Chapter 7.6, Problem 35E

Thirty-six percent of all Americans drink bottled water more than once a week (Natural Resources Defense Council, December 4, 2015). Suppose you have been hired by the Natural Resources Defense Council to investigate bottled water consumption in St. Paul. You plan to select a sample of St. Paulites to estimate the proportion who drink bottled water more than once a week. Assume the population proportion of St. Paulites who drink bottled water more than once a week is .36, the same as the overall proportion of Americans who drink bottled water more than once a week.
a. Suppose you select a sample of 540 St. Paulites. Show the sampling distribution of $\bar{p}$
b. Based upon a sample of 540 St. Paulites, what is the probability that the sample proportion will be within .04 of the population proportion?
c. Suppose you select a sample of 200 St. Paulites. Show the sampling distribution of $\bar{p}$ .
d. Based upon the smaller sample of only 200 St. Paulites, what is the probability that the sample proportion will be within .04 of the population proportion?
e. As measured by the increase in probability, how much do you gain in precision by the larger sample in parts (a) and (b) rather than the smaller sample in parts (c) and (d)?