4- Consider a test for mue(mean).
A) if the p-value is such that you can reject H_0 at the 1% level signifucance, can you always reject H_0 at 5% level significance? Explain why or why not.
B) if the p-value is such that you can reject H_0 at the 5% level of significance, can you always reject H_0 at the 1% level significance? Explain why or why not.
A) If the p-value obtained from the test is low enough that you can reject the null hypothesis (H_0) at the 1% level of significance, it means that the observed data is very unlikely to occur if the null hypothesis is true. In other words, the evidence against the null hypothesis is strong enough to conclude that the alternative hypothesis is true.
Now, when we test at a higher level of significance, such as 5%, the critical region is larger and the p-value threshold for rejection is higher. This means that there is a greater chance of observing the same data (or data more extreme) under the null hypothesis. Therefore, if the null hypothesis is rejected at the 1% level of significance, it will also be rejected at the 5% level of significance because the evidence against the null hypothesis is even stronger at the 1% level.
In summary, if you can reject H_0 at the 1% level of significance, you can always reject it at the 5% level because the evidence against the null hypothesis is more substantial.
B) Conversely, if you can reject H_0 at the 5% level of significance, it means that the observed data is unlikely to occur if the null hypothesis is true, but there is still a possibility (though small) that it could have occurred by chance.
When we test at a lower level of significance, such as 1%, the critical region is smaller and the p-value threshold for rejection is lower. This means that there is a lower chance of observing the same data (or data more extreme) under the null hypothesis. Therefore, if the null hypothesis is rejected at the 5% level of significance, it may not necessarily be rejected at the 1% level because the evidence may not be strong enough to reach the stricter threshold.
In summary, if you can reject H_0 at the 5% level of significance, it does not guarantee that you can always reject it at the 1% level because the evidence against the null hypothesis may not be sufficient at the more stringent level.