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To solve this problem, let’s go over each part:

### Step 1: Identify the Degrees of Freedom
The degrees of freedom (\( df \)) in this chi-square distribution calculation is determined by the sample size \( n \). Since we have \( n = 13 \), the degrees of freedom is:
\[
df = n - 1 = 13 - 1 = 12
\]

### Step 2: Find the Chi-Square Values
We need to find the critical chi-square values at the 0.975 and 0.025 significance levels with 12 degrees of freedom.

1. **For \( \chi^2_{0.975}(df) \)**:
   This value represents the chi-square critical value at the 97.5th percentile for 12 degrees of freedom.
   
2. **For \( \chi^2_{0.025}(df) \)**:
   This value represents the chi-square critical value at the 2.5th percentile for 12 degrees of freedom.

Using a chi-square table or calculator:
- \( \chi^2_{0.975}(12) \approx 4.404 \)
- \( \chi^2_{0.025}(12) \approx 23.337 \)

### Step 3: Interpret the Interval
The calculated interval is:
\[
\left[ \chi^2_{0.975}(12), \chi^2_{0.025}(12) \right] = [4.404, 23.337]
\]

### Step 4: Check if the Sample Chi-Square Value Falls in the Interval
The given sample chi-square value is \( \chi^2 = 25.191 \). Since \( 25.191 \) is outside the interval \([4.404, 23.337]\), it suggests that the population variance may not be reasonable.

Would you like more details, or have any questions about this process?

### Related Questions:
1. What does it mean if the sample chi-square value falls outside the critical interval?
2. How can we interpret chi-square values in the context of hypothesis testing?
3. How would the interval change if the sample size were different?
4. What assumptions are necessary for using the chi-square distribution in this scenario?
5. How can chi-square tests be used to test for variance in other contexts?

#### Tip:
When using chi-square tables, ensure you match the degrees of freedom and significance level precisely for accurate results.

Suppose you have a normally distributed population such that σ² = 6. Using the sampling distribution of S², we find that from a random sample of size n = 13, χ² = 25.191. Does our population variance seem reasonable?

Analyze a population variance using a chi-square test with a sample size of 13 and a calculated chi-square value of 25.191. Learn how to determine if the population variance is reasonable within a specified interval using critical values at 0.975 and 0.025 significance levels. Ideal for college-level statistics students.

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