Question

use the t - distribution table to find the critical value(s) for the indicated alternative hypotheses, level of significance $alpha$, and sample sizes $n_1$ and $n_2$. assume that the samples are random and independent, and the populations are normally distributed. complete parts (a) and (b).
$h_a: mu_1
eq mu_2, alpha = 0.20, n_1 = 6, n_2 = 8$
click the icon to view the t - distribution table.
(a) find the critical value(s) assuming that the population variances are equal
(type an integer or decimal rounded to three decimal places as needed. use a comma to separate answers as needed.)

Explanation:

Step1: Calculate degrees of freedom

When population variances are equal, the degrees of freedom \(df=n_1 + n_2-2\). Given \(n_1 = 6\) and \(n_2=8\), then \(df=6 + 8-2=12\).

Step2: Determine the critical value

The alternative hypothesis \(H_a:\mu_1
eq\mu_2\) is a two - tailed test. The level of significance \(\alpha = 0.20\), so the area in each tail is \(\frac{\alpha}{2}=0.10\).
Looking up the \(t\) - distribution table with \(df = 12\) and a tail area of \(0.10\), we find the critical value.

Answer:

\(\pm1.356\)