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ENGR 610 Applied Statistics Fall 2007 - Week 8

ENGR 610 Applied Statistics Fall 2007 - Week 8. Marshall University CITE Jack Smith. Overview for Today. Review Hypothesis Testing , Ch 9 Go over homework problem: 9.69, 9.71, 9.74 Design of Experiment , Ch 10 One-Factor Experiments Randomized Block Experiments Homework assignment.

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ENGR 610 Applied Statistics Fall 2007 - Week 8

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  1. ENGR 610Applied StatisticsFall 2007 - Week 8 Marshall University CITE Jack Smith

  2. Overview for Today • Review Hypothesis Testing, Ch 9 • Go over homework problem: 9.69, 9.71, 9.74 • Design of Experiment, Ch 10 • One-Factor Experiments • Randomized Block Experiments • Homework assignment

  3. Critical Regions • Critical value of test statistic (Z, t, F, 2,…) • Based on desired level of significance () • Acceptance (of null hypothesis) region • Rejection (alternative hypothesis) region • Two-tailed or one-tailed

  4. Z Test ( known) - Two-tailed • Critical value (Zc) based on chosen level of significance,  • Typically  = 0.05 (95% confidence), where Zc = 1.96 (area = 0.95/2 = 0.475) •  = 0.01 (99%) and 0.001 (99.9%) are also common, where Zc = 2.57 and 3.29 • Null hypothesis rejected if sample Z > Zc or < -Zc, where

  5. Z Test ( known) - One-tailed • Critical value (Zc) based on chosen level of significance,  • Typically  = 0.05 (95% confidence), but where Zc = 1.645 (area = 0.95 - 0.50 = 0.45) • Null hypothesis rejected if sample Z > Zc, where

  6. t Test ( unknown) - Two-tailed • Critical value (tc) based on chosen level of significance, , and degrees of freedom, n-1 • Typically  = 0.05 (95% confidence), where, for exampletc = 2.045 (upper area = 0.05/2 = 0.025), for n-1 = 29 • Null hypothesis rejected if sample t > tc or < -tc, where t

  7. Z Test on Proportion • Using normal approximation to binomial distribution

  8. p-value • Use probabilities corresponding to values of test statistic (Z, t,…) • Compare probability (p) directly to  instead of, say, t to tc • If the p-value  , accept null hypothesis • If the p-value < , reject null hypothesis • Does not assume any particular distribution (Z-normal, t, F, 2,…)

  9. Z Test for the Difference between Two Means • Random samples from independent groups with normal distributions and known1 and 2 • Any linear combination (e.g. the difference) of normal distributions (k, k) is also normal CLT: Populations the same

  10. t Test for the Difference between Two Means (Equal Variances) • Random samples from independent groups with normal distributions, but with equal and unknown1 and 2 • Using the pooled sample variance H0: µ1 = µ2

  11. t Test for the Difference between Two Means (Unequal Variances) • Random samples from independent groups with normal distributions, with unequal and unknown1 and 2 • Using the Satterthwaiteapproximation to the degrees of freedom (df) • Use Excel Data Analysis tool!

  12. F test for the Difference between Two Variances • Based on F Distribution - a ratio of 2 distributions, assuming normal distributions • FL(,n1-1,n2-1)  F  FU(,n1-1,n2-1), where FL(,n1-1,n2-1) = 1/FU(,n2-1,n1-1), and where FU is given in Table A.7 (using nearest df)

  13. Mean Test for Paired Data or Repeated Measures • Based on a one-sample test of the corresponding differences (Di) • Z Test for known population D • t Test for unknown D (with df = n-1) H0: D = 0

  14. 2 Test for the Difference among Two or More Proportions • Uses contingency table to compute • (fe)i = nip or ni(1-p) are the expected frequencies, where p = X/n, and (fo)i are the observed frequencies • For more than 1 factor, (fe)ij = nipj, where pj = Xj/n • Uses the upper-tail critical 2 value, with the df = number of groups – 1 • For more than 1 factor, df = (factors -1)*(groups-1) Sum over all cells

  15. Other Tests • 2 Test for the Difference between Variances • Follows directly from the 2 confidence interval for the variance (standard deviation) in Ch 8. • Very sensitive to non-Normal distributions, so not a robust test. • Wilcoxon Rank Sum Test between Two Medians

  16. Design of Experiments • R.A. Fisher (Rothamsted Ag Exp Station) • Study effects of multiple factors simultaneously • Randomization • Homogeneous blocking • One-Way ANOVA (Analysis of Variance) • One factor with different levels of “treatment” • Partitioning of variation - within and among treatment groups • Generalization of two-sample t Test • Two-Way ANOVA • One factor against randomized blocks (paired treatments) • Generalization of two-sample paired t Test

  17. One-Way ANOVA • ANOVA = Analysis of Variance • However, goal is to discern differences in means • One-Way ANOVA = One factor, multiple treatments (levels) • Randomly assign treatment groups • Partition total variation (sum of squares) • SST = SSA + SSW • SSA = variation among treatment groups • SSW = variation within treatment groups (across all groups) • Compare mean squares (variances): MS = SS / df • Perform F Test on MSA / MSW • H0: all treatment group means are equal • H1: at least one group mean is different

  18. Partitioning of Total Variation • Total variation • Within-group variation • Among-group variation (Grand mean) (Group mean) c = number of treatment groups n = total number of observations nj = observations for group j Xij = i-th observation for group j

  19. Mean Squares (Variances) • Total mean square (variance) • MST = SST / (n-1) • Within-group mean square • MSW = SSW / (n-c) • Among-group mean square • MSA = SSA / (c-1)

  20. F Test • F = MSA / MSW • Reject H0 if F > FU(,c-1,n-c) [or p<] • FU from Table A.7 • One-Way ANOVA Summary

  21. Tukey-Kramer Comparison of Means • Critical Studentized range (Q) test • qU(,c,n-c) from Table A.9 • Perform on each of the c(c-1)/2 pairs of group means • Analogous to t test using pooled variance for comparing two sample means with equal variances

  22. One-Way ANOVA Assumptions and Limitations • Assumptions for F test • Random and independent (unbiased) assignments • Normal distribution of experimental error • Homogeneity of variance within and across group (essential for pooling assumed in MSW) • Limitations of One-Factor Design • Inefficient use of experiments • Can not isolate interactions among factors

  23. Randomized Block Model • Matched or repeated measurements assigned to a block, with random assignment to treatment groups • Minimize within-block variation to maximize treatment effect • Further partition within-group variation • SSW = SSBL + SSE • SSBL = Among-block variation • SSE = Random variation (experimental error) • Total variation: SST = SSA + SSBL + SSE • Separate F tests for treatment and block effects • Two-way ANOVA, treatment groups vs blocks, but the focus is only on treatment effects

  24. Partitioning of Total Variation • Total variation • Among-group variation • Among-block variation (Grand mean) (Group mean) (Block mean)

  25. Partitioning, cont’d • Random error c = number of treatment groups r = number of blocks n = total number of observations (rc) Xij = i-th block observation for group j

  26. Mean Squares (Variances) • Total mean square (variance) • MST = SST / (rc-1) • Among-group mean square • MSA = SSA / (c-1) • Among-block mean square • MSBL = SSBL / (r-1) • Mean square error • MSE = SSE / (r-1)(c-1)

  27. F Test for Treatment Effects • F = MSA / MSE • Reject H0 if F > FU(,c-1,(r-1)(c-1)) • FU from Table A.7 • Two-Way ANOVA Summary

  28. F Test for Block Effects • F = MSBL / MSE • Reject H0 if F > FU(,r-1,(r-1)(c-1)) • FU from Table A.7 • Assumes no interaction between treatments and blocks • Used only to examine effectiveness of blocking in reducing experimental error • Compute relative efficiency (RE) to estimate leveraging effect of blocking on precision

  29. Estimated Relative Efficiency • Relative Efficiency • Estimates the number of observations in each treatment group needed to obtain the same precision for comparison of treatment group means as with randomized block design. • nj (without blocking)  RE*r (with blocking)

  30. Tukey-Kramer Comparison of Means • Critical Studentized range (Q) test • qU(,c,(r-1)(c-1)) from Table A.9 • Where group sizes (number of blocks, r) are equal • Perform on each of the c(c-1)/2 pairs of group means • Analogous to paired t test for the comparison of two-sample means (or one-sample test on differences)

  31. Homework • Work through Appendix 10.1 • Work and hand in Problems • 10.27 • 10.28 (except part c) • Read Chapter 11 • Design of Experiments: Factorial Designs

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