Confidence Interval of a Population
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.
Re: Confidence Interval of a Population
To show that they are unbiased, you need to show that the expectation of the sample mean is equal to the parameter.
In a Bernoulli distribution with n samples, you know that the mean of a Bernoulli is simply Successes/Outcomes = p where p is a constant in this particular example.
Now the sample mean is given by [X1 + X2 + ... + XN]/N but all the X's are independent random Bernoulli random variables.
To be unbiased, you need to show that E[x_bar] = p. The first thing to figure out is what the expectation of a single observation is and use that to prove the result.
The variance is pretty much the same, but for this you want to use the fact that the sum of independent Bernoulli's with the same parameter is Binomial distribution with parameters n and p. So you know that [X1 + X2 + ... + XN]/N = 1/N * Binomial(n,p) and Var[1/N * X] = 1/N^2 * Var[X].