Question

find the critical values, ( x_{r}^{2} ) and ( x_{l}^{2} ), for ( c = 0.99 ) and ( n = 10 ).

a. 2.088 and 21.666
b. 1.735 and 23.587
c. 2.558 and 23.209
d. 2.156 and 25.188

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 significance level

The confidence level \(c = 0.99\), so the significance level \(\alpha=1 - c=1 - 0.99 = 0.01\).

Step3: Find the right - tail and left - tail critical values

For the right - tail critical value \(\chi_{R}^{2}\), the area to the right is \(\frac{\alpha}{2}=0.005\). Looking up in the chi - square distribution table with \(df = 9\), \(\chi_{R}^{2}=23.209\).
For the left - tail critical value \(\chi_{L}^{2}\), the area to the right is \(1-\frac{\alpha}{2}=0.995\). Looking up in the chi - square distribution table with \(df = 9\), \(\chi_{L}^{2}=2.558\).

Answer:

C. \(2.558\) and \(23.209\)