In a random sample of six people, the mean driving distance to work was 22.6 miles and the standard deviation was 7.4 miles. Assume the population is normally distributed and use the t-distribution to find the margin of error and construct a $99 \%$ confidence interval for the population mean $\mu$. Interpret the results. 12.2 miles (Round to one decimal place as needed.) Construct a $99 \%$ confidence interval for the population mean. $(10.4,34.8)$ (Round to one decimal place as needed.) Interpret the results. Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or a decimal. Do not round.) A. With $\square \%$ confidence, it can be said that the population mean driving distance to work (in miles) is between the interval's endpoints. B. It can be said that $\square \%$ of the population has a driving distance to work (in miles) that is between the interval's endpoints. C. With $\square \%$ confidence, it can be said that most driving distances to work (in miles) in the population are between the interval's endpoints. D. $\square \%$ of all random samples of six people from the population will have a mean driving distance to work (in miles) that is between the interval's endpoints. View an example Get more help . Clear all Check answer ins , is an ces he shoe

Step 1 ：Given that the sample mean (\(\bar{x}\)) is 22.6 miles, the sample standard deviation (s) is 7.4 miles, and the sample size (n) is 6 people.

Step 2 ：We are asked to find a 99% confidence interval for the population mean (\(\mu\)). This can be calculated using the formula: \(\bar{x} \pm t \frac{s}{\sqrt{n}}\), where t is the t-score corresponding to the desired level of confidence.

Step 3 ：Since the sample size is 6, the degrees of freedom is 5 (since \(n - 1 = 5\)). The t-score for a 99% confidence interval with 5 degrees of freedom can be found using a t-distribution table or a statistical calculator. The t-score is approximately 4.032.

Step 4 ：Substituting the given values into the formula, we get the margin of error to be approximately 12.2 miles.

Step 5 ：Subtracting and adding the margin of error from the sample mean, we get the 99% confidence interval for the population mean to be approximately (10.4, 34.8) miles.

Step 6 ：Interpreting the results, we can say that with 99% confidence, the population mean driving distance to work is between 10.4 miles and 34.8 miles. This is because a confidence interval estimates the range of values within which the population parameter (in this case, the mean driving distance to work) is likely to fall.

Step 7 ：Final Answer: The 99% confidence interval for the population mean is \(\boxed{(10.4, 34.8)}\). This means that we are 99% confident that the true population mean driving distance to work is between 10.4 miles and 34.8 miles.