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.
Bank A
6.5
6.6
6.7
6.8
7.1
7.3
7.6
7.8
7.8
7.8
 
Bank Upper B
4.3
5.4
5.8
6.3
6.6
7.6
7.6
8.6
9.2
10.0
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Part 1
Construct a 95​% confidence interval for the population standard deviation sigma at Bank A.
  
enter your response here minless thansigma Subscript Bank Upper Aless than
  
enter your response here min
​(Round to two decimal places as​ needed.)

To construct a 95% confidence interval for the population standard deviation \( \sigma \) at Bank A, we need to follow these steps:

1. **Calculate the sample standard deviation**:
   First, compute the sample standard deviation \( s \) from the given data at Bank A.

2. **Degrees of freedom**:
   The degrees of freedom \( df \) for this calculation is \( n - 1 \), where \( n \) is the number of data points.

3. **Use the Chi-square distribution**:
   The 95% confidence interval for the population standard deviation \( \sigma \) is based on the Chi-square distribution. The formula for the confidence interval is:

   \[
   \left( \sqrt{\frac{(n-1)s^2}{\chi^2_{upper}}}, \sqrt{\frac{(n-1)s^2}{\chi^2_{lower}}} \right)
   \]
   where:
   - \( \chi^2_{upper} \) and \( \chi^2_{lower} \) are the critical values of the Chi-square distribution for the chosen confidence level.
   - \( s^2 \) is the sample variance.

Let's go through these steps.

### Step 1: Calculate the sample standard deviation for Bank A

Data for Bank A:  
6.5, 6.6, 6.7, 6.8, 7.1, 7.3, 7.6, 7.8, 7.8, 7.8

The sample mean \( \bar{x} \) is calculated as:

\[
\bar{x} = \frac{6.5 + 6.6 + 6.7 + 6.8 + 7.1 + 7.3 + 7.6 + 7.8 + 7.8 + 7.8}{10} = 7.2
\]

Now, the sample standard deviation \( s \) is:

\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
\]

\[
s = \sqrt{\frac{(6.5 - 7.2)^2 + (6.6 - 7.2)^2 + \dots + (7.8 - 7.2)^2}{9}} = 0.53
\]

### Step 2: Degrees of freedom

The degrees of freedom \( df \) is \( n - 1 = 10 - 1 = 9 \).

### Step 3: Use the Chi-square distribution

From the table of Chi-square critical values for a 95% confidence interval and \( df = 9 \):
- \( \chi^2_{upper} = 16.919 \)
- \( \chi^2_{lower} = 3.325 \)

### Step 4: Calculate the confidence interval

Now we can calculate the confidence interval for the population standard deviation \( \sigma \) as follows:

\[
\text{Lower bound} = \sqrt{\frac{(9)(0.53^2)}{16.919}} = \sqrt{\frac{2.53}{16.919}} = \sqrt{0.1496} = 0.39
\]

\[
\text{Upper bound} = \sqrt{\frac{(9)(0.53^2)}{3.325}} = \sqrt{\frac{2.53}{3.325}} = \sqrt{0.761} = 0.87
\]

### Final Answer:

The 95% confidence interval for the population standard deviation \( \sigma \) at Bank A is:

\[
0.39 \, \text{min} < \sigma_A < 0.87 \, \text{min}
\]

Would you like further details or explanations? Here are a few related questions to explore:

1. How do you interpret the confidence interval for standard deviation?
2. How would the confidence interval change if the sample size were larger?
3. Why do we use the Chi-square distribution for estimating standard deviation?
4. Can we construct a confidence interval for the mean as well?
5. What assumptions are needed to construct this confidence interval?

**Tip:** When calculating confidence intervals, always ensure that the data meets the assumptions of normality, as the Chi-square distribution is sensitive to this.

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. Bank A: 6.5, 6.6, 6.7, 6.8, 7.1, 7.3, 7.6, 7.8, 7.8, 7.8 Bank B: 4.3, 5.4, 5.8, 6.3, 6.6, 7.6, 7.6, 8.6, 9.2, 10.0 Construct a 95% confidence interval for the population standard deviation sigma at Bank A.

Learn how to construct a 95% confidence interval for the population standard deviation at Bank A using the Chi-Square distribution. With data points like 6.5, 6.6, 7.8, and others, this guide walks through key statistical concepts and formulas to provide a detailed solution.

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