A simple random sample of size \(n\) is drawn from a population that is known to be normally distributed. The sample variance, \(s^{2}\), is determined to be 12.4.

(a) Construct a \(90 \%\) confidence interval for \(\sigma^{2}\) if the sample size, \(n\), is 20 .

(b) Construct a 90\% confidence interval for \(\sigma^{2}\) if the sample size, \(n\), is 30 . How does increasing the sample size affect the width of the interval?

(c) Construct a \(98 \%\) confidence interval for \(\sigma^{2}\) if the sample size, \(n\), is 20.

Compare the results with those obtained in part (a). How does increasing the level of confidence affect the confidence interval?