Question

use the t-distribution table to find the critical value(s) for the indicated alte sample sizes ( n_1 ) and ( n_2 ). assume that the samples are random and indep 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
(type an integer or decimal rounded to three decimal places as neede

Explanation:

Step1: Determine Degrees of Freedom

For two - sample t - test, when population variances are equal (assuming from the problem context as it's a common case when not specified otherwise), the degrees of freedom \(df=n_1 + n_2-2\). Given \(n_1 = 10\) and \(n_2=2\), then \(df=10 + 2-2=10\).

Step2: Determine Significance Level for Each Tail

The alternative hypothesis is \(H_a:\mu_1
eq\mu_2\), so it's a two - tailed test. The significance level \(\alpha = 0.20\), so the significance level for each tail \(\alpha/2=\frac{0.20}{2}=0.10\).

Step3: Find Critical Values from t - distribution Table

We look for the t - value in the t - distribution table with \(df = 10\) and \(\alpha/2=0.10\) (for two - tailed test). From the t - distribution table, the critical values for a two - tailed test with \(df = 10\) and \(\alpha=0.20\) are \(t_{\alpha/2}=\pm1.372\).

Answer:

\(\pm1.372\)