Suppose that a $2^{2}$ factorial design studying factors A and B was conducted. An investigator fits the model

\[y_{i}=\beta_{0} x_{i 0}+\beta_{1} x_{i 1}+\beta_{2} x_{i 2}+\beta_{3} x_{i 3}+\epsilon_{i}\]

$i, \ldots, n$, to estimate the factorial effects of the study.

a. What is the value of $n$ ? Define $x_{i k}, k=0,1,2,3$.

b. Derive the least-squares estimates of $\beta_{k}, k=0,1,2,3$.

c. Show that the least-squares estimates of A, B, AB, are one-half the factorial effects.