Question

find the critical values, $x_{r}^{2}$ and $x_{l}^{2}$, for $c = 0.95$ and $n = 12$.
a. 3.053 and 24.725
b. 3.816 and 21.920
c. 2.603 and 19.675
d. 4.575 and 26.757

Explanation:

Step1: Calculate the degrees of freedom

The degrees of freedom \( df=n - 1\). Given \(n = 12\), then \(df=12-1 = 11\).

Step2: Calculate the right - tail and left - tail areas

The confidence level \(c = 0.95\), so the significance level \(\alpha=1 - c=1 - 0.95 = 0.05\).
The right - tail area \(\alpha_R=\frac{\alpha}{2}=0.025\), and the left - tail area \(\alpha_L = 1-\frac{\alpha}{2}=0.975\).

Step3: Find the critical values using the chi - square distribution table

Looking up the chi - square distribution table with \(df = 11\):
For the right - tail critical value \(X_{R}^{2}\) (with area \(0.025\) to the right), \(X_{R}^{2}=21.920\).
For the left - tail critical value \(X_{L}^{2}\) (with area \(0.975\) to the right), \(X_{L}^{2}=3.816\).

Answer:

B. \(3.816\) and \(21.920\)