a. Suppose that the cost of a survey is C = C0 + C=n, where C0 is a startup cost an...

a. Suppose that the cost of a survey is C = C₀ + C=n, where C₀ is a startup cost and C₁ is the cost per observation. For a given cost C, find the allocation n₁, . . . , 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 stratum—in some strata the observations might be relatively cheap and in others relatively expensive. The cost of a survey with an allocation n₁, . . . , n_L is

For a fixed total cost C, what choice of n₁, · · ·, 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.