Find the critical values. Round the answer to three decimal places.



For 

=α0.10

, the critical values are  and 






To find the critical values for a chi-square test at the \(\alpha = 0.10\) significance level with 31 degrees of freedom, we need to determine the critical values that correspond to the upper and lower tails of the chi-square distribution.

Given:
- \(\alpha = 0.10\)
- Degrees of freedom (\(df\)) = 31
- Since it is a two-tailed test, \(\alpha/2 = 0.05\) for each tail.

Using a chi-square distribution table or calculator, we look up the chi-square values for \(\alpha/2 = 0.05\) and \(1 - \alpha/2 = 0.95\) with 31 degrees of freedom.

1. **Lower critical value**: \(\chi^2_{0.05, 31}\)
2. **Upper critical value**: \(\chi^2_{0.95, 31}\)

Using a chi-square table or a calculator, we find:

- The critical value for the lower tail (\(\chi^2_{0.05, 31}\)): 

  \[
  \chi^2_{0.05, 31} \approx 18.493
  \]

- The critical value for the upper tail (\(\chi^2_{0.95, 31}\)): 

  \[
  \chi^2_{0.95, 31} \approx 45.722
  \]

**Conclusion:**

For \(\alpha = 0.10\) with 31 degrees of freedom, the critical values are:

- Lower critical value: \(\chi^2_{0.05, 31} \approx 18.493\)
- Upper critical value: \(\chi^2_{0.95, 31} \approx 45.722\)

Would you like more details or have any questions about this process?

Here are 8 related questions to expand your understanding:

1. How do you interpret the critical values in the context of hypothesis testing?
2. What steps are involved in finding critical values using a chi-square distribution table?
3. How would the critical values change with a different significance level, such as \(\alpha = 0.05\)?
4. What is the importance of degrees of freedom in determining chi-square critical values?
5. How can you use software tools to find chi-square critical values?
6. What are some common applications of the chi-square test for variance in real-world scenarios?
7. How does changing the sample size affect the critical values and the test outcome?
8. Can you explain the relationship between the chi-square distribution and the normal distribution?

**Tip:** Always ensure the degrees of freedom are correctly calculated as \(n - 1\) when using the chi-square distribution to find critical values for standard deviation or variance tests.

Learn how to find critical values for a chi-square test at α = 0.10 with 31 degrees of freedom. Understand the significance level and the chi-square distribution. Suitable for undergraduate students studying statistics.

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