Question

find the critical value(s) and rejection region(s) for a two - tailed chi - square test with a sample size n = 21 and level of significance α = 0.10. click the icon to view the chi - square distribution table. find the critical value(s). 10.851, 31.410 (round to three decimal places as needed. use a comma to separate answers as needed.) find the rejection region(s). choose the correct answer below. there are four options a, b, c, d with corresponding chi - square distribution graphs.

Explanation:

Step1: Determine Degrees of Freedom

For a chi - square test, the degrees of freedom \(df=n - 1\). Given \(n = 21\), so \(df=21 - 1=20\).

Step2: Find Critical Values for Two - tailed Test

For a two - tailed chi - square test with \(\alpha = 0.10\), we split \(\alpha\) into two tails. So \(\alpha/2=0.05\) and \(1-\alpha/2 = 0.95\).
We use the chi - square distribution table. For \(df = 20\) and \(\chi_{0.95}^{2}\) (left - tail critical value) and \(\chi_{0.05}^{2}\) (right - tail critical value).
From the chi - square distribution table, \(\chi_{0.95,20}^{2}=10.851\) and \(\chi_{0.05,20}^{2}=31.410\).

Step3: Determine Rejection Region

In a two - tailed chi - square test, the rejection regions are the areas in both tails of the chi - square distribution. That is, we reject the null hypothesis when \(\chi^{2}<\chi_{1 - \alpha/2}^{2}\) or \(\chi^{2}>\chi_{\alpha/2}^{2}\). Visually, this corresponds to having two shaded regions (one in the left tail and one in the right tail) of the chi - square distribution. Looking at the options, option B has two shaded regions (one on the left and one on the right) which matches the two - tailed rejection region.

Answer:

Critical values: \(10.851, 31.410\)
Rejection region: B