The problem asks us to find the value of $\chi^2_{0.65}$ for a chi-square distribution with 13 degrees of freedom, where the area to the right of this value is 0.65.

To do this, we use the chi-square distribution table or a chi-square calculator to find the corresponding value of $\chi^2$ for the given probability (0.65) and degrees of freedom (13). Essentially, we are looking for the value of $\chi^2$ such that the cumulative area to the right is 0.65 (or equivalently, the cumulative area to the left is 1 - 0.65 = 0.35).

Using a calculator or statistical software to compute this:

• The value of $\chi^2_{0.65}$ with 13 degrees of freedom is approximately 9.194.

Thus, $\chi^2_{0.65} \approx 9.194$, rounded to three decimal places.

Would you like further details on the chi-square calculation process or have any other questions?

Here are 5 related questions to consider:

1. How do you calculate the chi-square value manually?
2. What is the importance of degrees of freedom in a chi-square distribution?
3. How is a chi-square distribution different from a normal distribution?
4. How can we interpret the area under the chi-square curve?
5. What are some practical applications of the chi-square distribution?

Tip: Always ensure you're using the correct degrees of freedom when consulting chi-square tables or using statistical software, as it significantly affects the result.