In a random sample of $545$ accountants engaged in preparing county operating budgets for use in planning and control, $117$ indicated that estimates of cash flow were the most difficult element of the budget to derive.

a$.$ Test at the $5\%$ level the null hypothesis that at least $25\%$ of all accountants find cash flow estimates the most difficult estimates to derive.

b$.$ Based on the procedure used in part a$,$ what is the probability that the null hypothesis would be rejected if the true percentage of those finding cash flow estimates most difficult was each of the following?