The following sample of heights was taken from 9
 cans off the assembly line. Construct the 95%
 confidence interval for the population standard deviation for all cans that come off the assembly line. Round your answers to two decimal places.

12.4,12.4,12.2,11.9,11.6,11.4,12.6,11.9,12.0

To construct a 95% confidence interval for the population standard deviation, we need to use the Chi-Square distribution formula for the sample variance. The steps are as follows:

1. **Calculate the sample variance \( s^2 \)**:
   - First, find the sample mean \( \bar{x} \).
   - Then, calculate the sample variance \( s^2 \).

2. **Use the Chi-Square distribution**:
   - The confidence interval for the population variance \( \sigma^2 \) is given by:
     \[
     \left( \frac{(n - 1) s^2}{\chi^2_{\alpha/2}}, \frac{(n - 1) s^2}{\chi^2_{1 - \alpha/2}} \right)
     \]
   - Where:
     - \( n \) is the sample size.
     - \( s^2 \) is the sample variance.
     - \( \alpha \) is the significance level (for 95% confidence, \( \alpha = 0.05 \)).
     - \( \chi^2_{\alpha/2} \) and \( \chi^2_{1-\alpha/2} \) are the critical values of the Chi-Square distribution with \( n - 1 \) degrees of freedom.

3. **Find the confidence interval for \( \sigma \)**:
   - To get the confidence interval for the population standard deviation \( \sigma \), take the square root of the upper and lower bounds of the variance confidence interval.

Let's calculate it step-by-step.The 95% confidence interval for the population standard deviation is approximately \( (0.27, 0.75) \).

Would you like further details or have any questions?

Here are some related questions:
1. How is the Chi-Square distribution used in estimating variance?
2. What are the assumptions behind using a Chi-Square distribution for variance estimation?
3. How does the sample size affect the width of the confidence interval?
4. Can the confidence interval method be applied for small sample sizes?
5. What happens to the confidence interval if we increase the confidence level?

**Tip:** Always check if the sample data is normally distributed when constructing confidence intervals for variance or standard deviation.

The following sample of heights was taken from 9 cans off the assembly line. Construct the 95% confidence interval for the population standard deviation for all cans that come off the assembly line. Round your answers to two decimal places.
12.4,12.4,12.2,11.9,11.6,11.4,12.6,11.9,12.0

Learn how to construct a 95% confidence interval for the population standard deviation based on a sample of 9 heights from cans. This problem involves the use of Chi-Square distribution and provides a step-by-step calculation of the confidence interval. Suitable for college-level statistics or advanced high school students.

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