2. The following data shows the values of the response variable y and two factors A and B, each having two levels. "V" "A" "B" 1.94 "(" "(" 0.51 "0" "(" 1.31 "0" "(" 1.34 "0" "(" -0.01 "(" "1 " 1.13 "(" "1" 1.01 "(" "1" 1.12 "(" "1" 3.12 "1" "(" 2.29 "1" "( 4.84 "1" "(" 4.19 "1" "(" 2.58 "1" "1 2.81 "1 " "1 0.24 "1" "1 " 1.62 "1" "1" You may solve this prob- lem either manually by using a calculator or by using R. (a) Consider a two-way ANOVA model without interaction (call it MI): Y = Bo + BAIAl + PBIBI + 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 = Po + PAIAl + PBIBI + PAIBIAl . IBIT 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