Question

find the critical value and rejection region for the type of t - test with level of significance a and sa
two - tailed test, \\( \alpha = 0.01, n = 25 \\)
o a. \\( t _ { 0 } = - 2.797, t _ { 0 } = 2.797 ; t < - 2.797, t > 2.797 \\)
o b. \\( t _ { 0 } = 2.797 ; t > 2.797 \\)
o c. \\( t _ { 0 } = - 2.492, t _ { 0 } = 2.492 ; t < - 2.492, t > 2.492 \\)
o d. \\( t _ { 0 } = - 2.787, t _ { 0 } = 2.787 ; t < - 2.787, t > 2.787 \\)

Explanation:

Step1: Determine the degrees of freedom

The degrees of freedom \(df=n - 1\). Given \(n = 25\), then \(df=25-1 = 24\).

Step2: Find the critical value for a two - tailed test

For a two - tailed \(t\) - test with \(\alpha=0.01\) and \(df = 24\), we look up the value in the \(t\) - distribution table. The critical values are \(t_{0}=-t_{\alpha/2}\) and \(t_{0}=t_{\alpha/2}\), where \(\alpha/2=0.005\). From the \(t\) - distribution table, \(t_{0.005,24}=2.797\).

Step3: Determine the rejection region

For a two - tailed test, the rejection region is \(t<-t_{\alpha/2}\) or \(t > t_{\alpha/2}\). Substituting \(t_{\alpha/2}=2.797\), the rejection region is \(t < - 2.797\) or \(t>2.797\).

Answer:

A. \(t_{0}=-2.797,t_{0}=2.797;t < - 2.797,t>2.797\)