Find the critical values 

𝜒21 − 𝛼/2

 and 

𝜒2𝛼/2

 for the given confidence level and sample size. (Round your answers to three decimal places.)

confidence level 99%, 

*n* = 90

𝜒21 − 𝛼/2

= 

𝜒2𝛼/2

= 

To find the critical values for a chi-square distribution for a 99% confidence level with a sample size of \( n = 90 \), we need to determine:

1. **Degrees of Freedom** \( df = n - 1 = 90 - 1 = 89 \)
2. **Confidence Level** \( 99\% \) (which implies a significance level \( \alpha = 1 - 0.99 = 0.01 \))

Given that we're working with a two-tailed confidence interval:

1. \( \chi^2_{1 - \alpha/2} \) is the chi-square value where the right tail has \( \alpha/2 \) probability.
2. \( \chi^2_{\alpha/2} \) is the chi-square value where the left tail has \( \alpha/2 \) probability.

Since \( \alpha = 0.01 \):
- \( \alpha/2 = 0.005 \), meaning we need the 0.995 and 0.005 quantiles for the chi-square distribution with 89 degrees of freedom.

### Calculation

Using chi-square tables or a statistical calculator for a chi-square distribution with \( df = 89 \):

1. **\( \chi^2_{1 - \alpha/2} = \chi^2_{0.995} \)**: This corresponds to the 99.5th percentile.
2. **\( \chi^2_{\alpha/2} = \chi^2_{0.005} \)**: This corresponds to the 0.5th percentile.

After rounding to three decimal places, we get:

\[
\chi^2_{1 - \alpha/2} = \chi^2_{0.995} \approx 124.116
\]
\[
\chi^2_{\alpha/2} = \chi^2_{0.005} \approx 59.342
\]

### Final Answers

\[
\chi^2_{1 - \alpha/2} = 124.116
\]
\[
\chi^2_{\alpha/2} = 59.342
\]

Would you like further details on chi-square distribution properties or help with similar problems?

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Here are some follow-up questions:
1. How are critical values used in hypothesis testing?
2. What does it mean to have a 99% confidence level in this context?
3. How would changing the sample size affect the critical values?
4. What are chi-square distribution properties?
5. How do we interpret the two-tailed critical region?

**Tip:** Chi-square distributions are skewed to the right, especially with lower degrees of freedom. Higher degrees reduce this skew.

Find the critical values 𝜒²(1 - 𝛼/2) and 𝜒²(𝛼/2) for the given confidence level and sample size. (Round your answers to three decimal places.) Confidence level: 99%, n = 90

Learn how to find the critical values for a chi-square distribution at a 99% confidence level with a sample size of 90. This solution covers calculating degrees of freedom, significance levels, and identifying critical values for the chi-square distribution. Ideal for students in advanced statistics or undergraduate courses.

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