48. In a study, the reported distances of two small groups of randomly selected GPS units are measured. The difference in sample means (μ₠âˆ' μ₂) is âˆ'0.12 miles. A t-test of the hypothesis Hâ‚€: μ₠= μ₂ versus Ha: μ₠≠ μ₂ is set up. The test statistic is âˆ'2.84, and the critical values are âˆ'2.228 and 2.228. What is the appropriate conclusion?
Answer: A
Reject H0 based on evidence that the mean reported distance difference is not 0 miles.
The t-test results indicate a test statistic of -2.84, which falls outside the critical values of -2.228 and 2.228. This provides sufficient evidence to reject the null hypothesis, suggesting that the mean difference in reported distances is significantly different from zero.
A) Reject H0 based on evidence that the mean reported distance difference is not 0 miles.
This option is correct because the calculated test statistic of -2.84 is less than the critical value of -2.228, leading to rejection of the null hypothesis (H0). This indicates that there is significant evidence to conclude that the mean difference in reported distances is not zero.
B) Fail to reject H0 based on the fact that the mean difference of -0.12 miles is less than 0 miles.
This option is incorrect. The conclusion to fail to reject H0 cannot be based solely on the mean difference being less than zero. The relevant comparison is between the test statistic and the critical values, which in this case shows that the test statistic is significant.
C) Fail to reject H0 because there is insufficient evidence that the mean reported distance difference is not 0 miles.
This option is incorrect. The evidence provided by the test statistic of -2.84 is indeed sufficient to reject H0. The statistical significance indicates that there is strong evidence against the null hypothesis, contrary to what this option suggests.
D) Reject H0 based on evidence that the mean reported distance difference is equal to 0 miles.
This option is incorrect because rejecting H0 does not imply that the mean difference is equal to zero; rather, it indicates that the mean difference is significantly different from zero. Thus, this statement misrepresents the conclusion drawn from the test.
Conclusion
The correct conclusion is to reject H0, as the test statistic of -2.84 is beyond the critical threshold, affirming that the mean distance difference is significantly different from zero. Other options either misunderstand the implications of the test results or incorrectly reason about the evidence available, failing to recognize the significance of the findings.