The response variable \(Y_{i j}\) in a \(2^{2}\) design can also be expressed as a regression model

\[Y_{i j}=\mu+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{12} x_{1} x_{2}+\varepsilon_{i j}\]

where the \(\varepsilon_{i j}\) are independent normal random variables and each has mean 0 and variance \(\sigma^{2}\).

Because \(\beta_{1}\) is a regression coefficient, it quantifies the change in the expected response when \(x_{1}\) 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 \(\bar{Y}_{1}, \bar{Y}_{2}, \bar{Y}_{3}\), and \(\bar{Y}_{4}\).

(b) Show that the expected value of the main effect of Factor \(A\) is \(2 \beta_{1}\).

(c) Show that the expected value of the \(A B\) interaction effect is \(2 \beta_{12}\).