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Chapter 8: Problem 29

The travel-to-work time for residents of the 15 largest cities in the United States is reported in the 2003 Information Please Almanac. Suppose that a preliminary simple random sample of residents of San Francisco is used to develop a planning value of 6.25 minutes for the population standard deviation. a. If we want to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 2 minutes, what sample size should be used? Assume \(95 \%\) confidence b. If we want to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 1 minute, what sample size should be used? Assume \(95 \%\) confidence

Short Answer

Expert verified
a. To estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 2 minutes, a sample size of \(37\) should be used. b. To estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 1 minute, a sample size of \(145\) should be used.

Step by step solution

01

Find the Z-score

For a \(95\%\) confidence interval, the Z-score is \(Z_\frac{α}{2} = Z_{0.975}\), which is approximately \(1.96\). You can find this value using a standard normal table or a calculator.
02

Determine the Margin of Error

For part a, the margin of error is 2 minutes. For part b, the margin of error is 1 minute.
03

Solve for the Sample Size

Rearrange the formula for the margin of error to solve for the sample size, n: \(n = (\frac{Z_\frac{α}{2} * σ}{Margin\ of\ Error})^2\)
04

Calculate the Sample Size for Part A

Use the given values to calculate the sample size for a margin of error of 2 minutes: \(n = (\frac{1.96 * 6.25}{2})^2\) \(n \approx 36.03\) Round up to the nearest whole number because you can't have a fraction of a sample. So, the required sample size for part a is 37.
05

Calculate the Sample Size for Part B

Use the given values to calculate the sample size for a margin of error of 1 minute: \(n = (\frac{1.96 * 6.25}{1})^2\) \(n \approx 144.12\) Round up to the nearest whole number because you can't have a fraction of a sample. So, the required sample size for part b is 145.

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