Question: Continuing the previous problem, suppose that the officials in this city want to estimate the difference between the proportions, labeled p1 and p2, of taxpayers living in neighborhood 1 whose annual real estate tax payments exceed $2000 and the similar proportion for taxpayers living in neighborhood 2.
a. What sample size (randomly selected from all taxpayers residing in each of neighborhoods 1 and 2) is required to generate a 90% confidence interval for this difference between proportions with a half length of 0.10? Assume for now that p1 and p2 are both close to 0.5.
b. We assumed that the two population proportions in part a are both close to 0.5. Use a two-way data table to find the required (common) sample size when each of the population proportions is allowed to vary from 0.1 to 0.9 in increments of 0.1. Comment on the sensitivity of the required sample size to the magnitudes of the population proportions. "The probability that the population mean is between 123.7 and 155.2 is 0.95." Why is this, strictly speaking, an invalid statement? How would you rephrase it to make it a valid statement?