Question

use the t-distribution table to find the critical value(s) for the indicated alternative hypo sample sizes ( n_1 ) and ( n_2 ). assume that the samples are random and independent, and distributed. complete parts (a) and (b).
( h_a: mu_1 < mu_2, alpha = 0.005, n_1 = 12, n_2 = 10 )
click the icon to view the t-distribution table.
(a) find the critical value(s) assuming that the population variances are equal.

• 2.845

(type an integer or decimal rounded to three decimal places as needed. use a comm
(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 comm

Explanation:

Step1: Determine Degrees of Freedom (Unequal Variances)

For unequal variances, we use the smaller of \( n_1 - 1 \) and \( n_2 - 1 \). Here, \( n_1 = 12 \), so \( n_1 - 1 = 11 \); \( n_2 = 10 \), so \( n_2 - 1 = 9 \). The smaller is 9.

Step2: Find Critical Value for Left - Tailed Test

We have a left - tailed test (\( H_a: \mu_1 < \mu_2 \)) with \( \alpha = 0.005 \) and degrees of freedom \( df = 9 \). Using the t - distribution table, the critical value \( t_{\alpha, df} \) for \( \alpha = 0.005 \) and \( df = 9 \) is - 3.250 (since it's a left - tailed test, the critical value is negative).

Answer:

-3.250