A stratified sample is being designed to estimate the prevalence p of a rare characteristic, say the proportion of residents in Milwaukee, Wisconsin, who have Lyme disease. Stratum 1, with N1 units, has a high prevalence of the characteristic; stratum 2, with N2 units, has low prevalence. Assume that the cost to sample a unit (for example, the cost to select a person for the sample and determine whether he or she has Lyme disease) is the same for each stratum, and that at most 2000 units are to be sampled.
a. Let p1 and p2 be the proportions in stratum 1 and stratum 2 with the rare characteristic.
If p1 = 0.10, p2 = 0.03, and N1/N = 0.4, what are n1 and n2 under optimal allocation?
b. If p1 = 0.10, p2 = 0.03, and N1/N = 0.4, what is V(ˆpstr) under proportional allocation? Under optimal allocation? What is the variance if you take an SRS of 2000 units from the population?
c. (Use a spreadsheet for this part of the exercise.) Now fix p = 0.05. Let p1 range from 0.05 to 0.50, and N1/N range from 0.01 to 0.50 (these two values then determine the value of p2). For each combination of p1 and N1/N, find the optimal allocation, and the variance under both proportional allocation and optimal allocation. Also find the variance from an SRS of 2000 units. When does the optimal allocation give a substantial increase in precision when compared to proportional allocation? When compared to an SRS?

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