a. Suppose that the cost of a survey is C = C0 + C1n,

Chapter , Problem 54

(choose chapter or problem)

a. Suppose that the cost of a survey is \(C=C_{0}+C_{1} n\), where \(C_0\) is a startup cost and \(C_1\) is the cost per observation. For a given cost C, find the allocation \(n_{1}, \ldots, n_{L}\) to L strata that is optimal in the sense that it minimizes the variance of the estimate of the population mean subject to the cost constraint.

b. Suppose that the cost of an observation varies from stratum to stratumin some strata the observations might be relatively cheap and in others relatively expensive. The cost of a survey with an allocation \(n_{1}, \ldots, n_{L}\) is

\(C=C_{0}+\sum_{l=1}^{L} C_{l} n_{l}\)

For a fixed total cost C, what choice of \(n_{1}, \cdots, n_{L}\) minimizes the variance?

c. Assuming that the cost function is as given in part (b), for a fixed variance, find \(n_{l}\) to minimize cost.