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
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### 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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11fb6ed7-9250-48c6-8339-b5d076b64c81%2Fb18b81a8-3173-4325-8f24-8f31aa81242a%2Fzdgupxp.jpeg&w=3840&q=75)
Transcribed Image Text:### 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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