Find the critical​ values,

Upper X Subscript Upper R Superscript 2X2R

and

Upper X Subscript Upper L Superscript 2X2L​,

for

cequals=0.95

and

nequals=12.

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Part 1

A.

3.053 and 24.725

B.

3.816 and 21.920

C.

2.603 and 19.675

D.

4.575 and 26.757

To determine the critical values \( \chi_{R}^2 \) and \( \chi_{L}^2 \) for a confidence level \( c = 0.95 \) and sample size \( n = 12 \), we use the chi-square distribution.

### Steps:

1. **Degrees of Freedom**: The degrees of freedom \( df \) is calculated as \( df = n - 1 \).  
   \( df = 12 - 1 = 11 \).

2. **Find the Critical Values**:
   - The confidence level is \( c = 0.95 \), so \( \alpha = 1 - 0.95 = 0.05 \).
   - The chi-square distribution is divided into two tails:
     - **Upper tail critical value** \( \chi_{R}^2 \) corresponds to \( \alpha/2 \) (i.e., 0.025) on the right tail.
     - **Lower tail critical value** \( \chi_{L}^2 \) corresponds to \( 1 - \alpha/2 \) (i.e., 0.975) on the left tail.

Using the chi-square distribution table or calculator:

- For \( df = 11 \) and \( \alpha/2 = 0.025 \), the upper critical value \( \chi_{R}^2 \approx 21.920 \).
- For \( df = 11 \) and \( 1 - \alpha/2 = 0.975 \), the lower critical value \( \chi_{L}^2 \approx 3.816 \).

### Answer:
The correct option is **B. 3.816 and 21.920**.

Would you like more details or have any questions?

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Here are 8 related questions you might be interested in:

1. What is a chi-square distribution, and how is it used?
2. How do you find chi-square critical values using a table?
3. What does the degree of freedom mean in chi-square tests?
4. How does changing the confidence level affect the critical values?
5. What are the applications of chi-square tests in statistics?
6. Can you explain the difference between one-tailed and two-tailed tests?
7. How do you interpret the results of a chi-square test in hypothesis testing?
8. What is the relationship between chi-square distribution and normal distribution?

**Tip**: Always check whether your test is one-tailed or two-tailed when determining critical values.

Learn how to find chi-square critical values for a confidence level of 0.95 and sample size of 12 using the chi-square distribution. Understand the calculation of degrees of freedom and the significance of critical values in statistical testing.

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