Find the critical values chi squared Subscript Upper Lχ2L and chi squared Subscript Upper Rχ2R for the given confidence level c and sample size n.
cequals=0.80.8​,  nequals=2525
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Part 1
chi squared Subscript Upper Lχ2Lequals=enter your response here ​(Round to three decimal places as​ needed.)

To find the critical values \(\chi^2_L\) and \(\chi^2_R\) for a given confidence level \(c\) and sample size \(n\), we need to determine the critical points from the chi-square distribution.

Given:
- Confidence level \(c = 0.8\)
- Sample size \(n = 25\)

The degrees of freedom (\(df\)) is calculated as \(n - 1\):
\[ df = n - 1 = 25 - 1 = 24 \]

The confidence level of 0.8 corresponds to the middle 80% of the chi-square distribution. This means that 10% of the distribution is in each tail (since 100% - 80% = 20%, and this is divided equally between the two tails).

We need to find the critical values for the chi-square distribution at the lower 10% and upper 90% tails:
- \(\chi^2_L\) is the 10th percentile (0.1 quantile).
- \(\chi^2_R\) is the 90th percentile (0.9 quantile).

Using a chi-square distribution table or calculator for \(df = 24\):

1. For the lower tail critical value (\(\chi^2_L\)):
   \[
   \chi^2_L = \chi^2_{0.1, 24}
   \]

2. For the upper tail critical value (\(\chi^2_R\)):
   \[
   \chi^2_R = \chi^2_{0.9, 24}
   \]

Using a chi-square calculator or table for \(df = 24\):

- \(\chi^2_{0.1, 24} \approx 15.659\) (rounded to three decimal places)
- \(\chi^2_{0.9, 24} \approx 35.172\) (rounded to three decimal places)

Therefore:
\[
\chi^2_L \approx 15.659
\]
\[
\chi^2_R \approx 35.172
\]

Would you like more details or have any questions? Here are some related questions:

1. How do you find the degrees of freedom for a chi-square test?
2. What is the chi-square distribution used for?
3. How do you interpret the critical values in a chi-square test?
4. What are the properties of the chi-square distribution?
5. How do you find the chi-square critical value using a statistical table?
6. What is the relationship between confidence level and critical values in a chi-square test?
7. How do changes in sample size affect the critical values in a chi-square test?
8. What are the applications of the chi-square test in real-world scenarios?

**Tip:** Always double-check your calculations and use accurate tables or calculators to ensure precision in your statistical analysis.

Learn how to find the chi-squared critical values \(\chi^2_L\) and \(\chi^2_R\) for a confidence level of 0.8 and a sample size of 25. This problem involves statistical inference using the chi-square distribution, providing critical values essential for hypothesis testing and confidence interval calculations.

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