5.1 For a factor X with d categories, the one-factor mean function is E(YIU2, ..., Ud) = Bo + ByU2 + ... + BaUd (5.17) where U, is a dummy variable equal to 1 for the jth level of the factor and 0 otherwise. 5.1.1 Show that , = Bo is the mean for the first level of X and that U; = Po + B; is the mean for all the remaining levels, j = 2, . . ., d. 5.1.2 It is convenient to use two subscripts to index the observations, so ya is the ith observation in level j of the factor, j = 1, ..., d and i = 1, . . ., n;. The total sample size is n = En;. The residual sum of squares function can then be written as RSS(B) = > > (), - Bo - B,U2 - ... -Bald) j=1 i=1 Find the OLS estimates of the Bs, and then show that the OLS esti- mates of the group means are u, = V1, j = 1, . . ., d, where y, is the average of the ys for the jth level of X. 5.1.3 Show that the residual sum of squares can be written RSS = _ (m, - D)SD} j=1 where SD; is the standard deviation of the responses for the jth level of X. What is the df for RSS? 5.1.4 If all the n, are equal, show that (1) the standard errors of Bz, ..., Ba are all equal, and (2) the standard error of Bo is equal to the stan- dard error of each of Bo + B;, j = 2, .. . . d