Given the data below, what is the 90% confidence interval for the variance? 22, 23, 19, 17, 29, 25

Question

Quizplus · Accepted Answer

First, we need to find the sample variance:
s^2 = Sum((x_i - x̄)^2)/(n-1) = (22-22.67)^2 + (23-22.67)^2 + (19-22.67)^2 + (17-22.67)^2 + (29-22.67)^2 + (25-22.67)^2 / 5 = 20.47
Next, we need to use a chi-squared distribution to find the critical values. Since we have n=6 observations, we have n-1=5 degrees of freedom. We want the 90% confidence interval, which means alpha=0.1 and we split it into two tails of equal size, so alpha/2=0.05 in each tail. Looking at a chi-squared distribution table with 5 degrees of freedom, we find that the critical values are 9.24 and 0.83.
Finally, we use the formula for the confidence interval:
[C(n-1)*s^2 / chi2(a/2,n-1), C(n-1)*s^2 / chi2(1-a/2,n-1)]
Plugging in our values, we get:
[5*20.47/9.24, 5*20.47/0.83] = [11.10, 105.10]
So the 90% confidence interval for the variance is (approximately) [11.10, 105.10]. Therefore, the best choice is C.

Given the Data Below, What Is the 90% Confidence Interval