2
$\begingroup$

Let's assume we have a right one-sided test with a p-value of 0.03 and a positive test statistical value. Now let's perform a two-sided test with the same datums. Are we going to reject H0? The significance level is 5%, the test could be either Z or T with one population, so the distribution is symmetric.

$\endgroup$
13
  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Jan 13, 2023 at 21:24
  • 2
    $\begingroup$ What's the significance level? Is the test statistic continuous? Is its distribution symmetric? $\endgroup$
    – Glen_b
    Jan 14, 2023 at 9:46
  • $\begingroup$ What test do you have in mind ? $\endgroup$ Jan 14, 2023 at 15:02
  • $\begingroup$ Sorry, the significance level is 5%, I test could be either Z or T with one population, so the distribution is symmetric. $\endgroup$
    – Nati
    Jan 16, 2023 at 14:46
  • 1
    $\begingroup$ It's just that when you are conducting a two-sided test, you have to use both tails. Because you don't know if you should use the right tail or the left tail. $\endgroup$ Jan 17, 2023 at 20:31

1 Answer 1

2
$\begingroup$

The two-sided p-value in the question will be 0.06, insignificant at 5% level, as two probabilities 0.03 have to be added from both sides. But this was apparently clear already.

The more interesting question is, how is it possible and how can it be reasonable that the same data reject the $H_0$ when testing one-sided, but don't reject when testing two-sided?

The reason is that in general the significance level is actually a performance characteristic of the test. The idea is that if you test a lot of times at level 5% in situations in which the null hypothesis is true, 5% of the tests will (wrongly) reject the null hypothesis.

Now if you test one-sided, you can only reject if the test statistic is too large (say). You will therefore reject if the statistic is so high that it is in the region of the expected 5% highest values. If you test two-sided, you will reject both if the tests statistic is too large and if it is too small. Now it should be clear that you cannot always reject the two-sided test if the one-sided tests rejects. Because if you did so, as the two-sided test is symmetric, you'd need to reject as often on the negative side, and that would give you 10% rejections, but you want to test at 5% level, so that'd be too much.

In fact, in order to reach a rejection probability of 5% for the two-sided test, you can only afford to reject half the time on the positive side than the one-sided test would reject. The two-sided test has more options to reject (namely on both sides), so it needs to reject less often on either side in order to achieve the same overall level of 5%.

"Translated" into p-values this means that the p-value, if observing on the right side, should be smaller than 5% only half the time for the two-sided test than for the one-sided test. This is achieved by multiplying the one-sided p-value by 2.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.