45. A researcher measured the reported distances of two small groups of randomly selected GPS units. They found that the difference in sample means (μ₠âˆ' μ₂) is 0.22 miles. Which conclusion is appropriate from a t-test of the hypothesis Hâ‚€: μ₠= μ₂ versus Ha: μ₠≠ μ₂ if the p-value is 0.025 and the significance level is α = 0.05?
Answer: B
Reject H0 based on evidence that the mean reported distance difference is not 0 miles.
The p-value of 0.025 is less than the significance level of α = 0.05, indicating that there is statistically significant evidence to reject the null hypothesis H0. This suggests that the mean difference of 0.22 miles is indeed significant enough to conclude that the two groups differ in their reported distances.
A) Fail to reject H0 based on the fact that the mean difference of 0.22 miles is greater than 0 miles.
This option is incorrect because the decision to reject or fail to reject the null hypothesis is based on the p-value in relation to the significance level, not merely on the observed mean difference. Since the p-value is less than α, there is sufficient evidence to reject H0, regardless of the mean difference being greater than 0.
B) Reject H0 based on evidence that the mean reported distance difference is not 0 miles.
This option is correct as the p-value of 0.025 indicates that the observed difference in means is statistically significant. Since this p-value is less than the significance level of 0.05, it supports rejecting the null hypothesis in favor of the alternative hypothesis that the mean difference is not equal to zero.
C) Fail to reject H0 because there is insufficient evidence that the mean reported distance difference is not 0 miles.
This option is incorrect because the p-value provides sufficient evidence to reject H0. A p-value of 0.025 indicates that the likelihood of observing such a mean difference, assuming the null hypothesis is true, is low, thus warranting the rejection of H0.
D) Reject H0 based on evidence that the mean reported distance difference is equal to 0 miles.
This option is incorrect as it misrepresents the evidence. Rejecting H0 implies that we believe the mean difference is not equal to 0, which contradicts the assertion made in this option. The rejection is based on significant evidence suggesting the mean difference is indeed not zero.
Conclusion
The correct conclusion is to reject H0 based on the p-value being less than the significance level, indicating significant evidence that the mean reported distance difference is not 0 miles. All other options either misinterpret the statistical evidence or incorrectly state the relationship between the mean difference and the null hypothesis, leading to erroneous conclusions.