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The confidence interval estimate is

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​mi/hless than<sigmaσless than<enter your response here

​mi/h.

​(Round to one decimal place as​ needed.)

To construct a 95% confidence interval for the population standard deviation \(\sigma\), based on the given sample data, we'll need to follow these steps:

### Step 1: Calculate the sample variance and standard deviation
First, calculate the sample mean (\(\bar{x}\)) and then the sample variance (\(s^2\)) and sample standard deviation (\(s\)).

### Step 2: Use the Chi-Square Distribution
The confidence interval for the population standard deviation is given by the formula:

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

Where:
- \(n\) is the sample size,
- \(s^2\) is the sample variance,
- \(\chi^2_{\alpha/2}\) and \(\chi^2_{1-\alpha/2}\) are critical values from the Chi-Square distribution with \(n-1\) degrees of freedom.

Let me calculate these values.The sample standard deviation is approximately \( 4.2 \, \text{mi/h} \). The 95% confidence interval for the population standard deviation is:

\[
2.99 \, \text{mi/h} \, < \sigma < 7.17 \, \text{mi/h}
\]

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How does increasing the sample size affect the width of the confidence interval?
2. What are the assumptions behind using the Chi-Square distribution for confidence intervals?
3. How is the Chi-Square distribution related to the normal distribution?
4. How would the confidence interval change if we used a different confidence level, such as 90% or 99%?
5. What are some applications of confidence intervals in real-world statistics?

**Tip**: Confidence intervals provide a range of plausible values for a parameter, offering more insight than a single point estimate.

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.

Learn how to calculate a 95% confidence interval for the population standard deviation based on traffic speeds measured on a busy highway. This example uses a Chi-Square distribution with critical values and demonstrates key statistical concepts such as confidence intervals and standard deviation. Suitable for students in Grades 11-12 or introductory college statistics courses.

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