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 critical value(s). (round to three decimal places as needed. use a comma to separate answers as needed.)

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: Identify the Test Type and Significance Level

We have a left - tailed test with \(\alpha = 0.05\). For a left - tailed chi - square test, we need to find the critical value \(\chi^{2}_{\alpha,df}\) such that \(P(\chi^{2}<\chi^{2}_{\alpha,df})=\alpha\). This is equivalent to finding the value in the chi - square distribution table with degrees of freedom \(df = 18\) and the area to the left of the critical value equal to \(0.05\).
Looking at the chi - square distribution table, for \(df = 18\) and the cumulative probability (area to the left) of \(0.05\), we find that \(\chi^{2}_{0.05,18}=9.390\) (using chi - square distribution table or statistical software).
The rejection region for a left - tailed chi - square test is all values of \(\chi^{2}\) such that \(\chi^{2}<\chi^{2}_{\alpha,df}\), so the rejection region is \(\chi^{2}<9.390\).

Answer:

The critical value is \(9.390\), and the rejection region is \(\chi^{2}<9.390\)