Question

suppose you want to test the claim that $\mu_1 = \mu_2$. two samples are randomly selected from sample statistics are given below. assume that $\sigma_1^2 = \sigma_2^2$. at a level of significance of $\alpha = 0.$$h_0$?
$n_1 = 14, n_2 = 12, \bar{x}_1 = 3, \bar{x}_2 = 4, s_1 = 2.5, s_2 = 2.8$
\bigcirc a. reject $h_0$ if the standardized test statistic is less than $-2.492$ or greater than $2.4$
\bigcirc b. reject $h_0$ if the standardized test statistic is less than $-2.064$ or greater than $2.0$
\bigcirc c. reject $h_0$ if the standardized test statistic is less than $-1.711$ or greater than $1.7$
\bigcirc d. reject $h_0$ if the standardized test statistic is less than $-1.318$ or greater than $1.3$

Explanation:

Step1: Identify Test Type

This is a two - sample t - test (since \(\sigma_1^2=\sigma_2^2\) is assumed, we use pooled t - test) for testing \(H_0:\mu_1 = \mu_2\). The degrees of freedom \(df=n_1 + n_2-2\).
Given \(n_1 = 14\) and \(n_2=12\), so \(df=14 + 12-2=24\).

Step2: Determine Significance Level and Critical Value

The significance level \(\alpha = 0.05\) (since it's a two - tailed test, as we are testing \(\mu_1=\mu_2\) and the alternative hypothesis is \(H_1:\mu_1
eq\mu_2\) for a two - tailed test).
For a two - tailed t - test with \(df = 24\) and \(\alpha=0.05\), we look up the critical value in the t - distribution table. The critical values are \(t_{\alpha/2,df}=t_{0.025,24}\).
From the t - table, \(t_{0.025,24}=2.064\). So the rejection region is \(t < - 2.064\) or \(t>2.064\) (since it's a two - tailed test, we have two critical values, positive and negative of the same magnitude).

Step3: Match with Options

Option B says "Reject \(H_0\) if the standardized test statistic is less than - 2.064 or greater than 2.064" (assuming the "2.0" in the option is a typo and should be "2.064" as per our calculation).

Answer:

B. Reject \(H_0\) if the standardized test statistic is less than - 2.064 or greater than 2.064