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 19.8 .

(a) Construct a \(95 \%\) confidence interval for \(\sigma^{2}\) if the sample size, \(n\), is 10 .

(b) Construct a \(95 \%\) confidence interval for \(\sigma^{2}\) if the sample size, \(n\), is 25 . How does increasing the sample size affect the width of the interval?

(c) Construct a \(99 \%\) confidence interval for \(\sigma^{2}\) if the sample size, \(n\), is 10.

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