Question

find the critical value(s) and rejection region(s) for a left - tailed chi - square test with a sample size n = 19 and level of significance α = 0.05.
click the icon to view the chi - square distribution table.

find the rejection region(s). choose the correct answer below.

○ a.
image of a chi - square distribution curve with a shaded region
○ b.
image of a chi - square distribution curve with a shaded region
○ c.
image of a chi - square distribution curve with a shaded region
○ d.
image of a chi - square distribution curve with a shaded region

Explanation:

Step1: Determine Degrees of Freedom

For a chi - square test, the degrees of freedom \(df=n - 1\). Given \(n = 19\), so \(df=19 - 1=18\).

Step2: Recall Left - Tailed Chi - Square Test

In a left - tailed chi - square test, the rejection region is in the left - hand tail of the chi - square distribution. The chi - square distribution is right - skewed, so a left - tailed test will have the rejection region where \(\chi^{2}\) is less than the critical value.

Step3: Analyze the Graphs

• Option A: The shaded region is in the middle, which is not a left - tailed or right - tailed region (more like a two - tailed or a non - standard region for a chi - square test).
• Option B: It shows two shaded regions (left and right), which is for a two - tailed test.
• Option C: The shaded region is in the left - hand tail of the chi - square distribution, which is consistent with a left - tailed test.
• Option D: The shaded region is in the right - hand tail, which is for a right - tailed test.

Answer:

C