《ENGINEERING STATISTICS》

Chapter ⅠHistograms and Empirical Distributions1

1.1 Introduction1

1.2 Empirical Distributions2

1.3 Measures of Central Tendency7

1.4 Measures of Variation8

1.5 Computation of the Mean and Standard Deviation from the Frequency Table9

Chapter Ⅱ Random Variables and Probability Distributions13

2.1Introduction13

2.2 Set of All Possible Outcomes of the Experiment14

2.3 Random Variables16

2.4 Probability and Probability Distributions19

2.5 Discrete Probability Distributions22

2.6 Continuous Probability Distributions25

2.7 Random Sample29

2.8 Expectation30

2.9 Moments31

2.10 Some Properties of Random Variables35

Chapter Ⅲ The Normal Distribution40

3.1Definitions40

3.2The Mean and Variance of the Normal Distribution41

3.2.1 Evaluation of the Mean and Variance43

3.3 Tables of the Normal Integral44

3.4 Combinations of Normally Distributed Variables48

3.5 The Standardized Normal Random Variable49

3.6 The Distribution of the Sample Mean50

3.7 Tolerances51

3.8 Tolerances in Complex Items59

3.9 The Central Limit Theorem64

Chapter Ⅳ Other Probability Distributions70

4.1Introduction70

4.2The Chi-Square Distribution71

4.2.1 The Chi-Square Random Variable71

4.2.2 The Addition Theorem74

4.2.3 The Distribution of the Sample Variance75

4.3 The t-Distribution78

4.3.1 The t-Random Variable78

4.3.2 The Distribution of ？81

4.3.3 The Distribution of the Difference Between Two Sample Means82

4.4The F Distribution84

4.4.1 The F Random Variable84

4.4.2 The Distribution of the Ratio of Two Sample Variances86

4.5The Binomial Distribution87

4.5.1 The Binomial Random Variable87

4.5.2 Tables of the Binomial Probability Distribution89

4.5.3 The Normal Approximation to the Binomial90

4.5.4 The Arc Sine Transformation91

4.5.5 The Poisson Approximation to the Binomial92

Chapter Ⅴ Significance Tests96

5.1Introduction96

5.2 The Operating Characteristic Curve98

5.3 One- and Two-Sided Procedures103

5.4 Statistical Decision Theory109

Chapter ⅥTests of the Hypothesis about a Single Parameter111

6.1Tests of the Hypothesis that the Mean of a Normal Distribution Has a Specified Value when the Standard Deviation Is Known111

6.1.1 Choice of an OC Curve111

6.1.2 Tables and Charts for Determining Decision Rules113

6.1.2.1 Tables and Charts for Two-Sided Procedures113

6.1.2.2 Summary for Two-Sided Procedures116

6.1.2.3 Tables and Charts for One-Sided Procedures117

6.1.2.4 Summary for One-Sided Procedures119

6.1.2.5 Tables and Charts for OC Curves120

6.1.3 Analytical Determination of Decision Rules122

6.1.3.1 Acceptance Regions and Sample Sizes122

6.1.3.2 The OC Curve125

6.1.4 Example126

6.2 Test of the Hypothesis that the Mean of a Normal Distribution Has a Specified Value when the Standard Deviation Is Unknown127

6.2.1 The Choice of an OC Curve127

6.2.2 Tables and Charts for Carrying Out t Tests129

6.2.2.1 Tables and Charts for Two-Sided Procedures130

6.2.2.2 Summary for Two Sided Procedures131

6.2.2.3 Tables and Charts for One-Sided Procedures131

6.2.2.4 Summary for One-Sided Procedures134

6.2.2.5 Tables and Charts for OC Curves134

6.2.3 Examples of t Tests136

6.3Test of the Hypothesis that the Standard Deviation of a Normal Distribution Has a Specified Value137

6.3.1 Choice of an OC Curve137

6.3.2Charts and Tables to Design Tests of Dispersion138

6.3.2.1 Tables and Charts for Two-Sided Procedures138

6.3.2.2 Summary for Two-Sided Procedure Using Tables and Charts140

6.3.2.3 Tables and Charts for One-Sided Procedures140

6.3.2.4 Summary for One-Sided Procedures Using Tables and Charts143

6.3.2.5 Tables and Charts for Operating Character-istic Curves144

6.3.3 Analytical Treatment for Chi-Square Tests145

6.3.4 Example147

Chapter Ⅶ Tests of Hypotheses about Two Parameters156

7.1 Test of the Hypothesis that the Means of Two Normal Distribu-tions Are Equal when Both Standard Deviations Are Known156

7.1.1Choice of an OC Curve156

7.1.2 Tables and Charts for Determining Decision Rules157

7.1.2.1 Tables and Charts for Two-Sided Procedures157

7.1.2.2 Summary for Two-Sided Procedures Using Tables and Charts159

7.1.2.3 Summary for One-Sided Procedures Using Tables and Charts159

7.1.2.4 Tables and Charts for Operating Character-istic Curves161

7.1.3Analytical Determination of Decision Rules162

7.1.3.1 Acceptance Regions and Sample Sizes162

7.1.3.2 The Operating Characteristic Curve164

7.1.4 Example165

7.2 Test of the Hypothesis that the Means of Two Normal Distribu-tions Are Equal Assuming that the Standard Deviations Are Unknown but Equal166

7.2.1 Choice of an OC Curve166

7.2.2Tables and Charts for Carrying out Two Sample t Tests167

7.2.2.1 Tables and Charts for Two-Sided Procedures168

7.2.2.2 Summary for Two-Sided Procedures Using Tables and Charts169

7.2.2.3 Summary for One-Sided Procedures Using Tables and Charts169

7.2.2.4 Tables and Charts for Operating Character-istic Curves170

7.2.3 Example172

7.3 Test of the Hypothesis that the Means of Two Normal Distribu-tions Are Equal Assuming that the Standard Deviations Are Unknown and not Necessarily Equal173

7.3.1 Test Procedure173

7.3.2 Example174

7.4 Test for Equality of Means when the Observations Are Paired175

7.4.1 Test Procedure175

7.4.2 Example178

7.5 Non-parametric Tests179

7.5.1 The Sign Test179

7.5.2 The Wilcoxon Signed Rank Test179

7.5.3 A Test for Two Independent Samples184

7.6 Test of the Hypothesis that the Standard Deviations of Two Normal Distributions Are Equal186

7.6.1 Choice of an OC Curve186

7.6.2 Charts and Tables for Carrying out F Tests187

7.6.2.1 Tables and Charts for Two-Sided Procedures187

7.6.2.2 Summary for Two-Sided Procedures Using Tables and Charts189

7.6.2.3 Tables and Charts for One-Sided Procedures189

7.6.2.4 Summary for One-Sided Procedures Using Tables and Charts191

7.6.2.5 Tables and Charts for Operating Character-istic Curves192

7.6.3 Analytical Treatment for Tests193

7.6.4 Example195

7.7 Cochran’s Test for the Homogeneity of Variances198

Chapter ⅧEstimation211

8.1 Introduction211

8.2 Point Estimation211

8.3 Optimal Estimates214

8.4 Confidence Interval Estimation215

8.5Confidence Interval for the Mean of a Normal Distribution when the Standard Deviation Is Known216

8.5.1 Example217

8.6Confidence Interval For the Mean of a Normal Distribution when the Standard Deviation Is Unknown217

8.6.1 Example218

8.7Confidence Interval for the Standard Deviation of a Normal Distribution219

8.7.1 Example219

8.8Confidence Interval for the Differen between the Means of Two Normal Distributions when the Standard Deviations Are Both Known220

8.8.1 Example221

8.9Confidence Interval for the Difference between the Means of Two Normal Distributions where the Standard Deviations Are Both Unknown but Equal221

8.9.1 Example222

8.10 Confidence Interval for the Ratio of Standard Deviations of Two Normal Distributions223

8.10.1 Example224

8.11 A Table of Point Estimates and Interval Estimates224

8.12 Statistical Tolerance Limits224

8.12.1 Example228

8.13 One-Sided Statistical Tolerance Limits229

8.13.1 Example229

8.14 Distribution-Free Tolerance Limits229

Chapter ⅨFitting Straight Lines238

9.1 Introduction238

9.2 Types of Linear Relationships242

9.3Least Squares Estimates of the Slope and Intercept243

9.3.1 Formulation of the Problem and Results243

9.3.2 Theory245

9.4Confidence Interval Estimates of the Slope and Intercept246

9.4.1 Formulation of the Problem and Results246

9.4.2 Theory248

9.5Point Estimates and Confidence Interval Estimates of the Average Value of y for a Given x249

9.5.1 Formulation of the Problem and Results249

9.5.2 Theory250

9.6 Point Estimates and Interval Estimates of the Independent Variable x Associated with an Observation on the Dependent Variable y251

9.7Prediction Interval for a Future Observation on the Dependent Variable253

9.7.1 Formulation of the Problem and Results253

9.7.2 Theory254

9.8 Tests of Hypotheses about the Slope and Intercept255

9.9 Estimation of the Slope B when A is Known to be Zero257

9.10 Ascertaining Linearity259

9.11 Transforming to a Straight Line261

9.12 Work Sheets for Fitting Straight Lines264

9.13 Illustrative Examples264

9.14 Correlation273

Chapter ⅩAnalysis of Variance286

10.1 Introduction286

10.2 Model for the One-Way Classification287

10.2.1 Fixed Effects Model287

10.2.2 Random Effects Model289

10.2.3 Further Examples of Fixed Effects and of the Random Effects Models290

10.2.4 Computational Procedure: One-Way Classification290

10.2.5The Analysis of Varian Procedu292

10.2.5.1 A Heuristic Justification292

10.2.5.2 The Partition Theorem293

10.2.6 Analysis of the Fixed Effects Model: One-Way Clas-sification294

10.2.7 The OC Curve of the Analysis of Variance for the Fixed Effects Model299

10.2.8 Example Using the Fixed Effects Model305

10.2.9 Analysis of the Random Effects Model307

10.2.10 The OC Curve for the Random Effects Model307

10.2.11 Example Using the Random Effects Model313

10.2.12 Randomization Tests in the Analysis of Variance314

10.3Two-Way Analysis of Variance， One Observation per Combina-tion315

10.3.1 Fixed Effects Model316

10.3.2 Random Effects Model319

10.3.3 Mixed Fixed Effects and Random Effects Model319

10.3.4 Computational Procedure， Two-Way Classification， One Observation per Combination320

10.3.5 Analysis of the Fixed Effects Model， Two-Way Classification， One Observation per Combination321

10.3.6 The OC Curve of the Analysis of Variance for the Fixed Effects Model: Two-Way Classification， One Observation per Combination324

10.3.7 Example Using the Fixed Effects Model325

10.3.8 Analysis of the Random Effects Model: Two-Way Classification， One Observation per Combination327

10.3.9 The OC Curve for the Random Effects Model: Two-Way Classification328

10.3.10 Example Using the Random Effects Model329

10.3.11 Analysis of the Mixed Effects Model， Two-Way Classification， One Observation per Combination330

10.3.12 The OC Curve of the Analysis of Variance for the Mixed Effects Model， Two-Way Classification， One Observation per Cell331

10.3.13 Example Using the Mixed Effects Model332

10.4Two-Way Analysis of Variance， n Observations per Combination332

10.4.1 Description of the Various Models332

10.4.2 Computational Procedure， Two-Way Classification， n Observations per Cell334

10.4.3 Analysis of the Fixed Effects Model， Two-WayClassification， n Observations per Combination335

10.4.4 The OC Curve of the Analysis of Variance for the Fixed Effects Model， Two-Way Classification， n Observations per Cell339

10.4.5 Example Using the Fixed Effects Model， Two-Way Classification， Three Observations per Combination340

10.4.6 Analysis of the Random Effects Model， Two-Way Classification， n Observations per Combination342

10.4.7 The OC Curve of the Random Effects Model， Two-Way Classification， n Observations per Combination344

10.4.8 Example Using the Random Effects Model345

10.4.9 Analysis of the Mixed Effects Model， Two-Way Classification， n Observations per Cell345

10.4.10 The OC Curve of the Analysis of Variance for the Mixed Effects Model， Two-Way Classification， One Observation per Cell347

10.4.11 Example Using the Mixed Effects Model347

10.5 Summary of Models and Tests349

Chapter ⅩⅠAnalysis of Enumeration Data365

11.1 Enumeration Data365

11.2 Chi-Square Tests365

11.3 The Hypothesis Completely Specifies the Theoretical Frequency366

11.3.1 Dichotomous Data367

11.4 Test of Independence in a Two-Way Table369

11.4.1 Computing Form for Test of Independence in a 2 by 2 Table370

11.5 Comparison of Two Percentages371

11.6 Confidence Intervals for Proportion372

11.6.1 Exact Confidence Intervals for p373

11.6.2 Normal Approximations to Confidence Intervals373

Chapter ⅩⅡStatistical Quality Control: Control Charts378

12.1 Introduction378

12.2 Obtaining Data From Rational Subgroups378

12.3Control Chart for Variables: ? - Charts379

12.3.1 Statistical Concepts379

12.3.2 Estimates of ?’381

12.3.3 Estimate of ?’ by ?381

12.3.4 Estimate of ?’ by ?383

12.3.5 Starting a Control Chart for ?383

12.3.6 Relation Between Natural Tolerance Limits and Specification Limits384

12.3.7 Interpretation of Control Charts for ?385

12.4R Charts and ? Charts387

12.4.1 Statistical Concepts387

12.4.2 Setting up a control chart for R or ?388

12.5 Example of ? and R Chart389

12.6Control Chart For Fraction Defective391

12.6.1 Relation Between Control Charts Based on Variables Data and Charts Based on Attributes Data391

12.6.2 Statistical Theory391

12.6.3 Starting the Control Chart393

12.6.4 Continuing the p Chart394

12.6.5 Example395

12.7Control Charts For Defects395

12.7.1 Difference Between a Defect and a Defective395

12.7.2 Statistical Theory396

12.7.3 Starting and Continuing the c Chart396

12.7.4 Example396

Chapter ⅩⅢSampling lnspection402

13.1The Problem of Sampling Inspection402

13.1.1 Introduction402

13.1.2 Drawing the Sample403

13.2Lot-by-Lot Sampling Inspection by Attributes404

13.2.1Single Sampling Plans404

13.2.1.1 Single Sampling404

13.2.1.2 Choosing a Sampling Plan406

13.2.1.3 Calculation of OC Curves for Single Sampling Plans407

13.2.1.4 Example407

13.2.2Double Sampling Plans413

13.2.2.1 Double Sampling413

13.2.2.2 OC Curves for Double Sampling Plans413

13.2.2.3 Example414

13.2.3 Multiple Sampling Plans415

13.2.4Classification of Sampling Plans416

13.2.4.1 Classification By AQL416

13.2.4.2 Classification By LTPD416

13.2.4.3 Classification By Point of Control416

13.2.4.4 Classification By AOQL417

13.2.5Dodge-Romig Tables418

13.2.5.1 Single Sampling Lot Tolerance Tables418

13.2.5.2 Double Sampling Lot Tolerance Tables421

13.2.5.3 Single Sampling AOQL Tables421

13.2.5.4 Double Sampling AOQL Tables421

13.2.6Military Standard 105A424

13.2.6.1 History424

13.2.6.2 Classification of Defects425

13.2.6.3 Acceptable Quality Levels425

13.2.6.4 Normal， Tightened， and Reduced Inspection426

13.2.6.5 Sampling Plans427

13.2.7 Designing your Own Attribute Plan432

13.2.7.1Computing the OC Curve of a Single Sampling Plan457

13.2.7.2 Finding a Sampling Plan Whose OC Curve Passes Through Two Points457

13.2.7.3 Design of Item by Item Sequential Plans464

13.3 Lot-By-Lot Sampling Inspection By Variables467

13.3.1 Introduction467

13.3.2 General Inspection Criteria468

13.3.3Estimates of the Percent Defective470

13.3.3.1 Estimate of the Percent Defective when the Standard Deviation Is Unknown but Estimated by the Sample Standard Deviation470

13.3.3.2 Estimate of the Percent Defective when the Standard Deviation Is Unknown but Estimated by the Average Range471

13.3.3.3 Estimate of the Percent Defective when the Standard Deviation Is Known489

13.3.4 Comparison of Variables Procedures with M and k491

13.3.5The Military Standard for Inspection by Variables，MIL-STD-414492

13.3.5.1 Introduction492

13.3.5.2 Section A— General Description of Sampling Plans493

13.3.5.3 Section B — Variability Unknown，Standard Deviation Method494

13.3.5.4 Section C — Variability Unknown， Range Method510

13.3.5.5 Section D — Variability Known511

13.3.5.6 Example Using MIL-STD-414512

13.4Continuous Sampling Inspection513

13.4.1 Introduction513

13.4.2 Dodge Continuous Sampling Plans513

13.4.3 Multi-Level Sampling Plans537

13.4.4 The Dodge CSP-1 Plan without Control541

13.4.5 Wald-Wolfowitz Continuous Sampling Plans542

13.4.6 Girshick Continuous Sampling Plan543

13.4.7 Plans Which Provide for Termination of Production544

Appendix553

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