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To solve this problem, we need to find the critical value \( k \) for a chi-square distribution with 12 degrees of freedom, such that the probability \( P(X < k) = 0.025 \).

Given:
- \( X \sim \chi^2(12) \)
- \( P(X < k) = 0.025 \)

### Steps:
1. Since we want \( P(X < k) = 0.025 \), we are looking for the 2.5th percentile of the chi-square distribution with 12 degrees of freedom.
2. This value \( k \) can be found using chi-square distribution tables or statistical software that can compute chi-square critical values.

Using a chi-square table or a statistical calculator, we can find that:
\[
k \approx 4.404
\]

### Answer:
\[
k = 4.404
\]

Let me know if you want more details or if you have any questions!

Here are some related questions to expand on this concept:
1. How would you find \( k \) if the degrees of freedom were different, say 10?
2. How does the shape of the chi-square distribution change as the degrees of freedom increase?
3. What is the 97.5th percentile for a chi-square distribution with 12 degrees of freedom?
4. How does this calculation change if we were looking for \( P(X > k) = 0.025 \)?
5. Why is the chi-square distribution often used in hypothesis testing?

**Tip:** For chi-square distributions, values near the lower tail (small p-values) are always much smaller than those in the upper tail, especially with larger degrees of freedom.

Suppose X ~ χ²(12). Find k such that P(X < k) = 0.025. Round your answer to 3 decimals.

Learn how to find the critical value k such that P(X < k) = 0.025 for a chi-square distribution with 12 degrees of freedom. This problem covers probability concepts, chi-square distribution properties, and percentile calculations.

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