The annual earnings of 13 randomly selected computer software engineers have a sample standard deviation of $\$ 3637$. Assume the sample is from a normally distributed population. Construct a confidence interval for the population variance $\sigma^{2}$ and the population standard deviation $\sigma$. Use a $99 \%$ level of confidence. Interpret the results. What is the confidence interval for the population variance $\sigma^{2}$ ? (5609043, 51640317) (Round to the nearest integer as needed) Interpret the results. Select the correct choice below and fill in the answer box(es) to complete your choice. (Round to the nearest integer as needed.) A. With $99 \%$ confidence, you can say that the population variance is greater than C. With $1 \%$ confidence, you can say that the population variance is between and B. With $99 \%$ confidence, you can say that the population variance is between 5609043 and 51640317 D. With $1 \%$ confidence, you can say that the population variance is less than What is the confidence interval for the population standard deviation $\sigma$ ? ( $\square . \square$ ) (Round to the nearest integer as needed.) Get more help . Clear all

Step 1 ：Given values are: sample size \(n = 13\), sample standard deviation \(s = 3637\), and confidence level \(0.99\).

Step 2 ：Calculate the degrees of freedom as \(df = n - 1 = 13 - 1 = 12\).

Step 3 ：Calculate the alpha value as \(alpha = 1 - confidence\_level = 1 - 0.99 = 0.01\).

Step 4 ：Calculate the chi-square values for the lower and upper bounds of the confidence interval. The lower bound is \(chi2\_lower = 3.073823638089334\) and the upper bound is \(chi2\_upper = 28.299518822046025\).

Step 5 ：Calculate the confidence interval for the population variance. The lower bound is \(variance\_lower = 5609043\) and the upper bound is \(variance\_upper = 51640317\).

Step 6 ：Calculate the confidence interval for the population standard deviation. The lower bound is \(std\_dev\_lower = 2368\) and the upper bound is \(std\_dev\_upper = 7186\).

Step 7 ：Final Answer: The confidence interval for the population variance \(\sigma^{2}\) is \(\boxed{(5609043, 51640317)}\). The confidence interval for the population standard deviation \(\sigma\) is \(\boxed{(2368, 7186)}\). This means that we are 99% confident that the true population variance lies between 5609043 and 51640317, and the true population standard deviation lies between 2368 and 7186.