2. The following data shows the values of the response variable y and two factors A and

B, each having two levels.

You may solve this problem either manually by using a calculator or by using R.

(a) Consider a two-way ANOVA model without interaction (call it M1):

Y = BO+BAILAI + PBI/BI + e.

(M1)

Give the design matrix for this model, and find the least square estimates for Po, BAI, and

BA2. Find the residual sum of squares for this model, and the associated number of de-

grees of freedom.

(b) Consider a two-way ANOVA model with interaction (call it M2):

Y = BO+BAILAI + PB₁/B1 + BAIBI¹A1 · ĪB₁ + e.

(M2)

Give the design matrix for this model, and find the least square estimates for the coeffi-

cients. Find the residual sum of squares for this model, and the associated number of

degrees of freedom.

(c) Is the interaction between the two factors statistically significant? State the null and

the alternative hypothesis for this model, and explain your reasoning for the conclusion.