Let's walk through the steps to determine whether to reject the null hypothesis for a right-tailed z-test given the provided information:
1. Identify the Significance Level ([tex]\(\alpha\)[/tex]):
- The significance level [tex]\(\alpha\)[/tex] is 0.025. This is the threshold for determining whether the test statistic is extreme enough to reject the null hypothesis.
2. Determine the Test Statistic:
- The test statistic is given as [tex]\( z = 3 \)[/tex].
3. Find the Critical Value for a Right-Tailed Test:
- For a right-tailed test, the critical value is determined by finding the z-score that corresponds to the cumulative probability of [tex]\( 1 - \alpha \)[/tex].
- Given [tex]\(\alpha = 0.025\)[/tex], we need the z-score such that the area to the left of it under the standard normal curve is [tex]\( 1 - 0.025 = 0.975 \)[/tex].
4. Compare the Test Statistic with the Critical Value:
- The critical value for [tex]\(\alpha = 0.025\)[/tex] in a right-tailed test is approximately [tex]\( 1.96 \)[/tex].
- Now, compare the test statistic ([tex]\( z = 3 \)[/tex]) to the critical value ([tex]\( z_{crit} = 1.96 \)[/tex]).
5. Conclusion:
- If the test statistic exceeds the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
- Here, the test statistic [tex]\( z = 3 \)[/tex] is greater than the critical value [tex]\( z_{crit} = 1.96 \)[/tex].
Therefore, you Reject the Null Hypothesis based on the test statistic [tex]\( z = 3 \)[/tex] and the critical value [tex]\( 1.96 \)[/tex].
