2. Consider the two-way analysis of variance (ANOVA) model with two factors A and B with levels a and b respectively. Let Xij, i = 1, 2, .., a and j = 1, 2, ..., b denote the response for factor A at level i and factor B at level j. Denote the sample size by n = ab. Assume that all X;;s are independent normally distributed random variables with common variance of. Denote the mean of Xij by uij which is referred to as the mean of the (i, ))th cell. The two-way ANOVA model can be written as: Mij = u+ ai+ Bj, where Zi=1 0; = 0 and >;=1 Bj = 0. The mean in the (i, j)th cell is due to the additive effects of the levels, i of factor A and j of factor B, over the average (constant) ji . Let a; = Mi - ji, i = 1, 2, .., a and Bj = uj - u, j = 1,2, .., b and u = u . Derive the following hypothesis tests in detail: (a) HOA: di = a2 = . = da = 0 versus HIA: a; # 0 for some i; (10 marks) (b) HOB: B1 = B2 = ... = Bb = 0 versus HIB : B; # 0 for some j. (10 marks)