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?

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.