6. If we reject H0:µ = 10 in favor of Ha:µ 10 at a given level of significance with a positive value of the test statistic z, then if we test H0:µ = 10 versus Ha:µ > 10 using the same sample, Ha:µ > 10 will ______ be rejected at the same significance level. (Points : 1)
Always
Sometimes
Never
To understand the situation, let's break it down step by step.
First, we have the null hypothesis H0:µ = 10, which means that we assume the population mean (µ) is equal to 10. The alternate hypothesis is Ha:µ > 10, which means we are testing whether the population mean is greater than 10.
Now, suppose we reject the null hypothesis H0:µ = 10 in favor of the alternate hypothesis Ha:µ > 10 at a given level of significance with a positive value of the test statistic z. This means that we have sufficient evidence to conclude that the population mean is indeed greater than 10.
If we continue using the same sample and test H0:µ = 10 versus Ha:µ > 10 (with the same null and alternate hypotheses), the question is asking whether Ha:µ > 10 will be rejected at the same significance level.
The answer depends on whether the positive value of the test statistic z in the first test falls within the critical region of the second test, which determines whether the null hypothesis can be rejected.
- If the positive value of the test statistic z is much larger than the critical value for the second test at the given level of significance, then Ha:µ > 10 will be rejected. In this case, we have even stronger evidence to conclude that the population mean is greater than 10.
- If the positive value of the test statistic z is not as large as the critical value for the second test at the given level of significance, then Ha:µ > 10 will not be rejected. In this case, we don't have enough evidence to conclude that the population mean is significantly greater than 10.
Therefore, the answer is sometimes. Depending on the specific values and the level of significance, Ha:µ > 10 may or may not be rejected when using the same sample.