To find the area of the shaded region under the chi-square distribution with 6 degrees of freedom to the right of 4, we need to compute the complement of the cumulative distribution function (CDF) of the chi-square distribution at $x = 4$.

This is essentially a "right-tail probability," which can be computed as:

$P(X > 4) = 1 - P(X \leq 4)$

where $P(X \leq 4)$ is the CDF value of the chi-square distribution at $x = 4$ with 6 degrees of freedom.

I'll now calculate this for you.The area of the shaded region under the chi-square distribution curve to the right of $x = 4$ is approximately 0.677, rounded to three decimal places.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How is the chi-square distribution used in hypothesis testing?
2. What is the significance of degrees of freedom in the chi-square distribution?
3. How can we interpret the area under a chi-square curve in real-world scenarios?
4. How do you calculate left-tailed probabilities for a chi-square distribution?
5. How does the chi-square distribution compare to a normal distribution?

Tip: For chi-square distributions, higher degrees of freedom shift the curve rightwards, making it more symmetric.