12) Find the critical values, $X_{R}^{2}$ and $X_{L}^{2}$, for $c=0.90$ and $n=15$. 12) A) 4.660 and 29.131 B) 5.629 and 26.119 C) 4.075 and 31.319 D) 6.571 and 23.685
Solution
Step 1 ：Given that the confidence level, \(c = 0.90\), and the sample size, \(n = 15\).
Step 2 ：The degrees of freedom, \(df\), is calculated as \(n - 1 = 15 - 1 = 14\).
Step 3 ：The significance level, \(\alpha\), is calculated as \(1 - c = 1 - 0.90 = 0.10\).
Step 4 ：The critical values, \(X_{R}^{2}\) and \(X_{L}^{2}\), are calculated using the chi-square distribution with \(df = 14\).
Step 5 ：\(X_{R}^{2}\) is the value such that the area to the left under the chi-square distribution is \(\alpha / 2 = 0.10 / 2 = 0.05\).
Step 6 ：\(X_{L}^{2}\) is the value such that the area to the right under the chi-square distribution is \(\alpha / 2 = 0.10 / 2 = 0.05\).
Step 7 ：By calculating, we find that \(X_{R}^{2} \approx 6.571\) and \(X_{L}^{2} \approx 23.685\).
Step 8 ：So, the critical values for \(c = 0.90\) and \(n = 15\) are approximately 6.571 and 23.685.
Step 9 ：Final Answer: \(\boxed{D)} 6.571 \text{ and } 23.685\)
