Question

find the critical values, $x_{r}^{2}$ and $x_{l}^{2}$, for $c = 0.99$ and $n = 10$.
o a. 2.088 and 21.666
o b. 1.735 and 23.587
o c. 2.156 and 25.188
o d. 2.558 and 23.209

Explanation:

Step1: Calculate the degrees of freedom

The degrees of freedom \(df=n - 1\). Given \(n = 10\), then \(df=10-1 = 9\).

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

The confidence level \(c = 0.99\). The right - tail area \(\alpha_R=\frac{1 - c}{2}=\frac{1 - 0.99}{2}=0.005\), and the left - tail area \(\alpha_L = 1-\frac{1 - c}{2}=1 - 0.005=0.995\).

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

Looking up the chi - square distribution table with \(df = 9\):
For the right - tailed critical value \(X_{R}^{2}\), with \(\alpha=0.005\) and \(df = 9\), \(X_{R}^{2}=23.589\approx23.587\) (due to table precision).
For the left - tailed critical value \(X_{L}^{2}\), with \(\alpha = 0.995\) and \(df=9\), \(X_{L}^{2}=1.735\).

Answer:

B. 1.735 and 23.587