To interpret the results, we need to understand how confidence intervals for the standard deviations of waiting times differ between the single-line and multiple-line systems.

Let's address each step and the implications of the confidence intervals:

1. 95% Confidence Interval for Population Standard Deviation at Bank A (Single-Line System)
   Construct the 95% confidence interval for the standard deviation ($\sigma_{A}$) for Bank A’s single-line system. Use the given data (or calculate from sample data if provided).

2. 95% Confidence Interval for Population Standard Deviation at Bank B (Multiple-Line System)
   Similarly, calculate the 95% confidence interval for the standard deviation ($\sigma_{B}$) for Bank B’s multiple-line system.

3. Comparing Confidence Intervals to Evaluate Variation
   Once we have the confidence intervals, we can interpret them to see if one system has significantly lower variation. Lower variation in waiting times suggests a more consistent experience for customers.

4. 99% Confidence Interval for Population Standard Deviation at Bank A (Single-Line System)
   Construct a 99% confidence interval for $\sigma_{A}$. This interval will be wider than the 95% interval, providing more conservative bounds to confirm if the variation is consistently lower.

5. 99% Confidence Interval for Population Standard Deviation at Bank B (Multiple-Line System)
   Similarly, construct a 99% confidence interval for $\sigma_{B}$.

Analysis Based on Confidence Intervals

If the confidence intervals for Bank A’s standard deviation ($\sigma_{A}$) are lower than those for Bank B’s standard deviation ($\sigma_{B}$), it suggests that Bank A's single-line system has less variation in waiting times. This would imply a more predictable and consistent waiting experience for customers, making the single-line system potentially better from a variation perspective.

Answer Options Interpretation

1. Option A: If the confidence intervals confirm that the variation is significantly lower for the single-line system and it offers a more consistent experience, then Option A is correct.

2. Option B: This would suggest that, although the variation is lower for the multiple-line system, the single-line is preferred overall. This doesn’t align with typical findings in queue management, where a single line is often more consistent, so this option might be less likely based on expected results.

3. Option C: If the single-line system has lower variation, but somehow the multiple-line system is still preferred, Option C might be selected in cases where other factors (like customer preference) play a role.

4. Option D: If the multiple-line system has both lower variation and is the preferred choice, Option D would be the correct choice.

Based on the description and typical findings in queue management, Option A is likely correct if the single-line system shows significantly lower variation and thus appears to be a better arrangement.

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Would you like to go over the calculation steps for the confidence intervals or explore specific assumptions about the systems?

Here are some further questions for practice:

1. How is a confidence interval for the population standard deviation typically constructed?
2. What factors influence the width of a confidence interval?
3. How would a larger sample size affect the confidence intervals calculated?
4. What role does the level of confidence (e.g., 95% vs. 99%) play in decision-making for systems like this?
5. How might customer preference affect the choice between single-line and multiple-line systems?

Tip: In comparing systems, always consider both average times and variation to fully understand customer experience.