19. In a study, the GPAs of two small groups of randomly selected students are measured. The difference in sample means (μ1 - μ2) is -0.12. A t-test of the hypothesis H0: μ1 = μ2 versus Ha: μ1 ≠ μ2 is set up. The test statistic is -2.0 and the critical values are -2.998 and 2.998. What is the appropriate conclusion?
Answer: A
Fail to reject H0 because there is insufficient evidence that the mean GPA difference is not 0.
The conclusion based on the given test statistic of -2.0, which falls outside the critical values of -2.998 and 2.998, indicates that there is insufficient evidence to reject the null hypothesis. Therefore, we conclude that the mean GPA difference is not statistically significant and we fail to reject H0.
A) Fail to reject H0 because there is insufficient evidence that the mean GPA difference is not 0.
This option is correct as the test statistic of -2.0 is not in the critical region defined by the values -2.998 and 2.998. Since the test statistic does not exceed the critical thresholds, we do not have strong enough evidence to claim that the mean GPA difference is significantly different from 0.
B) Reject H0 based on evidence that the mean GPA difference is not 0.
This option is incorrect because the test statistic of -2.0 does not fall within the critical region. Rejecting H0 would require the test statistic to be greater than 2.998 or less than -2.998, which it is not. Therefore, we cannot conclude that the mean GPA difference is significantly different from 0.
C) Fail to reject H0 based on the fact that the mean difference of -0.12 is less than 0.
This option is incorrect because the conclusion to fail to reject the null hypothesis should be based on the statistical evidence from the test statistic and critical values, not merely on the value of the mean difference. The mean difference being less than 0 does not provide sufficient evidence to draw a conclusion regarding the hypothesis test.
D) Reject H0 based on evidence that the mean GPA difference is equal to 0.
This option is incorrect as it misinterprets the statistical evidence. Rejecting H0 would imply that there is strong evidence against the null hypothesis, which is not supported by the test statistic of -2.0 falling inside the acceptance region. The evidence does not support the claim that the mean GPA difference is equal to 0.
Conclusion
The appropriate conclusion is to fail to reject H0 because the test statistic does not fall into the critical region defined by the test's critical values. All other options either misinterpret the statistical evidence or draw incorrect conclusions based on the mean difference rather than the results of the hypothesis test. Thus, option A correctly reflects the outcome of the analysis based on the provided data.