The difference between the one-tailed and two-tailed p-values reflects the choice of the direction of the test, which can impact the results and interpretation. When designing a study, it is important to consider whether you are testing for a difference in a specific direction or in either direction, and to choose the appropriate type of test accordingly.

The one-tailed test places the entire rejection region in one tail of the sampling distribution increasing the width of the rejection region in one or the other tail rather than dividing the rejection region over both tails of the sampling distribution in a two-tailed test. Thus, the rejection region is much larger at one end of the sampling distribution in the one-tailed test making it entirely possible to reject with the one-tailed test, but not with the two-tailed test.

It is a common case, for the difference of sample in one side than the other. Regards.

It is best to know if you need to use a one-tails or two-tailed test before running the analysis. Two tailed tests address the question "is there a difference" while a one tailed test is for the question "is treatment A greater than treatment B." Just a caution, selecting the question based on a significant outcome would be p-hacking.

Thank you so much. All these answers were very helpful.

One- vs. two-sided tests are relevant only in the context of NP-tests. In Fisher's significance tests, only H0 is tested, and being unable to reject H0 does NOT imply to accept H0. "Sidedness" is not of any concern. This is one of the spooky poppycock produced in the illogical NHST-framework (a conglomerate of NP and Fisher).

Calculating the statistical significance of the observed data under a (null-)model has no "side". Either you have sufficient data to statistically discern the estimate from the hypothesized value or you don't. It does not matter on which side the estimate is from the hypothesized value. You are free to chose the level of significance needed to reach a conclusion (although there typically is a community agreement on that level).

If you specify a "side" for the test, then you should have specified a specific alternative (e.g. HB: the mean difference is (at least) 4.2). And for this specific alternative you should have made a sample size calculation to ensure a reasonable power (typically > 0.8). Then, and only then, you get a proper one-sided acceptance region for HB, based on your desired type-I and type-II error rates, and you would accept HB when the test value is in the acceptance region, and otherwise you would accept HA.

Jochen Wilhelm : « sideness » also exists for Fisher approach, since the computation of the p-value is very different if you consider H0 µ = µ0 (two-sided test), µ = µ0 (one-sided, other side).

Note that p-value for composite null hypothesis is defined as the maximum of p-values for all point hypothesis of the null, this is why in practice the p-value for µ

Yes, Emmanuel Curis , I just found references to Fishers work that state that he used one-sided p-values. It was never so clear to me:

https://www.onesided.org/articles/fisher-neyman-pearson-advocates-one-sided-tests-confidence-intervals.php