This is a statistics problem about sampling.

3. [32 points] A forester wants to estimate the proportion of trees in a certain county that are infected with a particular disease. There are 10 well defined forest areas in the county; these areas can be subdivided into plots of approximately the same size. Two-stage cluster sampling is used: 4 areas (clusters) are chosen with 6 plots (elements) randomly selected from each area. The sample data are summarized as follows. Number of Number of Average number of Sample variance Area plots plots sampled infected trees, go 1 12 6 13.67 18.20 2 15 7.00 5.60 14 11.50 5.80 21 4.00 6.40 (1) [8 points] Estimate the total number of plots in the county. (2) [8 points] Place a bound on the error of estimation in Question (1). 7 (3) [8 points] Estimate the average number of infected trees per plot in the county