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.
For some reason the numbers are together example 1=5 not 15
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To calculate the confidence interval of the sample mean of the ratings, we need to know the sample mean, sample standard deviation, and the desired level of confidence.
Step 1: Calculate the sample mean
Add up all the ratings and divide by the number of observations:
(5+6+7+3+8+5+6+3+4+7) / 10 = 54 / 10 = 5.4
So, the sample mean is 5.4.
Step 2: Calculate the sample standard deviation
To calculate the sample standard deviation, we first need to calculate the sample variance.
Calculate the squared difference between each observation and the sample mean, add them up, and divide by (n - 1), where n is the number of observations.
For each observation, subtract the sample mean and square the difference:
(5-5.4)^2 = 0.16
(6-5.4)^2 = 0.36
(7-5.4)^2 = 2.56
(3-5.4)^2 = 6.76
(8-5.4)^2 = 6.76
(5-5.4)^2 = 0.16
(6-5.4)^2 = 0.36
(3-5.4)^2 = 6.76
(4-5.4)^2 = 1.96
(7-5.4)^2 = 2.56
Sum up all the squared differences and divide by (n - 1):
(0.16 + 0.36 + 2.56 + 6.76 + 6.76 + 0.16 + 0.36 + 6.76 + 1.96 + 2.56) / (10 - 1) = 27.72 / 9 = 3.08
Now calculate the square root of the sample variance:
√(3.08) ≈ 1.76
So, the sample standard deviation is approximately 1.76.
Step 3: Determine the desired level of confidence
Let's assume we want a 95% confidence interval. This means that we want to be 95% confident that the calculated interval contains the true population mean rating.
Step 4: Calculate the margin of error
The margin of error depends on the sample standard deviation and the desired level of confidence. We need the critical value from the t-distribution table with (n - 1) degrees of freedom.
For a 95% confidence level and (n - 1) = 9 degrees of freedom, the critical value is approximately 2.262.
The margin of error equals the critical value times the standard deviation divided by the square root of the sample size:
Margin of error = 2.262 * (1.76 / √10) ≈ 1.12
Step 5: Calculate the confidence interval
The confidence interval can be calculated by subtracting and adding the margin of error to the sample mean:
Lower limit = Sample mean - Margin of error
Upper limit = Sample mean + Margin of error
Lower limit = 5.4 - 1.12 ≈ 4.28
Upper limit = 5.4 + 1.12 ≈ 6.52
So, the confidence interval of the sample mean of the ratings is approximately (4.28, 6.52) at a 95% confidence level.