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.