Question

use the t-distribution table to find the critical value(s) for the indicated alternative hypotheses, level sample sizes ( n_1 ) and ( n_2 ). assume that the samples are random and independent, and the population distributed. complete parts (a) and (b).
( h_a: mu_1
eq mu_2, alpha = 0.20, n_1 = 10, n_2 = 2 )
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

Explanation:

Step1: Determine Degrees of Freedom

For equal variances, the degrees of freedom \( df = n_1 + n_2 - 2 \). Substituting \( n_1 = 10 \) and \( n_2 = 2 \), we get \( df = 10 + 2 - 2 = 10 \).

Step2: Identify Significance Level and Tail

The alternative hypothesis is two - tailed (\( H_a:\mu_1
eq\mu_2 \)) with \( \alpha = 0.20 \). So, the significance level for each tail is \( \frac{\alpha}{2}=\frac{0.20}{2}=0.10 \).

Step3: Find Critical Values from t - table

Using the t - distribution table with \( df = 10 \) and one - tail area \( 0.10 \), we look for the value in the row corresponding to \( df = 10 \) and the column corresponding to \( \alpha = 0.10 \) (for one - tail). The critical values for a two - tailed test are \( \pm t_{\alpha/2,df} \). From the t - table, \( t_{0.10,10}=1.372 \). So the critical values are \( - 1.372,1.372 \).

Answer:

-1.372, 1.372