There are 1000 students in the University of A. A sample of 10 observations was obtained regarding the rating of the service quality of dining hall. Here is the sample:

Observation Rating
1 5
2 6
3 7
4 3
5 8
6 5
7 6
8 3
9 4
10 7

calculate the confidence interval of the sample mean of the ratings.

To calculate the confidence interval of the sample mean of the ratings, you can follow these steps:

Step 1: Calculate the sample mean (x̄) of the ratings.
Sum up all the ratings and divide the total by the number of observations:
x̄ = (5 + 6 + 7 + 3 + 8 + 5 + 6 + 3 + 4 + 7) / 10 = 54 / 10 = 5.4

Step 2: Calculate the sample standard deviation (s) of the ratings.
Subtract the mean from each observation, square the results, and calculate the average of the squared differences. Finally, take the square root of the average to get the sample standard deviation:
s = √[(5-5.4)² + (6-5.4)² + (7-5.4)² + (3-5.4)² + (8-5.4)² + (5-5.4)² + (6-5.4)² + (3-5.4)² + (4-5.4)² + (7-5.4)²] / 9 ≈ 1.56

Step 3: Determine the desired confidence level (e.g., 95% confidence level).
The confidence level expresses the level of confidence you have in the estimated interval. It is typically given as a percentage, such as 90% or 95%.

Step 4: Determine the critical value (z) for the chosen confidence level.
The critical value depends on the sample size and the desired confidence level. If the population standard deviation (σ) is known, you can use the z-table to find the critical value. Otherwise, you can use the t-table with the degrees of freedom (n-1) to estimate it. In this case, we don't have the population standard deviation, so we will use the t-table.

Since the sample size is 10, the degrees of freedom (df) are 10-1 = 9.
For a 95% confidence level, we need to find the critical value for α/2 = 0.025 (2.5% on each tail). Looking at the t-table with 9 degrees of freedom, the critical value is approximately 2.262.

Step 5: Calculate the margin of error (E) using the critical value and sample standard deviation.
The margin of error measures the amount of uncertainty in our estimate. It is calculated by multiplying the critical value by the sample standard deviation and dividing it by the square root of the sample size:
E = (z * s) / √n = (2.262 * 1.56) / √10 ≈ 1.43

Step 6: Calculate the lower and upper bounds of the confidence interval.
The lower bound is obtained by subtracting the margin of error from the sample mean, and the upper bound is obtained by adding the margin of error to the sample mean:
Lower bound = x̄ - E = 5.4 - 1.43 ≈ 3.97
Upper bound = x̄ + E = 5.4 + 1.43 ≈ 6.83

Therefore, the 95% confidence interval for the sample mean of the ratings is approximately (3.97, 6.83).