Let Y

_{a }and Y

_{b} denote Bernoulli random variables from 2 different populations, denoted a and b. suppose that E (Y

_{a})= P

_{a }and E (Y

_{b}) = P

_{b}. A random sample of size n

_{a} is chosen from population a, with sample average denoted P^

_{a} and a random sample of size n

_{b} is chosen from the population b, with sample average denoted p^

_{b} Sample from population a is independent from sample from population b.

Show that E (P^

_{a}) = P

_{a }and var (P^

_{a})= P

_{a}(1- P

_{a})/ n

_{a.}
Show that E(P^

_{b}) = P

_{b }and var (P^

_{b})= P

_{b}(1- P

_{b})/ n

_{b.}
Show that var (P^

_{a }- P^

_{b})= P

_{a}(1- P

_{a})/ n

_{a} + P

_{b}(1- P

_{b})/ n

_{b}
Suppose that n

_{a }and n

_{b} are large. Show that a 95% confidence interval for P

_{a }- P

_{b }is given by (P^

_{a }- P^

_{b}) plus and minus 1.96 sqrt P^

_{a}(1- P^

_{a})/ n

_{a} + P^

_{b}(1- P^

_{b})/ n

_{b.}
How would you construct a 90% confidence interval for P

_{a }- P

_{b}?

I have missed the class when prof. taught these....