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, respectively. On the basis of past experience, it is estimated that the standard deviations of values 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 estimate, determine the total sample size needed under both proportional allocation and optimal allocation.