13.18 The response variable Yi j in a 22 design can also be expressed as a regression model Yi j = + 1 x1

13.18 The response variable Yi j in a 22 design can also be expressed as a regression model Yi j = μ + β1 x1 + β2 x2 + β12 x1 x2 + εi j where the εi j are independent normal random variables and each has mean 0 and variance σ2. Because β1 is a regression coefficient, it quantifies the change in the expected response when x1 is changed by one unit. The effects are calculated on a change from −1 to 1 or 2 units.

(a) Obtain the expected values of Y1 ,Y2 ,Y3, and Y4 .

(b) Show that the expected value of the main effect of Factor A is 2 β1.

(c) Show that the expected value of the AB interaction effect is 2 β12.