A stratified sample is being designed to estimate the prevalence p of a rare characteristic, say the
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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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