Consider a randomized pair design with $n$ units where two treatments are randomly assigned to each unit, resulting in a pair of observations $\left(X_{i}, Y_{i}\right)$, for $i=1, \ldots, n$ on each unit. Assume that $E\left[X_{i}\right]=\mu_{X}$, $E\left[Y_{i}\right]=\mu_{y}$, and $\operatorname{Var}\left(X_{i}\right)=\operatorname{Var}\left(Y_{i}\right)=\sigma^{2}$ for $i=1, \ldots, n$. Alternatively, we may consider an unpaired design where we assign two independent treatment groups to $2 n$ units.

a. Show that the ratio of the variances in the paired to the unpaired design is $1-ho$, where $ho$ is the correlation between $X_{i}$ and $Y_{i}$.

b. If $ho=0.5$, how many subjects are required in the unpaired design to yield the same precision as the paired design?