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RELIABILITY & LIFE TESTING HANDBOOK
VOLUME 2
TABLE OF CONTENTS
CHAPTER 1- RELIABILITY AND CONFIDENCE LIMITS
FOR ONE-SHOT ITEMS OR
BINOMIAL (BERNOULLI) TRIALS ...... 1
1.1- THE BINOMIAL DISTRIBUTION ...... 1
1.2- THE CUMULATIVE BINOMIAL DISTRIBUTION ...... 9
1.3- BINOMIAL CONFIDENCE LIMITS ...... 22
1.4- EXACT BINOMIAL CONFIDENCE LIMITS ...... 47
CHAPTER 2- COMPARING TWO LOTS THROUGH BINOMIAL
TESTING ...... 59
2.1- WHEN AND HOW TO APPLY ...... 59
CHAPTER 3- SUSPENDED-ITEMS TEST, AND ANALYSIS
OF FIELD AND WARRANTY DATA ...... 77
3.1- OBJECTIVES AND APPLICABILITY ...... 77
3.2- METHODOLOGY AND APPLICATIONS ...... 78
3.3- TEST STOPPED WHEN ONLY PART OF THE SAMPLE
HAS FAILED - CENSORED TESTS ...... 105
CHAPTER 4- SUDDEN-DEATH TESTING ...... 147
4.1- OBJECTIVES AND APPLICABILITY ...... 147
4.2- METHODOLOGY AND APPLICATIONS ...... 148
4.3- STEP-BY-STEP PROCEDURE FOR SUDDEN-DEATH
TESTING AND DATA ANALYSIS ...... 150
4.4- RELATIONSHIP BETWEEN THE SUDDEN-DEATH LINE
AND THE POPULATION LINE ...... 153
4.5- CONFIDENCE BOUNDS ABOUT THE SUDDEN-DEATH
TEST POPULATION LINE ...... 154
4.6- SUDDEN-DEATH TEST DURATION ...... 161
CHAPTER 5- NONPARAMETRIC TESTING ...... 181
5.1- OBJECTIVES ...... 181
5.2-THE SUCCESS-RUN TEST ...... 181
5.3-THE CL-RANK TEST ...... 185
5.4-THE PERCENT-SURVIVING TEST ...... 187
5.5- THE SIGN TEST ...... 198
5.6- THE RUN AND RANDOMNESS TESTS ...... 202
5.7- COMPARISON OF TWO POPULATION MEANS USING
THEIR RANGES ...... 217
5.8- THE BINOMIAL-PEARSON RELIABILITY TEST ...... 220
5.9- BINOMIAL-PEARSON OVERLOAD-STRESS TEST ...... 223
5.10-THE WILCOXON-WHITE RANK-SUM TEST FOR TWO
INDEPENDENT SAMPLES...... 224
CHAPTER 6- ACCEPT-REJECT TESTING WITH FIXED TEST
TIME OR FIXED TEST FAILURES WHILE
MEETING BOTH THE CONSUMER'S AND THE
PRODUCER'S RISK REQUIREMENTS FOR THE
EXPONENTIAL CASE ...... 267
6.1- INTRODUCTION ...... 267
6.2- FIGURES 6.1 AND 6.2 APPROACH ...... 268
6.3- FIGURE 6.3 APPROACH ...... 271
6.4- JUSTIFICATION FOR USING FIGURE 6.3 TO OBTAIN
CONFIDENCE LIMITS ON THE MEAN LIFE ...... 274
6.5- USES OF FIGURES 6.1 AND 6.2 ...... 276
6.6- TO FIND mL1 . ...... 277
6.7- TO FIND THE ALLOWABLE NUMBER OF FAILURES, a,
GIVEN mL1 ...... 278
6.8- TO FIND mU1 ...... 279
6.9- TO FIND P (R) GIVEN mU1 ...... 280
6.10-USES OF FIGURE 6.3 ...... 280
6.11-THE OPTIMUM NUMBER OF TEST UNITS FOR
ACCEPT-REJECT TESTING WITH ZERO FAILURES
WHILE MEETING THE CONSUMER'S RISK
REQUIREMENT FOR THE EXPONENTIAL CASE ...... 286
6.12- THE OPTIMUM NUMBER OF TEST UNITS FOR
ACCEPT-REJECT TESTING WITH ONE OR MORE
FAILURES WHILE MEETING THE CONSUMER'S RISK
REQUIREMENT FOR THE EXPONENTIAL CASE ...... 289
CHAPTER 7- SEQUENTIAL PROBABILITY RATIO TESTS
(SPRT) FOR THE EXPONENTIAL CASE ...... 303
7.1- INTRODUCTION ...... 303
7.2- SPRT ...... 304
7.3- SPRT FOR THE EXPONENTIAL CASE ...... 306
7.4- THE DISTRIBUTION OF THE SUCCESSIVE SUMS OF
THE TIMES TO FAILURE FROM AN EXPONENTIAL
SEQUENTIAL TEST PROCESS ...... 315
7.5- CALCULATION OF THE CONFIDENCE LIMITS ...... 321
7.6- THE OPERATING CHARACTERISTIC CURVE FOR THE
SEQUENTIAL TEST PLAN ...... 337
7.7- EXPECTED TEST TIME IN THE SEQUENTIAL TEST ...... 344
7.8- MIL-HDBK-781 ...... 348
7.9- THE OPTIMUM TEST SAMPLE SIZE ...... 360
CHAPTER 8- SEQUENTIAL TESTING ON THE SCALE
PARAMETER OF THE WEIBULL DISTRIBUTION ...... 367
8.1- ACCEPT AND REJECT CRITERIA ...... 367
CHAPTER 9- ACCEPT-REJECT TESTS FOR THE
BINOMIAL CASE ...... 373
9.1- WHEN AND HOW TO APPLY ...... 373
9.2- CHARTED AND TABULATED TWO-RISK TEST PLANS
FOR ACCEPT-REJECT BINOMIAL TESTS ...... 378
CHAPTER 10- SEQUENTIAL TESTING FOR THE
BINOMIAL CASE ...... 389
10.1- INTRODUCTION ...... 389
10.2- TEST REQUIREMENTS ...... 390
10.3- BINOMIAL SEQUENTIAL TEST PLAN TABLES ...... 396
10.4- METHOD FOR TRUNCATING THE SEQUENTIAL
TESTING PLANS FOR THE BINOMIAL CASE ...... 397
CHAPTER 11- BAYESIAN MTBF AND RELIABILITY
DEMONSTRATION TESTS ...... 415
11.1- BAYESIAN CONCEPTS ...... 415
11.2- GENERAL PROCEDURE FOR FINDING
THE POSTERIOR DISTRIBUTION ...... 430
11.3- THE EXPONENTIAL PRIOR DISTRIBUTION
WITH EXPONENTIAL AND POISSON CONDITIONALS ...... 434
11.4- THE BETA AND GAMMA PRIOR DISTRIBUTIONS ...... 447
11.5- ASSESSMENT OF PRIOR DISTRIBUTIONS ...... 454
11.6- TWO-RISK REQUIREMENTS CRITERIA ...... 510
11.7- TEST PLANS, PROCEDURES AND COMPARISONS ...... 514
11.8- BAYESIAN RELIABILITY / MTBF DEMONSTRATION-
KEY CONCEPTS SUMMARY ...... 539
11.9- SINGLE PRIOR BELIEF BAYESIAN RELIABILITY
DEMONSTRATION ...... 540
11.10- NONIDENTIFIABLE POSTERIOR DISTRIBUTIONS ...... 557
CHAPTER 12- ACCELERATED RELIABILITY AND
LIFE TESTS ...... 605
12.1- WHAT IT IS, AND HOW IT IS APPLIED ...... 605
12.2- THE ARRHENIUS MODEL ...... 607
12.3- THE EYRING MODEL ...... 636
12.4- THE INVERSE POWER LAW MODEL ...... 647
12.5- THE COMBINATION MODEL ...... 650
12.6- THE GENERALIZED EYRING MODEL ...... 659
12.7- THE BAZOVSKY MODEL ...... 665
12.8- TEMPERATURE-HUMIDITY MODELS ...... 668
12.9- THE WEIBULL STRESS-LIFE MODEL ...... 670
CHAPTER 13- TEST SAMPLE SIZE DETERMINATION ...... 699
13.1- INTRODUCTION ...... 699
13.2- NORMAL DISTRIBUTION ...... 700
13.3- EXPONENTIAL DISTRIBUTION ...... 722
13.4- WEIBULL DISTRIBUTION ...... 738
13.5- BINOMIAL DISTRIBUTION ...... 759
APPENDICES ...... 779
APPENDIX A- RANK TABLES ...... 780
APPENDIX B- STANDARDIZED NORMAL DISTRIBUTION'S
AREA TABLES ...... 794
APPENDIX C- PERCENTAGE POINTS, F-DISTRIBUTION ...... 802
APPENDIX D- CRITICAL VALUES FOR VARIOUS
KOLMOGOROV-SMIRNOV (K-S) GOODNESS-
OF-FIT TESTS ...... 821
APPENDIX E- BINOMIAL OR ATTRIBUTE RELIABILITY TEST
CURVES FOR CONFIDENCE LEVELS, CL,
OF 50%, 80%, 90%, 95%, AND 99% ...... 831
INDEX ...... 837 ABOUT THE AUTHOR ...... ……… ...... 854