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. 2.156 and 25.188
c. 2.558 and 23.209
d. 1.735 and 23.587

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 total area in the two tails is \(1 - c=1 - 0.99 = 0.01\). The right - tail area \(\alpha_R=\frac{1 - c}{2}=\frac{0.01}{2}=0.005\), and the left - tail area \(\alpha_L = 1-\frac{1 - c}{2}=1 - 0.005=0.995\).

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

Looking up the chi - square distribution table with \(df = 9\):

• For the right - tailed critical value \(\chi_{R}^{2}\) (with area \(0.005\) to the right), \(\chi_{R}^{2}=23.589\approx23.587\) (due to table approximation).
• For the left - tailed critical value \(\chi_{L}^{2}\) (with area \(0.995\) to the left), \(\chi_{L}^{2}=1.735\).

Answer:

D. 1.735 and 23.587