12.27 Consider the model y ijk = mu+ alpha i + beta j + gamma k + epsilon ijk i = 1.2 j = 1, 2 k = 1, 2

(a) Write X' * X X' * y and the normal equations.

(b) Find a set of linearly independent estimable functions.

(c) Define appropriate side conditions, and find the resulting solution to the normal equations.

(d) Show that H_{0} / alpha_{1} = alpha_{2} is testable. Find hat beta' * X' * y = SS(mu, alpha, beta, gamma) and hat beta_{2}' * X_{2}' * y = SS(mu, beta, gamma) .

(e) Construct an ANOVA table for the test of H_{0} / alpha_{1} = alpha_{2}