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Applied Statistics Overview: Hypothesis Testing and Experimental Design

This overview covers key concepts in hypothesis testing and the design of experiments as part of an Applied Statistics course. Topics include critical regions, null and alternative hypotheses, two-tailed and one-tailed tests, as well as one-factor and randomized block experiments. We delve into Z-tests, t-tests, and ANOVA, focusing on critical values, significance levels, and the comparison of means. Homework problems are addressed, emphasizing the practical application of these statistical techniques.

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Applied Statistics Overview: Hypothesis Testing and Experimental Design

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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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