Question

find the critical value(s) and rejection region(s) for a left - tailed chi - square test with a sample size n = 17 and level of significance α = 0.05.
click the icon to view the chi - square distribution table.
find the critical value(s).
7.962
(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.
○ a. image of a chi - square distribution curve with a shaded region on the left
○ b. image of a chi - square distribution curve with two shaded regions
○ c. image of a chi - square distribution curve with a shaded region on the right
○ d. image of a chi - square distribution curve with a shaded region in the middle

Explanation:

Step1: Determine Degrees of Freedom

For a chi - square test, the degrees of freedom (\(df\)) is calculated as \(df=n - 1\), where \(n = 17\). So \(df=17 - 1=16\).

Step2: Find Critical Value for Left - Tailed Test

For a left - tailed chi - square test with \(\alpha = 0.05\) and \(df = 16\), we use the chi - square distribution table. The critical value \(\chi^{2}_{\alpha,df}\) for a left - tailed test is the value such that \(P(\chi^{2}<\chi^{2}_{\alpha,df})=\alpha\). Looking up in the chi - square table, for \(df = 16\) and \(\alpha=0.05\) (left - tailed, so we use the complement, i.e., we look for \(\chi^{2}_{1 - \alpha,df}\) where \(1-\alpha = 0.95\)), we find that \(\chi^{2}_{0.95,16}=7.962\).

Step3: Determine Rejection Region

In a left - tailed chi - square test, the rejection region is the set of all \(\chi^{2}\) values less than the critical value. Looking at the graphs, option A shows the rejection region (the blue shaded area) on the left - hand side (smaller values of \(\chi^{2}\)), which corresponds to the left - tailed rejection region.

Answer:

Critical value: \(7.962\)
Rejection region: A (the graph with the blue shaded area on the left - most part of the \(\chi^{2}\) distribution curve)