### Constructing a Confidence Interval for Population Standard Deviation**Problem Statement:**Listed below are speeds (mi/h) measured from traffic on a busy highway. This simple random sample was obtained at 3:30 P.M. on a weekday. Use the sample data to construct a 95% confidence interval estimate of the population standard deviation.**Traffic Speeds:**```64, 63, 63, 57, 63, 53, 59, 58, 59, 70, 61, 69```**Steps to Construct the Confidence Interval:**1. **Calculate the Sample Mean & Standard Deviation:** - First, you need to find the mean (average) of the sample data. - Then, calculate the standard deviation of the sample.2. **Find the Chi-Square Critical Values:** - Access the chi-square distribution table to find the critical values for chi-square at the 95% confidence level, based on the sample size (n-1 degrees of freedom).3. **Compute the Confidence Interval Estimate for the Population Standard Deviation:** The formula to calculate the confidence interval for the standard deviation $ \sigma $ is: \[ \sqrt{\frac{(n-1)s^2}{\chi^2_{upper}}} < \sigma < \sqrt{\frac{(n-1)s^2}{\chi^2_{lower}}} \] where: - $ n $ is the sample size. - $ s $ is the standard deviation of the sample. - $ \chi^2_{upper} $ and $ \chi^2_{lower} $ are the critical values from the chi-square distribution table for the upper and lower bounds, respectively.4. **Apply the Calculations:** - Plug in the values obtained into the formula above to get the confidence interval estimate.**Interactive Table:**- Click the icon to view the table of Chi-Square critical values. This will help you find the necessary chi-square values for your calculations.**Final Output:**- The confidence interval estimate is: \[ \text{[Minimum Value]} \text{ mi/h} < \sigma < \text{[Maximum Value]} \text{mi/h} \] Ensure to round the final results to one decimal place as needed.Teaching Tip

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Listed below are speeds (mi/h) measured from traffic on a busy highway. This simple random sample was obtained at 3:30 P.M. on a weekday. Use the sample data to construct a 95% confidence interval estimate of the population standard deviation. 64 63 63 57 63 53 59 58 59 70 61 69

Listed below are speeds (mi/h) measured from traffic on a busy highway. This simple random sample was obtained at 3:30 P.M. on a weekday. Use the sample data to construct a 95% confidence interval estimate of the population standard deviation. 64 63 63 57 63 53 59 58 59 70 61 69

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6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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Question

### Constructing a Confidence Interval for Population Standard Deviation

**Problem Statement:**
Listed below are speeds (mi/h) measured from traffic on a busy highway. This simple random sample was obtained at 3:30 P.M. on a weekday. Use the sample data to construct a 95% confidence interval estimate of the population standard deviation.

**Traffic Speeds:**
```
64, 63, 63, 57, 63, 53, 59, 58, 59, 70, 61, 69
```

**Steps to Construct the Confidence Interval:**

1. **Calculate the Sample Mean & Standard Deviation:**
- First, you need to find the mean (average) of the sample data.
- Then, calculate the standard deviation of the sample.

2. **Find the Chi-Square Critical Values:**
- Access the chi-square distribution table to find the critical values for chi-square at the 95% confidence level, based on the sample size (n-1 degrees of freedom).

3. **Compute the Confidence Interval Estimate for the Population Standard Deviation:**

The formula to calculate the confidence interval for the standard deviation $ \sigma $ is:
\[
\sqrt{\frac{(n-1)s^2}{\chi^2_{upper}}} < \sigma < \sqrt{\frac{(n-1)s^2}{\chi^2_{lower}}}
\]
where:
- $ n $ is the sample size.
- $ s $ is the standard deviation of the sample.
- $ \chi^2_{upper} $ and $ \chi^2_{lower} $ are the critical values from the chi-square distribution table for the upper and lower bounds, respectively.

4. **Apply the Calculations:**
- Plug in the values obtained into the formula above to get the confidence interval estimate.

**Interactive Table:**
- Click the icon to view the table of Chi-Square critical values. This will help you find the necessary chi-square values for your calculations.

**Final Output:**
- The confidence interval estimate is:
\[
\text{[Minimum Value]} \text{ mi/h} < \sigma < \text{[Maximum Value]} \text{mi/h}
\]

Ensure to round the final results to one decimal place as needed.

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