This question: 1 point(S) possible Submit test Listed below are student evaluation ratings of courses, where a rating of 5 is for "excellent." The ratings were obtained at one university in a state. Construct a confidence interval using a $99 \%$ confidence level. What does the confidence interval tell about the population of all college students in the state? \[ 3.5,3.1,4.0,4.6,3.0,3.9,3.7,4.5,4.5,4.2,4.3,3.6,3.4,3.9,3.7 \] What is the confidence interval for the population mean $\mu$ ? $\square<\mu<\square$ (Round to two decimal places as needed.) What does the confidence interval tell about the population of all college students in the state? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The results tell nothing about the population of all college students in the state, since the sample is from only one university. B. We are $99 \%$ confident that the interval from $\square$ to $\square$ actually contains the true mean evaluation rating. (Round to one decimal place as needed.) C. We are confident that $99 \%$ of all students gave evaluation ratings between $\square$ and $\square$. (Round to one decimal place as needed.)
Solution
Step 1 ：First, we need to calculate the sample mean, the sample standard deviation, and the sample size from the given ratings.
Step 2 ：The sample mean is calculated as the sum of all ratings divided by the number of ratings, which gives us a mean of 3.86.
Step 3 ：The sample standard deviation is calculated using the formula for standard deviation, which gives us a standard deviation of approximately 0.498.
Step 4 ：The sample size is simply the number of ratings, which is 15.
Step 5 ：Next, we need to calculate the confidence interval. The formula for a confidence interval is the sample mean plus or minus the z-score times the sample standard deviation divided by the square root of the sample size.
Step 6 ：The z-score for a 99% confidence level is approximately 2.576.
Step 7 ：Substituting the values into the formula, we get a confidence interval of (3.53, 4.19).
Step 8 ：Finally, we round the confidence interval to two decimal places, as specified in the question, to get a final confidence interval of (3.53, 4.19).
Step 9 ：The confidence interval for the population mean μ is \(\boxed{3.53}<\mu<\boxed{4.19}\). This means we are 99% confident that the interval from 3.53 to 4.19 actually contains the true mean evaluation rating.
