To draw: The conclusion and internet the decision.

Given:
A professor is concerned that the two sections of college algebra that he teaches are not performing at the same level. To test his claim, he looks at the mean exam score for a random sample of students from each of his classes. In Class1, the mean exam for 12 students is 78.7 with a standard deviation of 6.5. In Class 2, the mean exam core for 15 students is 81.1 with a standard deviation of 7.4. Assume that the population variances are equal
Procedure:
Rejection Regions for hypothesis tests for two population means. Reject the null hypothesis, $H_0$ if:
$t\le -t_{\alpha }$ for a left tailed test
$t\ge t_{\alpha }$ for a right tailed test
$|t|\le t_{\frac{\alpha }{2}}$ for a two tailed test
The alternative hypothesis contains “$\ne$“this is a two tailed test. Since the population variances are equal, the number of degrees of freedom for this test is
$n_1+n_2-2=12+15-2=25$
It is given that $\alpha =0.05$. The critical value for a two tailed test with 25 degrees of freedom and 0.05 level of significance is
$t\frac{\alpha }{2}=2.06$
The calculated value of the test statistic -0.88295 is less than the critical value, it does fall in the rejection region. The null hypothesis is rejected because $|t|\le t\frac{\alpha }{2}$
Thus, it is concluded that there is sufficient evidence at the 0.05 level of significance to support the claim that two sections of college algebra that he teaches are performing at the same level.