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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.

An experiment yielded n = 10 weight measurements (in kg) shown, assumed to have a normal distribution with unknown variance. Calculate the 90% two-sided confidence interval on the variance and the 90% confidence lower-bound on the variance.

Learn how to calculate the 90% confidence interval for variance based on a sample of weight measurements in kilograms. This problem covers key statistical concepts, including using the Chi-Square distribution for estimating variance with unknown population variance. Suitable for college-level statistics.

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