To solve this problem, we need to calculate a 90% confidence interval for the variance based on a sample of weight measurements. Let's go through each part step-by-step:

Given data

• Sample size ($n$) = 10
• Data points (in kg): $20.6, 20.6, 22.3, 20.8, 22.6, 22.2, 21.3, 24.3, 23.8, 21.2$
• The data is assumed to follow a normal distribution with unknown variance.

Steps for Solution

1. Calculate the sample variance

Let’s denote the sample variance as $s^2$, calculated using the formula: $s^2 = \frac{1}{n - 1} \sum_{i=1}^n (x_i - \bar{x})^2$ where $\bar{x}$ is the sample mean.

1. Calculate the mean ($\bar{x}$): $\bar{x} = \frac{20.6 + 20.6 + 22.3 + 20.8 + 22.6 + 22.2 + 21.3 + 24.3 + 23.8 + 21.2}{10}$ Simplifying this will give the sample mean.

2. Calculate $s^2$ using the data points.

2. Construct the 90% Confidence Interval for the Variance

For a 90% confidence interval, we use the Chi-Square distribution because we’re dealing with variance. The confidence interval for the population variance $\sigma^2$ is given by: $\left( \frac{(n - 1) s^2}{\chi^2_{1 - \alpha/2}}, \frac{(n - 1) s^2}{\chi^2_{\alpha/2}} \right)$ where:

• $\chi^2_{\alpha/2}$ and $\chi^2_{1 - \alpha/2}$ are the critical values from the Chi-Square distribution with $n - 1$ degrees of freedom.
• For a 90% confidence interval, $\alpha = 0.10$, so $\alpha/2 = 0.05$.

Using a Chi-Square table:

• For $n - 1 = 9$ degrees of freedom:
  • $\chi^2_{0.05} \approx 16.92$
  • $\chi^2_{0.95} \approx 3.33$

Then substitute $s^2$ and the Chi-Square values into the formula to get the confidence interval.

3. Calculate the Lower Bound of the 90% Confidence Interval for the Variance

For the lower bound, we’ll only consider: $\frac{(n - 1) s^2}{\chi^2_{1 - \alpha}}$ where $\chi^2_{0.90} \approx 3.33$.

Let me calculate these values for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can walk you through the remaining calculations step-by-step so you can complete them manually, or I can answer any other questions you may have.

Here are some related questions that might help deepen your understanding:

1. How is a Chi-Square distribution used in estimating variance?
2. What role does the sample size play in the confidence interval for variance?
3. How would the confidence interval change with a 95% level instead of 90%?
4. Why do we use sample variance instead of population variance in confidence interval calculations?
5. How can confidence intervals be applied in real-world scenarios outside of statistics?

Tip: When dealing with small sample sizes, always use $n - 1$ in the denominator for sample variance to get an unbiased estimate of population variance.