A chi-square distribution with 13

 degrees of freedom is graphed below. The region under the curve to the right of χ

2

0.65

 is shaded. The area of this region is 0.65

.

0.05

0.10

0.15

0.20


=

Area

0.65

χ

2

0.65

Find the value of χ

2

0.65

. Round your answer to three decimal places.

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.

A chi-square distribution with 13 degrees of freedom is graphed below. The region under the curve to the right of χ²₀.₆₅ is shaded. The area of this region is 0.65. Find the value of χ²₀.₆₅. Round your answer to three decimal places.

This problem demonstrates how to find the value of χ²₀.₆₅ for a chi-square distribution with 13 degrees of freedom, where the area to the right is 0.65. Using the chi-square distribution table or calculator, the solution is found to be approximately 9.194. The solution includes key mathematical concepts related to chi-square distributions and is suitable for college-level students.

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