a. Verify that the hypotheses for the three-factor factorial model may be written in matrix form as in (12.13).

b. Use the general form for the numerator sum of squares and the matrix expression for the hypotheses for factorial models to verify the expression for the numerator sums of squares in (12.31).

c. Verify that the reparameterized model for the two- and three-factor, cross- classification models is given by the general expressions in (12.16).

d. Verify that the results for the two- and three-factor model are special cases of the general results in Section 12.2.

e. For the two-factor model verify the equivalence of the two forms of the residual sum of squares given in (12.31) and (12.34). That is, verify that U.S.) = U-A-A-A2) na

#!# 12.9. Verify the expression for the expected value of the sum of squares in (12.37) and give algebraic expressions for the three factor hypotheses in terms of the cell means. Section 12.3