The manager of the dairy section of a large supermarket took a random sample of 250 egg cartons and found that 40 cartons had at least one broken egg. 

(a) Let p be the proportion of egg cartons with at least one broken egg out of the population of all egg cartons stocked by this store. Find a point estimate for p. 

(b) Find a 90% confidence interval for p. How many egg cartons would the manager need to examine to be 90% sure that the point estimate of the proportion of cartons with at least one broken egg is within 3% of the population proportion p?

(c) assuming a prior estimate of p equal to 0.16, 

(d) assuming no prior point estimate.