Difficulty: Easy

Average Score: 78%

Which two results occur when the null hypothesis is accepted using a t-test?

Rationale

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 in a t-test, it indicates that the test statistic did not exceed the critical value, suggesting that any observed difference between the sample means is not statistically significant. Thus, it leads to the conclusion that there is no significant difference between the two samples being compared.

A (The test statistic is less than the critical value.) CORRECT

This statement is true as it directly describes a key condition for accepting the null hypothesis. If the test statistic is less than the critical value, it implies insufficient evidence to reject the null hypothesis, supporting the idea that any observed differences are likely due to random variation rather than a true effect.

B (There appears to be no difference between the two samples.) CORRECT

This is also correct because accepting the null hypothesis means concluding that there is no significant difference between the two samples. This acceptance reflects a finding that any differences observed in sample means can be attributed to chance rather than a systematic effect.

C (There appears to be a difference between the two samples.) YOUR ANSWER

This choice is incorrect because accepting the null hypothesis explicitly indicates that there is no statistically significant difference between the samples. If there were a significant difference, the null hypothesis would be rejected.

D (The test statistic is greater than the critical value.) YOUR ANSWER

This statement contradicts the acceptance of the null hypothesis. If the test statistic were greater than the critical value, it would lead to rejecting the null hypothesis, suggesting a significant difference between the samples.

Conclusion

In summary, accepting the null hypothesis in a t-test leads to the conclusions that the test statistic is less than the critical value and there appears to be no difference between the two samples. This fundamental understanding of hypothesis testing ensures accurate interpretation of statistical results and informs decision-making based on empirical evidence.

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