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Construct 

aa 

99% 

confidence interval for the population standard deviation 

sigmaσ 

at Bank B.


To calculate a 99% confidence interval for the population standard deviation (\(\sigma\)) at Bank B, we will use the sample data you provided. The steps involve:

1. **Calculating the sample mean** (\(\bar{x}\)) and sample variance (\(s^2\)).
2. **Applying the chi-square distribution** to find the confidence interval bounds for the variance (\(\sigma^2\)).
3. **Deriving the confidence interval for the standard deviation**.

### Data for Bank B:
\[
4.2, 5.4, 5.9, 6.1, 6.6, 7.6, 7.6, 8.5, 9.4, 10.0
\]

Let's compute the necessary values:

1. **Sample size** (\(n\)) = 10 (since there are 10 values).
2. **Degrees of freedom** (\(df\)) = \(n - 1 = 9\).
3. **Sample mean** \(\bar{x}\) and **sample variance** \(s^2\) for the Bank B data.

Next, I'll calculate these and use the chi-square distribution to construct the 99% confidence interval. Give me a moment.The sample variance for Bank B is approximately \(3.35\), with a sample standard deviation of approximately \(1.83\).

For a 99% confidence interval for the population standard deviation (\(\sigma\)) at Bank B, the interval is:

\[
(1.13, 4.17)
\]

This means we are 99% confident that the true population standard deviation lies between 1.13 and 4.17.

Do you want further details on the process or have any questions? Here are 5 related questions to expand your understanding:

1. How would the confidence interval change if the confidence level were 95%?
2. How is the chi-square distribution used in estimating the variance?
3. How would the confidence interval change if we had a larger sample size?
4. Can this method be applied to non-normally distributed data?
5. How do different levels of variance affect the reliability of service times at Bank B?

**Tip**: The wider your confidence interval, the less certain you are about the estimate, but you're more likely to capture the true parameter.

Construct a 99% confidence interval for the population standard deviation σ at Bank B.

This solution calculates the 99% confidence interval for the population standard deviation of customer waiting times at Bank B. Using sample data, chi-square distribution, and standard deviation formulas, it constructs the interval between 1.13 and 4.17.

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