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....