To construct a 99% confidence interval for the population standard deviation (σ) of Bank B, we'll use the Chi-square distribution. The general steps are as follows:

Formula:

The formula for constructing a confidence interval for the population standard deviation is based on the sample variance ($s^2$) and the chi-square distribution:

$\left( \sqrt{\frac{(n - 1)s^2}{\chi^2_{\alpha/2}}}, \sqrt{\frac{(n - 1)s^2}{\chi^2_{1-\alpha/2}}} \right)$

Where:

• $n$ = sample size
• $s^2$ = sample variance
• $\chi^2_{\alpha/2}$ and $\chi^2_{1-\alpha/2}$ are the critical values from the chi-square distribution, corresponding to the upper and lower tails of the distribution.
• $\alpha$ = 1 - confidence level (for 99% confidence, $\alpha = 0.01$)

Steps to Compute the Confidence Interval:

1. Calculate the sample variance $s^2$ from your data (if you have it). If you only have the sample standard deviation, you can square it to get the variance.
2. Determine the sample size $n$.
3. Look up the chi-square critical values corresponding to $\alpha/2$ and $1-\alpha/2$ in a chi-square distribution table for $n-1$ degrees of freedom.
4. Substitute the values into the formula above to compute the confidence interval for $\sigma$.

Without Sample Data

If you provide the sample size (n) and the sample standard deviation or variance for Bank B, I can compute the exact interval.

Do you have the sample statistics for Bank B, or should we assume certain values?

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Would you like more details on any of the steps? Here are five related questions to expand your understanding:

1. How do you calculate the sample variance from raw data?
2. What is the chi-square distribution, and why is it used for standard deviation?
3. How does the confidence level affect the width of the confidence interval?
4. What are the critical values for common confidence intervals (e.g., 95%, 99%)?
5. How does the size of the sample affect the confidence interval for standard deviation?

Tip: Confidence intervals for standard deviation are wider for smaller sample sizes, so increasing the sample size will improve the precision of the estimate.