I am supposed to be finding the confidence interval for P(X=1) = EX = unemployment rate.

I have X = 1 if unemployed, 0 if employed. Y = 1 if no response, 0 if response.

I had to find unbiased estimators for theta1 = P(X=1,Y=0) and theta2 = P(Y=1). I also need to prove they are unbiased.

I am given n, number of questionnaires; n1 are collected back; Among the n1, m are unemployed.

I have theta1(hat) = m/n and theta2(hat) = 1 - n1/n.

I am not sure how to show these are unbiased and could use some direction.

Secondly, I need to find the variance of theta1(hat) and the 95% confidence interval for theta1(hat).

Any help is appreciated.