Question

The Center for Urban Transportation Research released a report stating that the average commuting distance in the United States is \(10.9 \mathrm{mi}\) (USA Today, August 13 , 1991). Suppose that this average is actually the mean of a random sample of 300 commuters and that the sample standard deviation is \(6.2 \mathrm{mi}\). Estimate the true mean commuting distance using a \(99 \%\) confidence interval.

Short answer

The estimated true mean commuting distance in the United States, using a 99% confidence interval is between 10.14 and 11.66 miles.

Step-by-step solution

Understand the problem

We are given that the sample size (n) is 300, the sample mean (\(\bar{x}\)) is 10.9 miles, and the sample standard deviation (s) is 6.2 miles. We are tasked with computing a 99% confidence interval for the population mean. This will require use of the confidence interval formula: \(\bar{x} \pm Z_{\frac{\alpha}{2}} \frac{s}{\sqrt{n}}\), where \(Z_{\frac{\alpha}{2}}\) represents the z-score associated with a 99% confidence level.

Determine the z-score

First we need to find the z-score that corresponds to a 99% confidence level. Since the normal distribution is symmetric, the remaining 1% (or 0.01) is split evenly into the two tails of the distribution, so we need to find the z-score that leaves 0.005 in the tail. Checking a standard z-table or using a z-score calculator shows that the z-score is approximately 2.58.

Apply the confidence interval formula

Now that we have the z-score, we can substitute the given values into the confidence interval formula: \(10.9 \pm 2.58 \frac{6.2}{\sqrt{300}}\). When we simplify the equation we obtain the confidence interval ranges (10.14, 11.66).

Interpret the confidence interval

The 99% confidence interval for the mean commuting distance in the United States is \( (10.14, 11.66) \). The interpretation of this interval is that, if we were to sample many groups of 300 commuters over and over again and calculate the 99% confidence interval for each sample, approximately 99% of these intervals would contain the true population mean.