The values listed below are waiting times (in minutes) of customers at two different banks. At Bank A, customers enter a single waiting line that feeds three teller windows. At Bank B, customers may enter any one of three different lines that have formed at three teller windows. Answer the following questions. \begin{tabular}{cccccccccccc} Bank A & 6.4 & 6.6 & 6.7 & 6.8 & 7.1 & 7.2 & 7.6 & 7.8 & 7.8 & 7.8 \\ \hline Bank B & 4.2 & 5.4 & 5.7 & 6.1 & 6.7 & 7.7 & 7.7 & 8.6 & 9.3 & 10.0 \end{tabular} E Click the icon to view the table of Chi-Square critical values. Construct a $99 \%$ confidence interval for the population standard deviation $\sigma$ at Bank $\mathrm{A}$. \[ 0.34 \min <\sigma_{\text {Bank } A}<1.24 \mathrm{~min} \] (Round to two decimal places as needed.) Construct a $99 \%$ confidence interval for the population standard deviation $\sigma$ at Bank B. $1.14 \min <\sigma_{\text {Bank } B}<4.20 \mathrm{~min}$ (Round to two decimal places as needed.) Interpret the results found in the previous parts. Do the confidence intervals suggest a difference in the variation among waiting times? Does the single-line system or the multiple-line system seem to be a better arrangement? A. The variation appears to be significantly lower with a multiple-line system. The multiple-line system appears to be better. B. The variation appears to be significantly lower with a single line system. The multiple-line system appears to be better. C. The variation appears to be significantly lower with a single line system. The single-line system appears to be better. D. The variation appears to be significantly lower with a multiple-line system. The single-line system appears to be better.
Solution
Step 1 ：The question is asking to interpret the results of the confidence intervals for the population standard deviation at Bank A and Bank B. We need to compare the confidence intervals of the two banks and determine which system has lower variation in waiting times.
Step 2 ：The confidence interval for Bank A is from 0.34 min to 1.24 min, and for Bank B is from 1.14 min to 4.20 min. If the confidence interval is smaller, it means the variation of waiting times is lower. Therefore, we can compare the width of the confidence intervals to determine which system has lower variation.
Step 3 ：Let's calculate the width of the confidence intervals for both banks. The width of the confidence interval for Bank A is approximately 0.90 min, and for Bank B is approximately 3.06 min.
Step 4 ：Since the width of the confidence interval for Bank A is smaller, it means the variation of waiting times is lower at Bank A. Therefore, the single-line system appears to be better.
Step 5 ：\(\boxed{\text{C. The variation appears to be significantly lower with a single line system. The single-line system appears to be better.}}\)
