https://file.aiquickdraw.com/mathbot/file/1729725103519ums1lgod.jpg

To find the critical values \( \chi^2_R \) and \( \chi^2_L \) for the given confidence level \( c = 0.90 \) and sample size \( n = 21 \), follow these steps:

1. **Determine the degrees of freedom (df)**: 
   \[
   \text{df} = n - 1 = 21 - 1 = 20
   \]

2. **Find the critical values from the chi-square distribution table**:
   - For a confidence level of \( c = 0.90 \), the area in each tail is \( \frac{1 - c}{2} = \frac{1 - 0.90}{2} = 0.05 \).
   - You need to find the chi-square values \( \chi^2_R \) and \( \chi^2_L \) that correspond to these areas.

3. **Using a chi-square table or calculator**, find:
   - \( \chi^2_R \) corresponding to the right tail with 0.05 in the upper tail and 20 degrees of freedom.
   - \( \chi^2_L \) corresponding to the left tail with 0.05 in the lower tail and 20 degrees of freedom.

Using the chi-square distribution table for 20 degrees of freedom:
- \( \chi^2_R = 31.410 \) (for the right tail with 0.05 in the upper tail)
- \( \chi^2_L = 10.851 \) (for the left tail with 0.05 in the lower tail)

Thus, the critical value for \( \chi^2_R \) is approximately **31.410**.

Would you like me to explain further or assist with the second critical value \( \chi^2_L \)? 

Here are 5 questions to expand this:
1. How is the confidence level related to the area in the tails of the distribution?
2. Why do we subtract 1 from the sample size to find the degrees of freedom?
3. What is the significance of the critical values in a chi-square test?
4. How do the critical values change with different degrees of freedom?
5. Can you explain how to interpret chi-square values in terms of hypothesis testing?

**Tip**: When using the chi-square table, ensure you're selecting the correct degrees of freedom and the appropriate tail areas for the confidence level.

Find the critical values χ²_R and χ²_L for the given confidence level c = 0.90 and sample size n = 21.

Learn how to find the critical values χ²_R and χ²_L for a 90% confidence interval with sample size n = 21. This solution covers chi-square distribution, degrees of freedom, and the importance of tail areas in statistics. Suitable for advanced high school or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation