1) While imprisoned, a mathematician suspects that a coin in his pocket is not fair. In particular, the probability that a head will be tossed is not known. Call this probability . In order to estimate , a test is run by tossing the coin 100 times and counting the number of times heads comes up. It turns out the mathematician obtains 57 heads.

(a) Use the data to give an estimate for .

(b) Find 95% confidence interval for .

(c) How many tosses of the coin are needed in order to be 99% confident that a future sample proportion, , will be within 0.05 of the true ?