Suppose the cost of an observation varies from stratum to stratum, where the cost of sampling a unit from stratum j is cj . Suppose there is a fixed start-up cost co to run the survey, so the total cost is co + c1n1 +· · ·+cJ nJ .
Now instead of fixing the total sample size n, we fix total survey cost C. Find the allocation of n1, ..., nJ that minimizes the variance of the estimator for the population mean. You may ignore the finite population correction factor.