The number of hours of reserve capacity of 10 randomly selected automotive batteries is shown to the right.
$\begin{array}{lllll}1.79 & 1.85 & 1.54 & 1.68 & 1.71 \\ 1.98 & 1.31 & 1.53 & 1.44 & 2.06\end{array}$
Assume the sample is taken from a normally distributed population. Construct $90 \%$ confidence intervals for (a) the population variance $\sigma^{2}$ and (b) the population standard deviation $\sigma$
Interpret the results. Select the correct choice below and fill in the answer box(es) to complete your choice. (Round to three decimal places as needed.)
A. With $10 \%$ confidence, it can belsaid that the population variance is greater than

C. With $10 \%$ confidence, it can be said that the population variance is between and
B. With $90 \%$ confidence, it can be said that the population variance is less than

D. With $90 \%$ confidence, it can be said that the population variance is between 0.030 and 0.154
(b) The confidence interval for the population standard deviation is (Round to three decimal places as needed.)

Step 1 ：Given the data of the number of hours of reserve capacity of 10 randomly selected automotive batteries, we first calculate the sample variance and standard deviation.

Step 2 ：The data is: \(1.79, 1.85, 1.54, 1.68, 1.71, 1.98, 1.31, 1.53, 1.44, 2.06\). The sample size (n) is 10.

Step 3 ：The mean of the data is \(1.689\).

Step 4 ：The sample variance (s2) is \(0.0569\) and the sample standard deviation (s) is \(0.2385\).

Step 5 ：Since the sample is taken from a normally distributed population, the sample variance follows a chi-square distribution. We can use this to construct the confidence interval for the population variance and standard deviation.

Step 6 ：The degrees of freedom (df) is \(n - 1 = 9\). The confidence level is \(90\%\).

Step 7 ：Using the chi-square distribution, the lower and upper chi-square values are \(3.3251\) and \(16.9190\) respectively.

Step 8 ：The lower and upper limits of the confidence interval for the population variance (\(\sigma^{2}\)) are \(0.0303\) and \(0.1540\) respectively.

Step 9 ：The lower and upper limits of the confidence interval for the population standard deviation (\(\sigma\)) are \(0.1740\) and \(0.3924\) respectively.

Step 10 ：\(\boxed{\text{Final Answer:}}\)

Step 11 ：(a) With \(90\%\) confidence, it can be said that the population variance is between \(0.030\) and \(0.154\).

Step 12 ：(b) The confidence interval for the population standard deviation is between \(0.174\) and \(0.392\).