The problem in the image involves calculating the area under the curve of a chi-square distribution with 13 degrees of freedom, specifically for values greater than 22. The shaded region is the tail to the right of 22.

To solve this, we need to find the cumulative probability $P(X > 22)$ for a chi-square distribution with 13 degrees of freedom. This is done using a chi-square cumulative distribution function (CDF) or its complement, the survival function (1 - CDF).

Using a chi-square table or a calculator for the chi-square distribution, we can compute the probability. Let me compute that for you.

The area to the right of $X = 22$ with 13 degrees of freedom is approximately:

$P(X > 22) \approx 0.054$

Thus, the area of the shaded region is about 0.054, rounded to three decimal places.

Would you like more details on how the chi-square distribution works or have any other questions?

Here are 5 related questions to expand your understanding:

1. What is the chi-square distribution used for in statistics?
2. How do you find the critical value of a chi-square distribution for a given confidence level?
3. How does changing the degrees of freedom affect the shape of the chi-square distribution?
4. Can you explain how the chi-square test for independence is performed?
5. How do you interpret a p-value obtained from a chi-square distribution?

Tip: The larger the degrees of freedom, the more symmetric the chi-square distribution becomes, approaching a normal distribution shape as the degrees increase.