Assume that $y_{i j}$ are i.i.d and follow a normal distribution with variance $\sigma^{2}$. Under the null hypothesis that a factorial effect 0 ,

a. Show that

\[\frac{\bar{y}_{+}-\bar{y}_{-}}{s / 2} \sim t_{2^{k}(m-1)}\]

in a $2^{k}$ design with $m$ replications.

b. Suppose that $y_{i j}$ are not normal. Why is it still reasonable to assume that the ratio of the factorial effect to its standard error follows a normal distribution?