Question

find the critical value(s) and rejection region(s) for a right - 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).
28.412
(round to three decimal places as needed. use a comma to separate answers as needed.)
find the rejection region(s). choose the correct graph below.
○ a.
○ b.
○ c.
○ d.

Explanation:

Step1: Determine Degrees of Freedom

For a chi - square test, the degrees of freedom (\(df\)) is calculated as \(df=n - 1\), where \(n = 21\). So \(df=21 - 1=20\).

Step2: Find Critical Value

We are conducting a right - tailed test with \(\alpha = 0.10\) and \(df = 20\). Using the chi - square distribution table (or a statistical software), we look for the value of \(\chi^{2}\) such that the area to the right of it under the chi - square curve with \(df = 20\) is \(0.10\). From the chi - square distribution table, \(\chi_{0.10,20}^{2}=28.412\).

Step3: Determine Rejection Region

In a right - tailed chi - square test, the rejection region is the set of all \(\chi^{2}\) values greater than the critical value. Looking at the graphs:

• Graph A: The shaded region is on the left, which is for a left - tailed test.
• Graph B: The shaded region is on the right (greater than the critical value), which is consistent with a right - tailed test.
• Graph C: The shaded region is in the middle, which is for a two - tailed test (rejection regions in both tails).
• Graph D: There are two shaded regions (left and right), which is for a two - tailed test.

Answer:

Critical value: \(28.412\)
Rejection region: The graph corresponding to option B (the graph where the shaded region is on the right - hand side of the chi - square distribution curve, representing values of \(\chi^{2}\) greater than the critical value \(28.412\)).