4 critical value qf 95 dfssa dfsse 1 3354131

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#critical value qf (. 95 , df_SSA, df_SSE) ## [1] 3.354131 #Statistics F_star = MSA / MSE 1 - pf (F_star, df_SSA, df_SSE) #p value ## [1] 0 Since F ú > F (0 . 95 , 2 , 27) we reject Ho and conclude at least one of dosages of active ingredient A has e ff ect on the relief time of patients. (d) Testing for main e ff ects (B) H 0 : 1 = · · · = b = 0 vs. H a : not all j equal zero. F ratio: F ú = MSB MSE H 0 F b 1 , ( n 1) ab . Decision rule: Compare F ú with F (1 ; b 1 , ( n 1) ab ) . F_star = MSB / MSE F_star ## [1] 1027.329 qf (. 95 , df_SSB, df_SSE) ## [1] 3.354131 1 - pf (F_star, df_SSB, df_SSE) ## [1] 0 Since F ú > F (0 . 95 , 2 , 27) we reject Ho and conclude at least one of dosages of active ingredient B has e ff ect on the relief time of patients. (e) Relate F-tests to estimated treatment means plot The formal F-tests conducted in parts (c,d) confirm the existence of dosage e ff ects of active ingredients A,B as well as an AxB interaction e ff ect. Exercise 19.32 Refer to Hay fever relief Problems 19.14 and 19.15. (a) Estimate μ 23 with a 95 percent confidence interval. Interpret your interval estimate. (b) Estimate D = μ 12 μ 11 with a 95 percent confidence interval. Interpret your interval estimate. 5
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(c) The analyst decided to study the nature of the interacting factor e ff ects by means of the following contrasts: L 1 = μ 12 + μ 13 2 μ 11 , L 4 = L 2 L 1 , L 2 = μ 22 + μ 23 2 μ 21 , L 5 = L 3 L 1 , L 3 = μ 32 + μ 33 2 μ 31 , L 6 = L 3 L 2 . Obtain confidence intervals for these contrasts; use the Sche ff e multiple comparison procedure with a 90 percent family confidence coe ffi cient. (d) How many pairwise comparisons are there between treatment means? Construct Tukey’s confidence intervals for D = μ 12 μ 11 at family-wise confidence coe ffi cient 90% . (a) A 95% Confidence interval for μ 23 Y 23 · ± t (1 2 , ( n 1) ab ) Ú MSE n Y23 = Y_fit[ 2 , 3 ] Y23 ## [1] 9.125 se_Y23 = sqrt (MSE / n) se_Y23 ## [1] 0.1226633 c (Y23 - qt (. 975 , (n - 1 ) * a * b) * se_Y23, Y23 + qt (. 975 , (n - 1 ) * a * b) * se_Y23) ## [1] 8.873316 9.376684 We are 95% confident that the mean relief in hours for patients taking dosages 2 and 3 from active ingredients A and B, respectively, is between 8 . 873 and 9 . 377 hours.
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