Another one of those review exercises questions!

A box contains an unknown number of white and black balls. We wish to estimate the proportion p of white balls in the box. To do so, we draw n successive balls with replacement. Let be the proportion of white balls obtained after n drawings.

(i) Show that for all ,

(ii) Using the result in part (i), find the smallest value of n such that with probability greater than 0.95, the proportion in the sample will estimate p within 0.1.

(iii) Same question as in (ii) using the central limit theorem.

Thank you again guys! Been great help, really!