3. You wish to estimate T, the total number of palm trees on a collection of 100 tropical islands. You know the area of each and that the sum of these areas is 100 sq. miles.

(a) Describe how you would conduct a p.p.s. sample with replacement of size 4 from the islands to estimate the total number of palm trees.

(b) You conduct the sample and the results are given in Table 1. Estimate r using a Hansen

Hurwitz estimator and estimate the standard error.

(c) Let pi be the probability that island i is chosen in a given draw from the population of islands. Show that for the sample described above, W, the probability that island i is included in the whole sample, is equal to 1 - (1 - Pi).

(d) Estimate using a Horvitz-Thompson estimator.