An auditor wants to estimate the mean value of a corporation's accounts receivable. The population is divided into four strata, containing 500, 400, 300, and 200 accounts, respec- tively. On the basis of past experience, it is estimated that the standard deviations of val- ues in these strata will be $150, $200, $300, and $400, respectively. If a 90% confidence interval for the overall population mean is to extend $25 on each side of the sample esti- mate, determine the total sample size needed under each of the following schemes:

(a) Proportional allocation.

(b) Optimal allocation,