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

• 1.714

(type an integer or decimal rounded to three decimal places as needed. use a comma to separate answers as needed.)
(b) find the critical value(s) assuming that the population variances are not 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 variances are equal

The formula for degrees of freedom when variances are equal is \(df=n_1 + n_2-2\).
Substitute \(n_1 = 17\) and \(n_2 = 8\) into the formula: \(df=17 + 8-2=23\).
Since \(H_a:\mu_1<\mu_2\), it is a left - tailed test with \(\alpha = 0.05\).

Step2: Calculate degrees of freedom when variances are not equal

The formula for degrees of freedom when variances are not equal is \(df=\min(n_1 - 1,n_2 - 1)\).
Substitute \(n_1 = 17\) and \(n_2 = 8\) into the formula: \(df=\min(17 - 1,8 - 1)=\min(16,7)=7\).
Since \(H_a:\mu_1<\mu_2\), it is a left - tailed test with \(\alpha = 0.05\).

Step3: Find critical value from t - distribution table

For a left - tailed test with \(\alpha = 0.05\) and \(df = 7\) (when variances are not equal), looking up the t - distribution table, the critical value is \(t=-1.895\).

Answer:

(a) \(-1.714\)
(b) \(-1.895\)