a. For the nested-factorial model in (13.27), verify the covariance structure in (13.28) and the matrix expressions in (13.29) and (13.31).

b. Verify the expressions for the expected mean squares in Table

#!# 13.7.

c. Verify the distributions in (13.34). 13.7 Littell (1987) examined data, shown below, from a semiconductor plant concerned with four modes of a process condition. For the experiment, 12 wafers were drawn from a lot and 3 wafers were assigned to each of the modes. Resistance was measured at 4 positions on each wafer.

a. Let denote the mean response for the ith mode and jth position and justify the assumption of the model =j+w(m) + pw(m) where w(m) ~~N(0,1)) and pw(m) ~ N(0. ). Relate your discussion to Examples 13.3. Data for Exercise 13.7 Position Mode Wafer 1 2 3 4 T 1 5.22 5.61 6.1 6.33 2 6.13 6.14 5.60 5.91 3 5.49 4.60 4.95 5.42 2 1 5.78 6.52 5.90 5.67 2 5.77 6.23 5.57 5.96 3 6.43 5.81 5.83 6.12 1 5.66 6.25 5.46 5.08 2 6.53 6.50 6.23 6.84 3 6.22 6.29 5.63 6.36 . 6.75 6.97 6.02 6.88 2 6.22 6.54 6.12 6.61 3 6.05 6.15 5.55 6.13

b. Using this data fit the linear model, complete the analysis of variance table, and test the usual fixed effect hypotheses.

c. Estimate the appropriate function of the variance components, and test for pairwise differences in the modes. Repeat for positions.

d. Describe the model and the analysis assuming that position is a random effect. Section 13.5