Weston completed a hypothesis test and found that he could not reject the null hypothesis. Which of the following is a valid reason for his conclusion?
A. The z-statistic is less than 0.
B. The z-statistic lies in the critical region.
C. The z-statistic is greater than 0.
D. The z-statistic lies outside the critical region.
Answer (2)
The null hypothesis is not rejected when the test statistic falls outside the critical region.
The critical region contains values that lead to the rejection of the null hypothesis.
If the z-statistic lies outside the critical region, there isn't enough evidence to reject the null hypothesis.
Therefore, the valid reason is: The z-statistic lies outside the critical region.
Explanation
Understanding Hypothesis Testing In hypothesis testing, we make a decision about the null hypothesis based on the value of a test statistic (here, the z-statistic) and a critical region. The critical region is a set of values for the test statistic that leads to the rejection of the null hypothesis. If the test statistic falls within the critical region, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
Analyzing the Options Let's analyze the options:
The z-statistic is less than 0: This doesn't automatically mean we cannot reject the null hypothesis. It depends on where the critical region is located. If the critical region includes negative values, we might still reject the null hypothesis.
The z-statistic lies in the critical region: If the z-statistic lies in the critical region, we reject the null hypothesis, not fail to reject it. So, this is incorrect.
The z-statistic is greater than 0: Similar to the first option, this alone doesn't determine whether we reject or fail to reject the null hypothesis. It depends on the location of the critical region.
The z-statistic lies outside the critical region: This is the correct reason. If the z-statistic is not in the critical region, we do not have enough evidence to reject the null hypothesis.
Conclusion Therefore, the valid reason for not rejecting the null hypothesis is that the z-statistic lies outside the critical region.
Examples
In medical research, a hypothesis test might be used to determine if a new drug is effective. The null hypothesis would be that the drug has no effect. If the test statistic (e.g., a z-statistic) falls outside the critical region, the researchers would fail to reject the null hypothesis, concluding that there isn't enough evidence to support the drug's effectiveness. This is crucial for making informed decisions about medical treatments.
The valid reason for Weston not rejecting the null hypothesis is that the z-statistic lies outside the critical region, indicating insufficient evidence to support a difference or effect. Therefore, option D is correct.
;
