Math 457

Solve please.

Exercise 49 (#7.117). Let Xoj (j = 1, ..., no) and Xij (i = 1, ....m, j = 1, ..., no) represent independent measurements on a standard and m competing new treatments. Suppose that X, is distributed as N(;, o?) with unknown ; and o' > 0, j = 1, ..., no, i = 0, 1, ..., m. For i = 0, 1, ..., m, let X; be the sample mean based on Xjj, j = 1, ..., no. Define 62 [(m + 1)(no - 1)]'EM. ER,(Xij - Xi.)'. (i) Show that the distribution of Ra = max I(Xi. - Mi) - (Xo. - Ho)1/a i=1 ,....m does not depend on any unknown parameter. (ii) Show that Dunnett's intervals m Eqxi - god Clal, Exit gad Clal i=0 i=1 1=0 i=1 for all co. C1, ..., Cm satisfying Ec; =0 are simultaneous confidence inter- vals for call; with confidence coefficient 1-a, where q, is the (1-a) th quantile of Rat.Exercise 48 (#7.111). Let (X1, ..., X,) be independently distributed as N(Bo + Biz;, 02), i = 1. ...,n, where Bo, 31, and of are unknown and zi's are known constants satisfying S = Ch (2 -2)' > 0, 1= n 121, z. Show that Is = Bo+Biz-0/2F2,n-2,aD(z), Bo+Bizto \\/ 2F2,n-2,aD(2) ZER, are simultaneous confidence intervals for Bo + 312, z ER, with confidence coefficient 1-a, where (Bo, B1 ) is the LSE of (Bo, 31), D(2) = (2-2)-/S= + n , and 62 = (n -2)-1 ER (Xi - Bo - Bizi)2