Let Ya and Yb denote Bernoulli random variables from 2 different populations, denoted a and b. suppose that E (Ya)= Pa and E (Yb) = Pb. A random sample of size na is chosen from population a, with sample average denoted P^a and a random sample of size nb 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) = Pa and var (P^a)= Pa(1- Pa)/ na.
Show that E(P^b) = Pb and var (P^b)= Pb(1- Pb)/ nb.
Show that var (P^a - P^b)= Pa(1- Pa)/ na + Pb(1- Pb)/ nb
Suppose that na and nb are large. Show that a 95% confidence interval for Pa - Pb is given by (P^a - P^b) plus and minus 1.96 sqrt P^a(1- P^a)/ na + P^b(1- P^b)/ nb.
How would you construct a 90% confidence interval for Pa - Pb?
I have missed the class when prof. taught these....