For a hypothetical survey to determine the number of pileated woodpecker nests, the study area is divided into N = 4 plots. For the ith plot in the population, yi is the number of nests, while xi is the number of "snags" (old trees that provide nesting habitat). The values for each population unit follow: y1 = 2, x1 = 20; y2 = 3, x2 = 25; y3 = 0, x3 = 0; y4 = 1, x4 = 15. Consider a simple random sampling design with sample size n = 2.
(a) Make a table listing every possible sample of size 2, the probability of selecting each sample, the estimate Ny of the population total for each sample, and the ratio estimate τˆr for each sample.
(b) Give the value and variance expected for the estimator Ny.
(c) Compute the exact mean and mean square error of the ratio estimator τˆr.