Design and analysis of experiments

1. An HMO accountant has determined that the mean monthly expenditures on pre- scription medications varies among three groups of patients (those aged 25-44 (group 1)), 45-64 (group 2), and 65+ (group 3), based on the F-test for the Completely Randomized Design. The model fit is: Mij = Hitlij, Eij mild N (0, 02 ). She wished to make comparisons among all three groups, simultaneously, at the or = 0.05 significance level. She obtains the following information from an extensive search of the records: n = n2 = n3 = 10, MSE = 720.0, Y1. = 160, Y2. = 125, Ya. = 175. (a) Give the ANOA table. Then test whether the true means differ (construct Ho and H1). (b) Use Bonferroni and Tukey methods to compare the true means among all pairs of groups with an overall experimentwise error of o = 0.05. Clearly state your conclusions. 2. An experiment is conducted to compare four treatments. Five replicates are ob- tained per treatment. Treatment 2 3 4 41 36 103 53 46 7 75 30 56 53 84 70 56 31 82 80 69 76 80 42 (a) Write out the cell means model,under the independent and normally dis- tributed errors assumption. (b) Identify all parameters and clearly state ranges of subscripts. (c) Give an unbiased estimate of the common standard deviation o. (d) Give the ANOVA table. (e) Test whether the underlying population means differ (show all parts of test, using a significance level of or = 0.05).3. In the single-factor ANOVA model, yj = utte; fori = 1, . .. , a and j =1, . .. ,n where E; T; = 0 and e; ~ N(0, 0?). Then prove the following results: (a) E(MSTreatment) = 0'+, Eh? (b) E(MSE) = 02. (c) Under Ho : 71 = ... = Ta =0, prove that F = MSTreatment/ MSE ~ Fa-1,a(n-1)