13. Which two results occur when the null hypothesis is accepted using a t-test?
Answer: A D
The test statistic is less than the critical value and there appears to be no difference between the two samples.
When the null hypothesis is accepted using a t-test, it indicates that the test statistic is less than the critical value, suggesting no significant difference between the groups being compared. This acceptance also supports the conclusion that there appears to be no difference between the two samples.
A) The test statistic is less than the critical value.
This statement is correct as it reflects the primary condition for accepting the null hypothesis. In hypothesis testing, if the t-test statistic is less than the critical value, it suggests that there is insufficient evidence to reject the null hypothesis, thereby supporting the idea that any observed differences are due to sampling variability.
B) There appears to be a difference between the two samples.
This statement is incorrect because accepting the null hypothesis implies that there is no statistically significant difference between the two samples. If there were a difference, the null hypothesis would be rejected instead.
C) The test statistic is greater than the critical value.
This statement is also incorrect. A test statistic greater than the critical value would lead to the rejection of the null hypothesis, indicating that there is a significant difference between the samples, which contradicts the premise of accepting the null hypothesis.
D) There appears to be no difference between the two samples.
This statement is correct as it reflects the conclusion drawn when the null hypothesis is accepted. It implies that the evidence does not suggest any significant difference exists between the groups being studied.
Conclusion
In conclusion, when the null hypothesis is accepted, it is confirmed that the test statistic is less than the critical value and that there appears to be no difference between the two samples. Options B and C are incorrect as they suggest the presence of a difference, which is contrary to the implications of accepting the null hypothesis. Thus, options A and D accurately represent the outcomes of such a statistical test.