A random sample of 20 observations is selected from a normally distributed population. The sample variance is 12. In the 95% confidence interval for the population variance, the upper limit will be:
A) 6.940
B) 7.564
C) 22.536
D) 25.599
E) 27.392

Question

Quizplus · Accepted Answer

We use the chi-square distribution with n-1 degrees of freedom to calculate the confidence interval.

The interval is given by:

( (n-1)*s^2 ) / chi-squared(alpha/2, n-1) <= sigma^2 <= ( (n-1)*s^2 ) / chi-squared(1-alpha/2, n-1)

where s^2 is the sample variance, n is the sample size, alpha is the significance level, and chi-squared is the chi-square distribution function.

Substituting the values gives:

(19*12) / chi-squared(0.025, 19) <= sigma^2 <= (19*12) / chi-squared(0.975, 19)

Using a chi-square table or calculator, we find that:

chi-squared(0.025, 19) = 32.8523
and 
chi-squared(0.975, 19) = 8.9065

Substituting and simplifying gives:

140.16 / 32.8523 <= sigma^2 <= 140.16 / 8.9065
4.2651 <= sigma^2 <= 15.7328

The upper limit of the interval is 15.7328, which corresponds to choice letter D, 25.599.

A Random Sample of 20 Observations Is Selected from a Normally