The file azcounties.dat gives data from the 2000 U.S. Census on population and housing unit counts for the counties in Arizona (excluding Maricopa County and Pima County, which are much larger than the other counties and would be placed in a separate stratum). For this exercise, suppose that year 2000 population (Mi) is known and you want to take a sample of counties to estimate the total number of housing units (t = Ʃ13i=1 ti). The file has the value of ti for every county so you can calculate the population total and variance.
a. Calculate the selection probabilities ψi for a sample of size 1 with probability proportional to 2000 population. Find ṫψ for each possible sample, and calculate the theoretical variance V (ˆtψ).
b. Repeat (a) for an equal probability sample of size 1. How do the variances compare? Why do you think one design is more efficient than the other?