Show that I i=1 (Y i+ Y ++) 2 = I i=1 Y 2 i+ IY 2 ++ and J

Suppose that the null hypothesis H0 in (11.7.11) is false. Show that E(S2 A) = (I 1)2 + J I i=1 2 i .

. Consider a two-way layout in which the values of E(Yij ) for i = 1,...,I and j = 1,...,J are as given in each of the following four matrices. For each matrix, state whether the effects of the factors are additive. a. Factor B Factor A 1 2 1 57 2 10 14 b. Factor B Factor A 1 2 1 36 2 47 c. Factor B Factor A 1 234 1 3 103 2 8 458 3 4 014 d. Factor B Factor A 123 4 1 123

Show that if the effects of the factors in a two-way layout are additive, then there exist unique numbers , 1,...,I , and 1,...,J that satisfy Eqs. (11.7.2) and (11.7.3). Hint: Let be the average of all i + j values, let i equal i minus the average of the is, and similarly for j .

Suppose that in a two-way layout, with I = 2 and J = 2, the values of E(Yij ) are as given in part (b) of Exercise 2. Determine the values of , 1, 2, 1, and 2 that satisfy Eqs. (11.7.2) and (11.7.3).

Suppose that in a two-way layout, with I = 3 and J = 4, the values of E(Yij ) are as given in part (c) of Exercise 2. Determine the values of , 1, 2, 3, and 1,...,4 that satisfy Eqs. (11.7.2) and (11.7.3

Verify that if , i, and j are defined by Eq. (11.7.6), then I i=1, i = J j=1 j = 0; E() = ; E(i) = i for i = 1,...,I ; and E( j ) = j for j = 1,...,J .

Show that if , i, and j are defined by Eq. (11.7.6), then Var() = 1 IJ 2, Var(i) = I 1 IJ 2 for i = 1, . . . , I, Var( j ) = J 1 IJ

Show that the right sides of Eqs. (11.7.9) and (11.7.8) are equal.

Show that in a two-way layout, for all values of i, j , i , and j (i and i= 1,...,I ; j and j = 1,...,J ), the following four random variables W1, W2, W3, and W4 are uncorrelated with one another: W1 = Yij Y i+ Y +j + Y ++, W2 = Y i + Y ++, W3 = Y +j  Y ++, W4 = Y +

Show that  I i=1 (Y i+ Y ++) 2 = I i=1 Y 2 i+ IY 2 ++ and  J j=1 (Y +j Y ++) 2 = J j=1 Y 2 +j J Y 2 +

Show that  I i=1  J j=1 (Yij Y i+ Y +j + Y ++) 2 = I i=1  J j=1 Y 2 ij J  I i=1 Y 2 i+ I  J j=1 Y 2 +j + IJY 2 +

In a study to compare the reflective properties of various paints and various plastic surfaces, three different types of paint were applied to specimens of five different types of plastic surfaces. Suppose that the observed results in appropriate coded units were as shown in Table 11.25. Determine the values of , 1, 2, 3, and 1,..., 5. Table 11.25 Data for Exercises 1215 Type of surface Type of paint 12345 1 14.5 13.6 16.3 23.2 19.4 2 14.6 16.2 14.8 16.8 17.3 3 16.2 14.0 15.5 18.7

For the conditions of Exercise 12 and the data in Table 11.25, determine the value of the least-squares estimate of E(Yij ) for i = 1, 2, 3, and j = 1,..., 5, and determine the value of 2

For the conditions of Exercise 12 and the data in Table 11.25, test the hypothesis that the reflective properties of the three different types of paint are the same.

For the conditions of Exercise 12 and the data in Table 11.25, test the hypothesis that the reflective properties of the five different types of plastic surfaces are the same.

Prove the claim in Theorem 11.7.3 about the distribution of U2 A+B