In an aerial survey of four plots selected by simple random sampling, the numbers of ducks detected were 44, 55, 4, and 16. Careful examination of photo imagery of the first and third of these plots (selected as a simple random subsample) revealed the actual presence of 56 and 6 ducks, respectively. The study area consists of 10 plots.
(a) Estimate the total number of ducks in the study area by using a ratio estimator. Estimate the variance of the estimator.
(b) Suppose that the photo analysis doubles the cost of observing a plot. Estimate the optimal subsampling fraction.