THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
In a stratified random sample of students on a campus,sample members were asked to rate,on a scale of 1 (poor)to 5 (excellent),opportunities for improving the college infrastructure.The results are shown in the table below.

-Find a 90% confidence interval for the mean rating given by all freshmen and sophomores on the campus.
A)3.98 ± 0.592
B)3.98 ± 0.229
C)3.98 ± 0.712
D)3.98 ± 0.456

Question

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
In a stratified random sample of students on a campus,sample members were asked to rate,on a scale of 1 (poor)to 5 (excellent),opportunities for improving the college infrastructure.The results are shown in the table below.
 
-Find a 90% confidence interval for the mean rating given by all freshmen and sophomores on the campus.
A)3.98 ± 0.592 
B)3.98 ± 0.229 
C)3.98 ± 0.712 
D)3.98 ± 0.456

Quizplus · Accepted Answer

The formula for finding the confidence interval for a stratified random sample is:
CI = X̄ ± tα/2 * (σ/√n)
where X̄ is the sample mean, tα/2 is the critical value from the t-distribution with n-1 degrees of freedom and a level of significance of α/2, σ is the population standard deviation, and n is the sample size.
From the table, we can see that the sample mean for freshmen and sophomores is 3.98. We do not know the population standard deviation, but we can estimate it using the sample standard deviation, which is 0.727. The sample size is 50.
Using a t-distribution table with 49 degrees of freedom (n-1) and a level of significance of 0.1 (since the confidence level is 90%), we find the critical value to be 1.677.
Plugging in these values into the formula, we get:
CI = 3.98 ± 1.677 * (0.727/√50)
CI = 3.98 ± 0.229
Therefore, the best choice is B, 3.98 ± 0.229.