The t Random Variable = 114
4.3.2 The Distribution of ($$\bar X$$ - μ)$$\sqrt n$$／S = 117
4.3.3 The Distribution of the Difference Between Two Sample Means = 118
4.4 The F Distribution = 120
4.4.1 The F Random Variable = 120
4.4.2 The Distribution of the Ratio of Two Sample Variances = 122
4.5 The Binomial Distribution = 123
4.5.1 The Binomial Random Variable = 123
4.5.2 Tables of the Binomial Probability Distribution = 126
4.5.3 The Normal Approximation to the Binomial = 127
4.5.4 The Arc Sine Transformation = 128
4.5.5 The Poisson Approximation to the Binomial = 128
Chapter 5 Decision Making = 134
5.1 Introduction = 134
5.2 Decision Making Without Experimentation = 134
5.2.1 Action Space = 135
5.2.2 States of Nature = 136
5.2.3 Loss Function = 137
5.2.4 Criteria for Choosing Among Actions = 138
5.2.5 Bayes Principle = 139
5.2.6 Evaluation of the Loss Function for the Rockwell Hardness Example = 141
5.2.7 Further Examples = 145
5.3 Decision Making With Experimentation = 148
5.3.1 Decision Procedures = 148
5.3.2 Risk Function = 150
5.3.3 Bayes Decision Procedures = 152
5.3.4 Calculation of the Posterior Distribution = 155
5.3.4.1 Derivation of Posterior Distribution = 158
5.3.5 Further Examples = 158
5.3.6 Assessment of the Bayesian Approach = 161
5.4 Significance Tests = 163
5.4.1 The Operating Characteristic Curve = 165
5.4.2 Comparison of the OC Curve with the Risk Function = 167
5.4.3 Comparison of OC Curves = 168
5.4.4 Tests of Hypotheses = 172
5.4.5 One- and Two-Sided Procedures = 174
Chapter 6 Tests of Hypotheses about a Single Parameter = 183
6.1 Test of the Hypothesis that the Mean of a Normal Distribution Has a Specified Value when the Standard Deviation Is Known = 183
6.1.1 Choice of an OC Curve = 183
6.1.2 Tables and Charts for Determining Decision Rules = 184
6.1.2.1 Tables and Charts for Two-Sided Procedures = 185
6.1.2.2 Summary for Two-Sided Procedures Using Tables and Charts = 188
6.1.2.3 Tables and Charts for One-Sided Procedures = 189
6.1.2.4 Summary for One-Sided Procedures Using Tables and Charts = 191
6.1.2.5 Tables and Charts for OC Curves = 192
6.1.3 Analytical Determination of Decision Rules = 192
6.1.3.1 Acceptance Regions and Sample Sizes = 192
6.1.3.2 The OC Curve = 196
6.1.4 Example = 197
6.2. Test of the Hypothesis that the Mean of a Normal Distribution Has a Specified Value when the Standard Deviation is Unknown = 198
6.2.1 Choice of an OC Curve = 198
6.2.2 Tables and Charts for Carrying Out t Tests = 199
6.2.2.1 Tables and Charts for Two-Sided Procedures = 200
6.2.2.2 Summary for Two-Sided Procedures = 200
6.2.2.3 Tables and Charts for One-Sided Procedures = 202
6.2.2.4 Summary for One-Sided Procedures = 204
6.2.2.5 Tables and Charts for OC Curves = 205
6.2.3. Examples of t Tests = 206
6.3 Test of the Hypothesis that the Standard Deviation of a Normal Distribution Has a Specified Value = 207
6.3.1 Choice of an OC Curve = 207
6.3.2 Charts and Tables to Design Tests of Dispersion = 208
6.3.2.1 Tables and Charts for Two-Sided Procedures = 208
6.3.2.2 Summary for Two-Sided Procedures Using Tables and Charts = 209
6.3.2.3 Tables and Charts for One-Sided Procedures = 210
6.3.2.4 Summary for One-Sided Procedures Using Tables and Charts = 213
6.3.2.5 Tables and Charts for OC Curves = 214
6.3.3 Analytical Treatment for Chi-Square Tests = 215
6.3.4 Example = 216
Chapter 7 Tests of Hypotheses about Two Parameters = 225
7.1 Test of the Hypothesis that the Means of Two Normal Distributions Are Equal when Both Standard Deviations Are Known = 225
7.1.1 Choice of an OC Curve = 225
7.1.2 Tables and Charts for Determining Decision Rules = 226
7.1.2.1 Tables and Charts for Two-Sided Procedures = 226
7.1.2.2 Summary for Two-Sided Procedures Using Tables and Charts = 228
7.1.2.3 Summary for One-Sided Procedures Using Tables and Charts = 229
7.1.2.4 Tables and Charts for OC Curves = 230
7.1.3 Analytical Determination of Decision Rules = 230
7.1.3.1. Acceptance Regions and Sample Sizes = 230
7.1.3.2 The OC Curve = 232
7.1.4 Example = 234
7.2 Test of the Hypothesis that the Means of Two Normal Distributions Are Equal, Assuming that the Standard Deviations Are Unknown but Equal = 235
7.2.1 Choice of an OC Curve = 235
7.2.2 Tables and Charts for Carrying out Two-Sample t Tests = 235
7.2.2.1 Tables and Charts for Two-Sided Procedures = 236
7.2.2.2 Summary for Two-Sided Procedures Using Tables and Charts = 237
7.2.2.3 Summary for One-Sided Procedures Using Tables and Charts = 237
7.2.2.4 Tables and Charts for OC Curves = 238
7.2.3 Example = 239
7.3 Test of the Hypothesis that the Means of Two Normal Distributions Are Equal, Assuming that the Standard Deviations Are Unknown and Not Necessarily Equal = 240
7.3.1 Test Procedure = 240
7.3.2 Example = 241
7.4 Test for Equality of Means when the Observations Are Paired = 242
7.4.1 Test Procedure = 242
7.4.2 Example = 244
7.5 Non-parametric Tests = 246
7.5.1 The Sign Test = 246
7.5.2 The Wilcoxon Signed Rank Test = 249
7.5.3 Wilcoxon Test for Two Independent Samples = 251
7.6 Test of the Hypothesis that the Standard Deviations of Two Normal Distributions Are Equal = 254
7.6.1 Choice of an OC Curve = 254
7.6.2 Charts and Tables for Carrying out F Tests = 255
7.6.2.1 Tables and Charts for Two-Sided Procedures = 255
7.6.2.2 Summary for Two-Sided Procedures Using Tables and Charts = 256
7.6.2.3 Tables and Charts for One-Sided Procedures = 257
7.6.2.4 Summary for One-Sided Procedures Using Tables and Charts = 258
7.6.2.5 Tables and Charts for OC Curves = 258
7.6.3 Analytical Treatment for Tests = 260
7.6.4 Example = 262
7.7 Cochran's Test for the Homogeneity of Variances = 263
Chapter 8 Estimation = 279
8.1 Introduction = 279
8.2 Point Estimation = 279
8.2.1 Comparison of Estimators = 280
8.2.2 Unbiased Estimators = 283
8.2.3 Consistent Estimators = 284
8.2.4 Efficient Unbiased Estimators = 285
8.2.5 Estimation by the Method of Maximum Likelihood = 286
8.2.6 Estimation by the Method of Moments = 290
8.2.7 Estimation by the Method of Bayes = 292
8.3 Confidence Interval Estimation = 294
8.3.1 Confidence Interval for the Mean of a Normal Distribution when the Standard Deviation Is Known = 295
8.3.2 Confidence Interval for the Mean of a Normal Distribution when the Standard Deviation Is Unknown = 296
8.3.3 Confidence Interval for the Standard Deviation of a Normal Distribution = 297
8.3.4 Confidence Interval for the Difference between the Means of Two Normal Distributions when the Standard Deviations Are Both Known = 298
8.3.5 Confidence Interval for the Difference between the Means of Two Normal Distributions where the Standard Deviations Are Both Unknown but Equal = 300
8.3.6 Confidence Interval for the Ratio of Standard Deviations of Two Normal Distributions = 301
8.3.7 A Table of Point Estimates and interval Estimates = 302
8.3.8 Approximate Confidence Intervals = 302
8.3.9 Simultaneous Confidence Intervals = 304
8.3.10 Bayesian Intervals = 308
8.4 Statistical Tolerance Limits = 309
8.4.1 Statistical Tolerance Limits Based on the Normal Distribution = 309
8.4.2 One-Sided Statistical Tolerance Limits Based on the Normal Distribution = 310
8.4.3 Distribution-Free Tolerance Limits = 310
Chapter 9 Fitting Straight Lines = 325
9.1 Introduction = 325
9.2 Types of Linear Relationships = 328
9.3 Least Squares Estimators of the Slope and Intercept = 331
9.3.1 Formulation of the Problem and Results = 331
9.3.2 Theory = 333
9.4 Confidence Interval Estimators of the Slope and Intercept = 336
9.4.1 Formulation of the Problem and Results = 336
9.4.2 Theory = 337
9.5 Point Estimators and Confidence Interval Estimators of the Average Value of Y for a Given x = 338
9.5.1 Formulation of the Problem and Results = 338
9.5.2 Theory = 339
9.6 Point Estimators and Interval Estimators of the Independent Variable x Associated with an Observation on the Dependent Variable Y = 340
9.7 Prediction Interval for a Future Observation on the Dependent Variable = 342
9.7.1 Formulation of the Problem and Results = 342
9.7.2 Theory = 343
9.8 Tests of Hypotheses about the Slope and Intercept = 345
9.9 Estimation of the Slope B when A Is Known to be Zero = 346
9.10 Ascertaining Linearity = 349
9.11 Transformation to a Straight Line = 351
9.12 Work Sheets for Fitting Straight Lines = 352
9.13 Illustrative Examples = 352
9.14 Correlation = 362
Chapter 10 Analysis of Variance = 377
10.1 Introduction = 377
10.2 Model for the One-Way Classification = 377
10.2.1 Fixed Effects Model = 377
10.2.2 Random Effects Model = 380
10.2.3 Further Examples of Fixed Effects and of the Random Effects Models = 380
10.2.4 Computational Procedure, One-Way Classification = 381
10.2.5 The Analysis of Variance Procedure = 383
10.2.5.1 A Heuristic Justification = 383
10.2.5.2 The Partition Theorem = 383
10.2.6 Analysis of the Fixed Effects Model, One-Way Classification = 385
10.2.7 The OC Curve of the Analysis of Variance for the Fixed Effects Model = 389
10.2.8 Example Using the Fixed Effects Model = 395
10.2.9 Analysis of the Random Effects Model = 396
10.2.10 The OC Curve for the Random Effects Model = 397
10.2.11 Example Using the Random Effects Model = 402
10.2.12 Randomization Tests in the Analysis of Variance = 403
10.3 Two-Way Analysis of Variance, One Observation per Combination = 405
10.3.1 Fixed Effects Model = 405
10.3.2 Random Effects Model = 408
10.3.3 Mixed Fixed Effects and Random Effects Model = 409
10.3.4 Computational Procedure, Two-Way Classification, One Observation per Combination = 409
10.3.5 Analysis of the Fixed Effects Model, Two-Way Classification, One Observation per Combination = 411
10.3.6 The OC Curve of the Analysis of Variance for the Fixed Effects Model, Two-Way Classification, One Observation per Combination = 413
10.3.7 Example Using the Fixed Effects Model = 414
10.3.8 Analysis of the Random Effects Model, Two-Way Classification, One Observation per Combination = 416
10.3.9 The OC Curve for the Random Effects Model, Two-Way Classification = 417
10.3.10 Example Using the Random Effects Model = 418
10.3.11 Analysis of the Mixed Effects Model, Two-Way Classification, One Observation per Combination = 418
10.3.12 The OC Curve of the Analysis of Variance for the Mixed Effects Model, Two-Way Classification, One Observation per Cell = 420
10.3.13 Example Using the Mixed Effects Model = 420
10.4 Two-Way Analysis of Variance, n Observations per Combination = 421
10.4.1 Description of the Various Models = 421
10.4.2 Computational Procedure, Two-Way Classification, n Observations per Combination = 423
10.4.3 Analysis of the Fixed Effects Model, Two-Way Classification, n Observations per Combination = 424
1O.4.4 The OC Curve of the Analysis of Variance for the Fixed Effects Model, Two-Way Classification, n Observations per Cell = 427
10.4.5 Example Using the Fixed Effects Model, Two-Way Classification, Three Observations per Combination = 428
10.4.6 Analysis of the Random Effects Model, Two-Way Classification, n Observations per Combination = 430
10.4.7 The OC Curve of the Random Effects Model, Two-Way Classification, n Observations per Combination = 431
10.4.8 Example Using the Random Effects Model = 432
10.4.9 Analysis of the Mixed Effects Model, Two-Way Classification, n Observations per Cell = 432
10.4.10 The OC Curve of the Analysis of Variance for the Mixed Effects Model, Two-Way Classification, n Observations per Cell = 434
10.4.11 Example Using the Mixed Effects Model = 434
10.5 Summary of Models and Tests = 436
Chapter ll Some Further Techniques of Data Analysis = 452
11.1 Introduction = 452
11.2 Qualitative Techniques for Determining the Form of a Distribution = 452
11.3 Quantitative Techniques for Determining the Form of a Distribution = 454
11.3.1 The Kolmogorov-Smirnov Test = 454
11.3.2 The Chi-Square Goodness of Fit Test = 458
11.4 Chi-Square Tests = 460
11.4.1 The Hypothesis Completely Specifies the Theoretical Frequency = 461
11.4.2 Dichotomous Data = 461
11.4.3 Test of Independence in a Two-Way Table = 463
11.4.4 Computing Form for Test of Independence in a 2 ×2 Table = 464
11.5 Comparison of Two Percentages = 465
11.6 Confidence Intervals for Proportion = 466
Chapter 12 Statistical Quality Control : Control Charts = 472
12.1 Introduction = 472
12.2 An Overview of Control Charts = 473
12.3 Control Chart for Variables : $$\bar X$$-Charts = 474
12.3.1 Statistical Concepts = 474
12.3.2 Estimate of $$\bar X'$$ = 475
12.3.3 Estimate of $$\sigma '$$ by $$\bar \sigma $$ = 476
12.3.4 Estimate of $$\sigma '$$ by $$\bar R$$ = 476
12.3.5 Starting a Control Chart for $$\bar X$$ = 478
12.3.6 Relation between Natural Tolerance Limits and Specification Limits = 479
12.3.7 Interpretation of Control Charts for $$\bar X$$ = 480
12.4 R Charts and σ Charts = 481
12.4.1 Statistical Concepts = 481
12.4.2 Setting up a Control Chart for R or σ = 483
12.5 Example of $$\bar X$$ and R Chart = 483
12.6 Control Chart For Fraction Defective = 485
12.6.1 Relation between Control Charts Based on Variables Data and Charts Based on Attributes Data = 485
12.6.2 Statistical Theory = 486
12.6.3 Starting the Control Chart = 487
12.6.4 Continuing the p Chart = 488
12.6.5 Example = 489
12.7 Control Charts For Defects = 489
12.7.1 Difference between a Defect and a Defective Item = 489
12.7.2 Statistical Theory = 490
12.7.3 Starting and Continuing the c Chart = 490
12.7.4 Example = 491
12.8 Further Developments on Control Charts = 492
12.8.1 The Signed Sequential Rank Control Chart = 493
12.8.2 The Cumulative Sum Control Chart = 495
Chapter 13 Sampling Inspection by Attributes = 503
13.1 The Problem of Sampling Inspection = 503
13.1.1 Introduction = 503
13.1.2 Drawing the Sample = 504
13.2 Lot-by-Lot Sampling Inspection by Attributes = 505
13.2.1 Single Sampling Plans = 505
13.2.1.1 Single Sampling = 505
13.2.1.2 Choosing a Sampling Plan = 507
13.2.1.3 Calculation of OC Curves for Single Sampling Plans = 507
13.2.1.4 Example = 508
13.2.2 Double Sampling Plans = 508
13.2.2.1 Double Sampling = 508
13.2.2.2 OC Curves for Double Sampling Plans = 509
13.2.2.3 Example = 509
13.2.3 Multiple Sampling Plans = 510
13.2.4 Classification of Sampling Plans = 511
13.2.4.1 Classification By AQL = 511
13.2.4.2 Classification By LTPD = 512
13.2.4.3 Classification By Point of Control = 512
13.2.4.4 Classification By AOQL = 512
13.2.5 Dodge-Romig Tables = 513
13.2.5.1 Single Sampling Lot Tolerance Tables = 514
13.2.5.2 Double Sampling Lot Tolerance Tables = 514
13.2.5.3 Single Sampling AOQL Tables = 518
13.2.5.4 Double Sampling AOQL Tables = 518
13.2.6 Military Standard 105D = 518
13.2.6.1 History = 518
13.2.6.2 Classification of Defects = 522
13.2.6.3 Acceptable Quality Levels = 522
13.2.6.4 Normal, Tightened, and Reduced Inspection = 523
13.2.6.5 Sampling Plans = 525
13.2.6.6 Summary of the Procedure To Be Followed in the Selection of a Sampling Plan from MIL-STD-105D = 527
13.2.7 Designing Your Own Attribute Plan = 527
13.2.7.1 Computing the OC curve of a Single Sampling plan = 527
13.2.7.2 Finding a Sampling Plan Whose OC Curve Passes through Two Points = 537
13.2.7.3 Design of Item-by-Item Sequential Plans = 537
13.2.8 A Bayesian Approach to Sampling Inspection = 545
13.2.8.1 Economic Structure = 545
13.2.8.2 A Decision Analysis Model = 546
13.2.8.3 Bayes Procedures = 547
13.3 Continuous Sampling Inspection = 550
13.3.1 Introduction = 550
13.3.2 Dodge Continuous Sampling Plans = 551
13.3.3 Multi-Level Sampling Plans = 552
13.3.4 The Dodge CSP-1 Plan without Control = 556
13.3.5 Wald-Wolfowitz Continuous Sampling Plans = 557
13.3.6 Girshick Continuous Sampling Plan = 558
13.3.7 Plans Which Provide for Termination of Production = 559
Chapter 14 Lot-by-Lot Sampling Inspection by Variables = 565
14.1 Introduction = 565
14.2 General Inspection Criteria = 566
14.3 Estimates of the Percent Defective = 568
14.3.1 Estimate of the Percent Defective when the Standard Deviation is Unknown but Estimated by the Sample Standard Deviation = 568
14.3.2 Estimate of the Percent Defective when the Standard Deviation is Unknown but Estimated by the Average Range = 569
14.3.3 Estimate of the Percent Defective when the Standard Deviation is Known = 571
14.4 Comparison of Variables Procedures with M and k = 574
14.5 The Military Standard for Inspection by Variables, MIL-STD-414 = 575
14.5.1 Introduction = 575
14.5.2 Section A - General Description of Sampling Plans = 576
14.5.3 Section B - Variability Unknown, Standard Deviation Method = 585
14.5.4 Section C - Variability Unknown, Range Method = 585
14.5.5 Section D - Variability Known = 590
14.5.6 Example Using MIL-STD-414 = 591
Appendix = 599
Answers to Selected Problems = 618
Index = 623
