University officials are planning to audit 1586 new appointments to estimate the proportion p who have been incorrectly processed by the payroll department.

(a) How large does the sample size need to be in order for X/n, the sample proportion, to have an 85% chance of lying within 0.03 of p?

(b) Past audits suggest that p will not be larger than 0.10. Using that information, recalculate the sample size asked for in part (a).