Question

find the critical $\chi^2$-value to test the claim $\sigma^2 \leq 3.2$ if $n = 20$ and $\alpha = 0.01$.

a. 30.144
b. 32.852
c. 36.191
d. 27.204

Explanation:

Step1: Determine the test type and degrees of freedom

This is a right - tailed test for the variance (since the claim is \(\sigma^{2}\leq3.2\), the alternative hypothesis is \(\sigma^{2}> 3.2\)). The degrees of freedom \(df=n - 1\), where \(n = 20\), so \(df=20 - 1=19\).

Step2: Find the critical \(\chi^{2}\) - value

We are given \(\alpha = 0.01\) and \(df = 19\). We use the chi - square distribution table (or a statistical calculator) to find the critical value \(\chi_{\alpha,df}^{2}=\chi_{0.01,19}^{2}\). Looking up in the chi - square table (or using a calculator function for chi - square critical values), we find that \(\chi_{0.01,19}^{2}=36.191\)? Wait, no, wait. Wait, let's re - check. Wait, for a right - tailed test with \(\alpha = 0.01\) and \(df=19\), the critical value. Wait, maybe I made a mistake. Wait, the claim is \(\sigma^{2}\leq3.2\), so the alternative is \(\sigma^{2}>3.2\), so it's a right - tailed test. The degrees of freedom is \(n - 1=19\). Let's check the chi - square table again. Wait, the chi - square table: for \(df = 19\) and \(\alpha=0.01\), the value is 36.191? Wait, no, wait, maybe I confused with another table. Wait, no, let's use the formula or a calculator. Wait, actually, the critical value for \(\chi^{2}\) with \(df = 19\) and \(\alpha=0.01\) (right - tailed) is 36.191? Wait, no, wait, no, wait. Wait, the options: option C is 36.191, but wait, maybe I made a mistake in degrees of freedom? Wait, \(n = 20\), so \(df=19\). Wait, let's check the chi - square critical values. Wait, the chi - square distribution: the critical value for \(\alpha = 0.01\) and \(df = 19\) is indeed 36.191? Wait, no, wait, no, I think I messed up. Wait, no, the claim is \(\sigma^{2}\leq3.2\), so the test is for variance. The critical value for a right - tailed test with \(\alpha = 0.01\) and \(df=n - 1 = 19\) is found from the chi - square table. Let's check the standard chi - square table. For \(df = 19\), the column for \(\alpha=0.01\) gives 36.191. But wait, let's check the options. Option C is 36.191. But wait, maybe I made a mistake. Wait, no, wait, let's re - calculate. Wait, no, the correct critical value for \(\chi^{2}\) with \(df = 19\) and \(\alpha=0.01\) (right - tailed) is 36.191. So the answer should be C.

Wait, no, wait, I think I made a mistake. Wait, the claim is \(\sigma^{2}\leq3.2\), so the null hypothesis \(H_{0}:\sigma^{2}\leq3.2\), alternative \(H_{1}:\sigma^{2}>3.2\). The critical value for \(\chi^{2}\) is calculated as follows. The degrees of freedom is \(n - 1=19\). The critical value \(\chi_{\alpha,df}^{2}=\chi_{0.01,19}^{2}\). Looking up in the chi - square table, for \(df = 19\) and \(\alpha = 0.01\), the value is 36.191. So the correct option is C.

Answer:

C. 36.191