Question

What would happen to the mean of the sampling distribution of p-hat if the Statistics major predicted the sampling distribution of p-hat for samples of size 500 instead of size 100?

Answer 1

Explanation:

If repeated sample sizes of large sizes are taken at random, and proportion P is calculated for samples the sample mean will have a normal distribution irrespective of the original distribution.

In other words, the sample proportion will follow a normal distribution with mean = p-hat and std deviation =
`\sqrt{(p(1-p))/(n) }`

This is a direct corrollary of central limit theorem for sample means.

Hence we have irrespective of sample size, sample proportion will have expected value same as p-hat.

So whether sample size is 500 or 100 the p hat will have the same distribution.