To construct the confidence interval for the population variance, we will use the chi-square distribution. The confidence interval for the population variance $\sigma^2$ is given by the formula:

$\left( \frac{(n-1) s^2}{\chi^2_{\frac{1-\alpha}{2}}}, \frac{(n-1) s^2}{\chi^2_{\frac{\alpha}{2}}} \right)$

Where:

• $n$ is the sample size,
• $s^2$ is the sample variance,
• $\alpha = 1 - c$ is the level of significance,
• $\chi^2_{\frac{1-\alpha}{2}}$ and $\chi^2_{\frac{\alpha}{2}}$ are the critical values of the chi-square distribution with $n-1$ degrees of freedom.

Given:

• $n = 8$,
• $s^2 = 10.7$,
• $c = 0.9$, so $\alpha = 1 - 0.9 = 0.1$,
• Degrees of freedom $df = n - 1 = 8 - 1 = 7$.

Step 1: Find the critical values of the chi-square distribution.

We need to find the chi-square values for $\chi^2_{0.95}$ and $\chi^2_{0.05}$, because:

• $\chi^2_{\frac{1-\alpha}{2}} = \chi^2_{0.95}$,
• $\chi^2_{\frac{\alpha}{2}} = \chi^2_{0.05}$.

From the chi-square table:

• $\chi^2_{0.95}$ for 7 degrees of freedom is 2.167,
• $\chi^2_{0.05}$ for 7 degrees of freedom is 14.067.

Step 2: Calculate the confidence interval for the population variance.

Now, we plug in the values:

$\text{Lower bound} = \frac{(8-1) \times 10.7}{14.067} = \frac{7 \times 10.7}{14.067} \approx 5.3$

$\text{Upper bound} = \frac{(8-1) \times 10.7}{2.167} = \frac{7 \times 10.7}{2.167} \approx 34.6$

Final Answer:

The 90% confidence interval for the population variance is approximately $(5.3, 34.6)$.

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Do you need more details on the steps or calculations? Here are five related questions to extend your understanding:

1. How do you find chi-square critical values without a table?
2. What is the confidence interval for the population standard deviation based on the same data?
3. How does the confidence level affect the width of the confidence interval?
4. How would the confidence interval change if the sample size was larger?
5. Can you construct a confidence interval for the population mean instead of the variance?

Tip: Always ensure that your data meets the assumptions of the chi-square distribution, which applies to normally distributed populations.