Given is a group of 82 people, 7 of those are at risk for some disorder. The overall goal is to model the proportion p of people with the disorder - and it is clear that this can be modeled by a Binomial distribution.
The task, however, goes on to say the following:
"With the central limit theorem (CLT) it follows that $\hat{p}$ follows approximately N(p, p$\frac{1}{n}$(1-p)). Plot the normal approximation for different n and p, compare the binomial distribution to the normal approximation for different n and p. Calculate the 95%-confidence interval (CI) for the approximate proportion".
I am very confused because it is not clear to me what $\hat{p}$ is - I assume it is the estimated distribution (and not just the success parameter). But would this not be N(np, np(1-p))? And if this is the estimated distribution, how can I construct a CI for it?