Refer to the previous exercise. Now consider the model permitting interaction.

Table 3.4 shows the resulting ANOVA table.

a. Argue intuitively and in analogy with results for one-way ANOVA that the SS values for factor A, factor B, and residual are as shown in the ANOVA table.

b. Based on the results in

(a) and what you know about the total of the SS values, show that the SS for interaction is as shown in the ANOVA table.

c. In the ANOVA table, show the df values for each source. Show the mean squares, and show how to construct test statistics for testing no interaction and for testing each main effect. Specify the null distribution for each test statistic.

Table 3.4 ANOVA Table for Normal Linear Model with Two-Way Layout Source df Sum of Squares Mean Square Fobs Mean 1 Nȳ2 A (rows) — cn ∑

i

(ȳi.. − ȳ)

2 — —

B (columns) — rn ∑

j

(ȳ.j. − ȳ)

2 — —

Interaction — n

∑

i

∑

j

(ȳij. − ȳi.. − ȳ.j. + ȳ)

2 — —

Residual (error) — ∑

i

∑

j

∑

k( yijk − ȳij.

)

2 —

Total N ∑r i=1

∑c j=1

∑n k=1 y2 ijk