Problem 6. In Lecture 15, we derived the confidence intervals for the paired difference ai - ar using the fact that ( Yi. - Yr.) - (ai - a;), ~ tritn, -2, for any pair i # r. Sir ( in + 7 ) Now let's generalize the above CIs and look at a contrast which is defined to be Et_1 tia; in which Etti =0. You can see that ai -ar is an specific example of a contrast. (1) Prove that EttiVi - Entidi ~ tn-ki VMSw Enti in which MSw = (n -k) 126, Em (Yj - Yi)2. (2) For the trout toxin example of Lecture 15, derive the Bonferroni simultaneous 95% CIs for the following contrasts: a1 - 02, a1 - 202 + 04, 303 - 01 - 202, 502 - 403 - 04, 202 - 01 - 03. (3) Scheffe (1959) coined a quite elegant approach to deriving simultan neous CIs on all contrasts. This procedure is valid for any number of contrast CIs concurrently via specifying a larger critical value c= (k - 1)Fk-1,n-k(@); that is, the overall probability is 1 - a that k Ctioie CtiVite MSw i i= 1 i= 1 simultaneously for all t = (t1, . .., th ) such that _ ti = 0. Calculate the Scheffe simultaneous 95% CIs for the contrasts in (2), and visualize the two sets of CIs side by side to compare