Question

find the critical value(s) and rejection region(s) for a two - tailed chi - square test with a sample size n = 18 and significance α = 0.10. click the icon to view the chi - square distribution table. find the critical value(s). (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. options with graphs: a, b, c, d (graphs of chi - square distributions with shaded regions).

Explanation:

Step1: Determine Degrees of Freedom

For a chi - square test, the degrees of freedom \(df=n - 1\). Given \(n = 18\), so \(df=18 - 1=17\).

Step2: Find Critical Values for Two - Tailed Test

For a two - tailed chi - square test with \(\alpha = 0.10\), we split the significance level into two tails. So \(\alpha/2=0.05\) and \(1-\alpha/2 = 0.95\).
We use the chi - square distribution table. For \(df = 17\) and \(\chi^{2}_{0.95}\) (lower tail) and \(\chi^{2}_{0.05}\) (upper tail).
From the chi - square table, \(\chi^{2}_{0.95,17}=10.085\) and \(\chi^{2}_{0.05,17}=27.587\).

Step3: Determine Rejection Region

In a two - tailed chi - square test, the rejection regions are \(\chi^{2}<\chi^{2}_{1 - \alpha/2}\) and \(\chi^{2}>\chi^{2}_{\alpha/2}\). Visually, this corresponds to two tails of the chi - square distribution. Looking at the options, option A has two shaded tails (one left and one right), which matches the two - tailed rejection region.

Answer:

Critical values: \(10.085, 27.587\)
Rejection region: Option A