I'm having trouble with this textbook question and would be extremely grateful if someone solved and showed me the steps to the solution of this question.

2. Y. and Yo are Bernoulli random variables from two different populations, denoted a and b. Suppose E(Y.) = p. and E(Yb) = p. A random sample of size n. is chosen from population a, with a sample average denoted p. and a random sample of size ng is chosen from population b, with a sample average denoted ps. Suppose the sample from population a is independent of the sample from population b. a. Show that E(p.) = P. and var(pa) = pa(1 - Pa)/ na. b. Show that var(p, - py) = Pall-pal + al-m). (Hint: Remember that the samples are independent.) c. Suppose na and my are large. Show that a 95% confidence interval for Pa - Pe is given by (Pa - Pb) + 1.96 Pa(1 - Pa) Pe(1 - pe) (1) d. Read the box "A Novel Way to Boost Retirement Savings" in Section 3.5 of the textbook. 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, pa - Ps