Question

use the t - distribution table to find the critical value(s) for the indicated alternative hypotheses, level of significance α, 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_1ltmu_2,alpha = 0.05,n_1 = 17,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 the degrees of freedom

When the population variances are equal, the degrees of freedom is \(df=n_1 + n_2-2\).
Given \(n_1 = 17\) and \(n_2=8\), then \(df=17 + 8-2=23\).

Step2: Determine the critical value

Since \(H_a:\mu_1<\mu_2\), this is a left - tailed test with \(\alpha = 0.05\).
Looking up the t - distribution table for \(df = 23\) and \(\alpha=0.05\) (left - tailed), the critical value is \(t=-1.714\).

Answer:

\(-1.714\)