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?
Fail to reject H0 because there is insufficient evidence that the mean GPA difference is not 0.
Since the test statistic of -2.0 does not exceed the critical value of -2.998, we do not have sufficient evidence to reject the null hypothesis H0: μ1 = μ2. This indicates that the observed difference in GPA means is not statistically significant at the chosen level of significance.
This choice is accurate because the test statistic of -2.0 is within the acceptance region defined by the critical values of -2.998 and 2.998. Therefore, we do not have enough evidence to conclude that there is a significant difference between the group means.
This option is incorrect because rejecting the null hypothesis requires the test statistic to fall outside the critical values. Since -2.0 does not exceed -2.998, we cannot claim that there is significant evidence against H0.
While it is true that the mean difference is -0.12, the reason for failing to reject H0 is not based on the sign of the difference but rather on the comparison of the test statistic to the critical values. The decision is made based on statistical significance, not merely the value of the mean difference.
This option misrepresents the findings since rejecting H0 suggests that there is evidence to support that the means differ significantly. However, our calculations indicate that we do not have sufficient evidence to conclude that the means are different from each other.
In this scenario, the analysis shows that the test statistic does not provide enough evidence to reject the null hypothesis. With a test statistic of -2.0 falling within the critical range, we conclude that there is insufficient evidence to support a significant difference in the mean GPAs of the two groups, affirming the null hypothesis that μ1 equals μ2.
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