# Construct a confidence interval given sample and population size for binary responses… I think

If 1,000,000 t-shirts exist in total, and you can track 25% of them (see if they have been purchased or not), and you find that of that 25% half have been purchased, you might believe that half of the total amount of t-shirts have been purchased. Given this 25% sample size, how much deviation from your measured purchase proportion can you expect in the entire population. I'm assuming this could be quantified somehow (with a confidence interval), but don't know how to go about it. Thanks for any help!

So 95 confidence interval is $$0.5 \pm 1.96\sqrt{0.00000075} =(49.83\%, 50.17\%)$$.
• For variance Var = f $S^2$, SE =$\sqrt f S$. I used f in the var, so do not need sqrt. – user158565 Dec 9 '18 at 21:57