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To solve this problem, we are asked to find the value \( k \) such that \( P(X > k) = 0.25 \), where \( X \sim \chi^2(5) \) (a chi-square distribution with 5 degrees of freedom).

### Steps:
1. For a chi-square distribution with 5 degrees of freedom, finding \( k \) such that \( P(X > k) = 0.25 \) is equivalent to finding the 75th percentile of the chi-square distribution with 5 degrees of freedom (since \( P(X > k) = 0.25 \) implies that \( k \) is at the 75th percentile).

2. You can find this using chi-square distribution tables or a statistical calculator that provides inverse chi-square values.

3. Using statistical software or tables, we find that for \( \chi^2(5) \) with an upper tail probability of 0.25, the value of \( k \) is approximately:

   \[
   k \approx 5.385
   \]

### Answer:
Thus, \( k = 5.385 \) (rounded to three decimal places).

Would you like further details on the calculation steps?

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1. What does it mean for a variable to follow a chi-square distribution?
2. How do we interpret the degrees of freedom in a chi-square distribution?
3. What is the difference between cumulative and upper-tail probabilities in distributions?
4. How would the value of \( k \) change if \( P(X > k) \) was 0.05 instead of 0.25?
5. How can we calculate chi-square percentiles without a table?

**Tip:** Chi-square distributions are commonly used in hypothesis testing, especially for variance and independence tests.

Suppose X ~ χ²(5). Find k such that P(X > k) = 0.25. Round your answer to 3 decimals.

Learn how to find the value of k such that P(X > k) = 0.25 for a chi-square distribution with 5 degrees of freedom. This problem covers key concepts such as chi-square distribution, percentiles, and inverse probability.

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