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? \[ 3.6,3.1,41,4.6,32,40,36,49,4.5,4.0,4.5,39,3.3,4.1,3.9 \text { ㅁ. } \] 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. We are confident that $99 \%$ of all students gave evaluation ratings between $\square$ and $\square$ (Round to one decimal place as needed) 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. The results tell nothing about the population of all college students in the state, since the sample is from only one university

Step 1 ：Given the student evaluation ratings of courses at a university, we are asked to construct a confidence interval using a 99% confidence level. The ratings are as follows: 3.6, 3.1, 4.1, 4.6, 3.2, 4.0, 3.6, 4.9, 4.5, 4.0, 4.5, 3.9, 3.3, 4.1, 3.9.

Step 2 ：First, we calculate the sample mean, which is the average of the ratings. This is done by adding all the ratings and dividing by the number of ratings. The sample mean is approximately 3.95.

Step 3 ：Next, we calculate the sample standard deviation, which measures the amount of variation or dispersion of the ratings. The sample standard deviation is approximately 0.53.

Step 4 ：We also note that the sample size, which is the number of ratings, is 15.

Step 5 ：We then find the Z-score for a 99% confidence level, which is approximately 2.58. The Z-score tells us how many standard deviations an element is from the mean.

Step 6 ：Using these values, we can calculate the confidence interval for the population mean. The confidence interval is given by the formula: sample mean ± Z-score * (sample standard deviation / square root of sample size). This gives us a confidence interval of approximately (3.60, 4.31).

Step 7 ：This means that we are 99% confident that the true mean student evaluation rating falls within this range. However, it's important to note that the confidence interval does not tell us anything about individual student evaluation ratings. It only provides information about the population mean.

Step 8 ：Also, the confidence interval does not tell us that 99% of all students gave evaluation ratings between 3.60 and 4.31. It only tells us that we are 99% confident that the true mean student evaluation rating falls within this range.

Step 9 ：Finally, the results of this analysis are based on a sample from one university, so they may not be representative of all college students in the state.

Step 10 ：The final answer is: The confidence interval for the population mean is \(\boxed{(3.60, 4.31)}\). We are \(\boxed{99\%}\) confident that the interval from \(\boxed{3.60}\) to \(\boxed{4.31}\) actually contains the true mean evaluation rating. The results tell nothing about the population of all college students in the state, since the sample is from only one university.