Elected officials in a California city are preparing the annual budget for their community. They would like to estimate how much their constituents living in this city are typically paying each year in real estate taxes. Given that there are over 100,000 homeowners in this city, the officials have decided to sample a representative subset of taxpayers and study their tax payments.
a. What sample size is required to generate a 95% confidence interval for the mean annual real estate tax payment with a half-length of $100? Assume that the best estimate of the population standard deviation σ is $535.
b. If a random sample of the size from part a is selected and a 95% confidence interval for the mean is calculated from this sample, will the half length of the confidence interval be equal to $100? Explain why or why not.
c. Now suppose that the officials want to construct a 95% confidence interval with a half-length of $75. What sample size is required to achieve this objective? Again, assume that the best estimate of the population standard deviation σ is $535. Explain the difference between this result and the result from part a.