(Y_{a}) and (Y_{b}) are Bernoulli random variables from two different populations, denoted (a) and (b). Suppose (Eleft(Y_{a} ight)=p_{a}) and (Eleft(Y_{b} ight)=p_{b}). A random sample of

\(Y_{a}\) and \(Y_{b}\) are Bernoulli random variables from two different populations, denoted \(a\) and \(b\). Suppose \(E\left(Y_{a}\right)=p_{a}\) and \(E\left(Y_{b}\right)=p_{b}\). A random sample of size \(n_{a}\) is chosen from population \(a\), with a sample average denoted \(\hat{p}_{a}\), and a random sample of size \(n_{b}\) is chosen from population \(b\), with a sample average denoted \(\hat{p}_{b}\). Suppose the sample from population \(a\) is independent of the sample from population \(b\).

a. Show that \(E\left(\hat{p}_{a}\right)=p_{a}\) and \(\operatorname{var}\left(\hat{p}_{a}\right)=p_{a}\left(1-p_{a}\right) / n_{a}\). Show that \(E\left(\hat{p}_{b}\right)=p_{b}\) and \(\operatorname{var}\left(\hat{p}_{b}\right)=p_{b}\left(1-p_{b}\right) / n_{b}\).

b. Show that \(\operatorname{var}\left(\hat{p}_{a}-\hat{p}_{b}\right)=\frac{p_{a}\left(1-p_{a}\right)}{n_{a}}+\frac{p_{b}\left(1-p_{b}\right)}{n_{b}}\). Remember that the samples are independent.

c. Suppose \(n_{a}\) and \(n_{b}\) are large. Show that a \(95 \%\) confidence interval for \(p_{a}-p_{b}\) is given by \(\left(\hat{p}_{a}-\hat{p}_{b}\right) \pm 1.96 \sqrt{\frac{\hat{p}_{a}\left(1-\hat{p}_{a}\right)}{n_{a}}+\frac{\hat{p}_{b}\left(1-\hat{p}_{b}\right)}{n_{b}}}\). How would you construct a \(90 \%\) confidence interval for \(p_{a}-p_{b}\) ?

d. Read the box "A Novel Way to Boost Retirement Savings" in Section 3.5. Let population \(a\) denote the opt-out (treatment) group and population \(b\) denote the opt-in (control) group. Construct a \(95 \%\) confidence interval for the treatment effect, \(p_{a}-p_{b}\).