APPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 651 APPENDIX B. BIBLIOGRAPHY 677 APPENDIX C. ANSWERS TO SELECTED EXERCISES 679

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1 APPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 1 Table I Summary of Common Probability Distributions 2 Table II Cumulative Standard Normal Distribution Table III Percentage Points, 2 of the Chi-Squared Distribution Table IV Percentage Points t, of the t-distribution Table V Percentage Points f, 1, 2 of the F-Distribution Chart VI Operating Characteristic Curves 2 Table VII Critical Values for the Sign Test 1 Table VIII Critical Values for the Wilcoxon Signed-Rank Test 1 Table IX Critical Values for the Wilcoxon Rank-Sum Test 2 Table X Factors for Constructing Variables Control Charts Table XI Factors for Tolerance Intervals APPENDIX B. BIBLIOGRAPHY APPENDIX C. ANSWERS TO SELECTED EXERCISES 9 9

3 Appendix A Statistical Tables and Charts 1

4 2 APPENDIX A STATISTICAL TABLES AND CHARTS Table I Summary of Common Probability Distributions Probability Section Name Distribution Mean Variance in Book Discrete Uniform 1 1b a2 1b a n, a b Binomial a n np np11 p2 - x b px 11 p2 n x, x, 1,..., n, p 1 Geometric 11 p2 x 1 p, 1 p 11 p2 p x 1, 2,..., p 1 Negative binomial a x 1 r 1 b 11 p2x r p r r p r11 p2 p x r, r 1, r 2,..., p 1 a K K b an x n x b Hypergeometric np, np11 p2 a N n - N 1 b a N n b where p K N x max1, n N K2, 1,... min1k, n2, K N, n N Poisson e x, x, 1, 2,..., x! -9 Continuous Uniform 1 1b a2 1b a2 2, a x b b a Normal 1 e x x,, Exponential e x, x, Erlang r x r 1 e x, x, r 1, 2,... 1r 12! r r Gamma x r 1 e x, x, r, 1r2 r r Weibull a x 1 b e 1x 2, a1 1 b 2 a1 2 b -1 x,, 2 c a1 1 bd 2 1 Lognormal e 2 2 e 2 2 1e x 22 exp a ln1x b

5 APPENDIX A z 1 1 1z2 P1Z z2 22 e 2 u2 du Φ (z) z Table II Cumulative Standard Normal Distribution z

6 APPENDIX A STATISTICAL TABLES AND CHARTS z 1 1 1z2 P1Z z2 22 e 2 u2 du Φ (z) z Table II Cumulative Standard Normal Distribution (continued) z

7 APPENDIX A α χ 2 α, ν Table III Percentage Points 2, of the Chi-Squared Distribution degrees of freedom.

8 APPENDIX A STATISTICAL TABLES AND CHARTS α t α, ν Table IV Percentage Points t, of the t-distribution degrees of freedom.

9 α f, 1, 2 Table V Percentage Points f,v1,v 2 of the F-Distribution f.2,v1,v2 Degrees of freedom for the denominator (v 2 ) v 2 v 1 Degrees of freedom for the numerator (v 1 )

10 Table V Percentage Points of the F-Distribution (continued) f.1,v1,v 2 Degrees of freedom for the denominator (v 2 ) v 1 Degrees of freedom for the numerator (v 1 ) v 2

11 Table V Percentage Points of the F-Distribution (continued) f.,v1,v 2 Degrees of freedom for the denominator (v 2 ) v 2 v 1 Degrees of freedom for the numerator (v 1 )

12 Table V Percentage Points of the F-Distribution (continued) f.2,v1,v 2 Degrees of freedom for the denominator (v 2 ) v 2 v 1 Degrees of freedom for the numerator (v 1 )

13 Table V Percentage Points of the F-Distribution (continued) f.1,v1,v 2 Degrees of freedom for the denominator (v 2 ) v 2 v 1 Degrees of freedom for the numerator (v 1 )

14 2 APPENDIX A STATISTICAL TABLES AND CHARTS Chart VI Operating Characteristic Curves 1.. Probability of accepting H n = d (a) O.C. curves for different values of n for the two-sided normal test for a level of significance Probability of accepting H.. n = d (b) O.C. curves for different values of n for the two-sided normal test for a level of significance.1. Source: Charts VIa, e, f, k, m, and q are reproduced with permission from Operating Characteristics for the Common Statistical Tests of Significance, by C. L. Ferris, F. E. Grubbs, and C. L. Weaver, Annals of Mathematical Statistics, June 19. Charts VIb, c, d, g, h, i, j, l, n, o, p, and r are reproduced with permission from Engineering Statistics, 2nd Edition, by A. H. Bowker and G. J. Lieberman, Prentice-Hall, 192.

15 APPENDIX A Chart VI Operating Characteristic Curves (continued) 1.. Probability of accepting H.. n = d (c) O.C. curves for different values of n for the one-sided normal test for a level of significance Probability of accepting H.. n = (d) O.C. curves for different values of n for the one-sided normal test for a level of significance.1. d

16 APPENDIX A STATISTICAL TABLES AND CHARTS Chart VI Operating Characteristic Curves (continued) 1.. n = 2 Probability of accepting H d (e) O.C. curves for different values of n for the two-sided t-test for a level of significance n = Probability of accepting H..... n = n = n =.2.1 n = 1 n = n = n = n = n = 2 n = 1 n = d ( f ) O.C. curves for different values of n for the two-sided t-test for a level of significance.1.

17 APPENDIX A Chart VI Operating Characteristic Curves (continued) Probability of accepting H n = n = n = n = 2 n = 1 n = 1 n = n = n = n =.1 n = 1 n = d (g) O.C. curves for different values of n for the one-sided t-test for a level of significance Probability of accepting H n = 1 n = n = n = n = 2 n = 1 n = 1 n = n = n = n =.1 n = d (h) O.C. curves for different values of n for the one-sided t-test for a level of significance.1.

18 APPENDIX A STATISTICAL TABLES AND CHARTS Chart VI Operating Characteristic Curves (continued) 1..9 n = 2. Probability of accepting H n = λ (i) O.C. curves for different values of n for the two-sided chi-square test for a level of significance n =. n = 2 Probability of accepting H λ ( j) O.C. curves for different values of n for the two-sided chi-square test for a level of significance.1.

19 APPENDIX A Chart VI Operating Characteristic Curves (continued) 1.. Probability of accepting H.. n = λ (k) O.C. curves for different values of n for the one-sided (upper tail) chi-square test for a level of significance.. 1. Probability of accepting H n = λ (l) O.C. curves for different values of n for the one-sided (upper tail) chi-square test for a level of significance.1.

20 9 APPENDIX A STATISTICAL TABLES AND CHARTS Chart VI Operating Characteristic Curves (continued) 1.. n = 2 Probability of accepting H (m) O.C. curves for different values of n for the one-sided (lower tail) chi-square test for a level of significance.. λ 1.. n = 2 Probability of accepting H λ (n) O.C. curves for different values of n for the one-sided (lower tail) chi-square test for a level of significance.1.

21 APPENDIX A 9 Chart VI Operating Characteristic Curves (continued) 1.. n 1 = n 2 = Probability of accepting H λ (o) O.C. curves for different values of n for the two-sided F-test for a level of significance.. 1. n 1 = n 2 =. Probability of accepting H λ ( p) O.C. curves for different values of n for the two-sided F-test for a level of significance.1.

22 APPENDIX A STATISTICAL TABLES AND CHARTS Chart VI Operating Characteristic Curves (continued) 1. n 1 = n 2 = 2. Probability of accepting H λ (q) O.C. curves for different values of n for the one-sided F-test for a level of significance.. 1. n 1 = n 2 = 2. Probability of accepting H λ 1. (r) O.C. curves for different values of n for the one-sided F-test for a level of significance.1.

23 APPENDIX A 1 Table VII Critical Values for the Sign Test r*.1..1 Two-sided tests.1..1 Two-sided tests n..2. One-sided tests n..2. One-sided tests Table VIII Critical Values for the Wilcoxon Signed-Rank Test w* Two-sided tests n* One-sided tests * If n 2, W (or W ) is approximately normally distributed with mean n(n 1) and variance n(n 1)(2n 1) 2.

24 2 APPENDIX A STATISTICAL TABLES AND CHARTS Table IX Critical Values for the Wilcoxon Rank-Sum Test w. n 1 * n *For n 1 and n 2, W 1 is approximately normally distributed with mean 2n 1 1 n 1 n 2 12 and variance n 1 n 2 (n 1 n 2 1) 12. Table IX Critical Values for the Wilcoxon Rank-Sum Test (continued) w.1 n 1 n

25 APPENDIX A Table X Factors for Constructing Variables Control Charts Factor for Control Limits X Chart R Chart S Chart n* A 1 A 2 d 2 D D c n * n 2: A 1 1n where n number of observations in sample.

26 APPENDIX A STATISTICAL TABLES AND CHARTS Table XI Factors for Tolerance Intervals Values of k for Two-Sided Intervals Confidence Level Percent Coverage

27 APPENDIX A Table XI Factors for Tolerance Intervals (continued) Values of k for One-Sided Intervals Confidence Level Percent Coverage

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