Question 9 of 15
A class survey in a large class for first-year college students asked, "About how many hours do you study during a typical week?" The mean response of the 463 students was $\bar{x}=13.7$ hours. Suppose that we know that the study time follows a Normal distribution with standard deviation $\sigma=7.4$ hours in the population of all first-year students at this university. The $99 \%$ confidence interval was found to be approximately 12.814 to 14.586 hours.

There were actually 464 responses to the class survey. One student had claimed to study 10,000 hours per week (10,000 is more than the number of hours in a year). We know that student is joking, so we left out this value. If we did a calculation without looking at the data, we would get $\bar{x}=35.2$ hours for all 464 students. Now what is the $99 \%$ confidence interval for the population mean? (Continue to use $\sigma=7.4$.) Give your answers to three decimal places.
I
lower bound:
upper bound:

Step 1 ：The question is asking for the 99% confidence interval for the population mean, given a sample mean and standard deviation. The formula for a confidence interval is: \[\bar{x} \pm Z \frac{\sigma}{\sqrt{n}}\] where \(\bar{x}\) is the sample mean, \(Z\) is the Z-score corresponding to the desired level of confidence, \(\sigma\) is the standard deviation, and \(n\) is the sample size.

Step 2 ：In this case, the sample mean is 35.2 hours, the standard deviation is 7.4 hours, and the sample size is 464. The Z-score for a 99% confidence interval is approximately 2.576.

Step 3 ：We can plug these values into the formula to find the lower and upper bounds of the confidence interval.

Step 4 ：Calculate the margin of error: \[Z \frac{\sigma}{\sqrt{n}} = 2.576 \frac{7.4}{\sqrt{464}} \approx 0.885\]

Step 5 ：Calculate the lower bound of the confidence interval: \[\bar{x} - \text{margin of error} = 35.2 - 0.885 = 34.315\]

Step 6 ：Calculate the upper bound of the confidence interval: \[\bar{x} + \text{margin of error} = 35.2 + 0.885 = 36.085\]

Step 7 ：Final Answer: The $99 \%$ confidence interval for the population mean is approximately \(\boxed{34.315}\) to \(\boxed{36.085}\) hours.