Question

find the critical values, $t_0$, to test the claim that $\mu_1 = \mu_2$. two samples are rano that are normal. the sample statistics are given below. assume that $\sigma_1^2 = \sigma_2^2$.$n_1 = 14, n_2 = 12, \bar{x}_1 = 6, \bar{x}_2 = 7, s_1 = 2.5, s_2 = 2.8$\\(\bigcirc\\) a. $\pm 1.711$\\(\bigcirc\\) b. $\pm 2.492$\\(\bigcirc\\) c. $\pm 1.318$\\(\bigcirc\\) d. $\pm 2.064$

Explanation:

Step1: Determine Degrees of Freedom

For two - sample t - test with equal variances ($\sigma_{1}^{2}=\sigma_{2}^{2}$), the degrees of freedom $df=n_{1}+n_{2}-2$. Given $n_{1} = 14$ and $n_{2}=12$, then $df=14 + 12-2=24$.

Step2: Determine the Test Type and Significance Level

We are testing the claim $\mu_{1}=\mu_{2}$, so it is a two - tailed test. For a two - tailed t - test with $df = 24$, we can use the t - distribution table or a calculator to find the critical value. The common significance level for such tests (if not specified) is $\alpha=0.05$. For a two - tailed test with $\alpha = 0.05$ and $df=24$, the critical value $t_{\alpha/2}$ is found from the t - table. Looking up in the t - distribution table, for $df = 24$ and two - tailed $\alpha=0.05$, the critical value is $\pm2.064$.

Answer:

D. $\pm2.064$