Suppose we are interested in estimating the proportion p of a population that has a certain disease. As in Section 2.3 let yi =1 if person i has the disease, and yi =0 if person i does not have the disease. Then Ṕ=.
a. Show, using the definition in (2.13), that

If the population is large and the sampling fraction is small, so that

Write (2.26) in terms of the CV for a sample of size 1. Histogram of the means of 1000 samples of size 300, taken with replacement from the data in Example 2.5.

b. Suppose that the fpc ≈ 1. Consider populations with p taking the successive values
0.001, 0.005, 0.01, 0.05, 0.10, 0.30, 0.50, 0.70, 0.90, 0.95, 0.99, 0.995, 0.999.
For each value of p, find the sample size needed to estimate the population proportion (a) with fixed margin of error 0.03, using (2.25), and (b) with relative error 0.03p, using (2.26). What happens to the sample sizes for small values of p?