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I have two groups and I conducted a two-sided $t$-test.

  • mean group 1: 3.79
  • mean group 2: 4.07
  • $p$-value: 0.0081

The $p$-value is less than $\alpha$ = 0.05, hence, I can reject the null hypothesis of equal means and conclude that there is a difference between the mean of the two groups.

Is the mean of group 2 for sure bigger than the mean of group 1? Or do I have to conduct a one-sided $t$-test?

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  • $\begingroup$ Is there a theoretical basis for your interest in one side? $\endgroup$ – Thomas Bilach Aug 19 at 15:37
  • $\begingroup$ (1) Nothing in statistics is "for sure." (2) What would the result of the one-sided t-test be? (You can compute it, by hand, in a fraction of a second from the information given.) $\endgroup$ – whuber Aug 19 at 16:28
  • $\begingroup$ I want to compare the two groups and see if they differ in one variable. Right now I know that they are different, but does this result mean that the variable of interest is higher in group 2 compared to group 1. $\endgroup$ – user283542 Aug 19 at 16:29
  • $\begingroup$ The type of hypothesis you analyze is something you determine a priori. Now that you've done the test, you can estimate the difference between the means and report a confidence interval for that difference. Since you reject the null, the confidence interval for group2 mean - group 1 mean will only contain positive values, so you can be reasonably confident that the group 2 has a larger mean. $\endgroup$ – Demetri Pananos Aug 19 at 16:30
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It is actually simple:

Your two-sided t-test rejects the null hyothesis of equal population means at level $\alpha$. In addition to this, you see that sample mean 2 is larger than sample mean 1. Then the one sided conclusion "true mean 2 is larger than true mean 1" follows at level $\alpha$.

So your reasoning is basically correct and you even don't have to perform a one-sided test for this. Consulting the sample is sufficient for this step.

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It appears you're modifying your statistical test after looking at the data; this is not good practice.

Is the mean of group 2 for sure bigger than the mean of group 1? Or do I have to conduct a one-sided t-test?

The sample mean for group 1 is clearly smaller than the sample mean for group 2 (i.e., $\bar{x}_{1} < \bar{x}_{2}$). But you should not let your observation of the sample statistics drive your decision to use a one- or two-sided alternative. Based upon your question, it appears you are somewhat agnostic about directionality. I assume this, in part, by your decision to use a two-sided $t$-test. Even if the test was performed as a result of the default behavior of most software packages, a two-sided approach is preferred as you cannot articulate a strong theoretical basis for only being interested in one-side.

Again, you should not allow your observation of the test statistic to determine the direction of the test. Under a two-sided alternative, you split $\alpha$ evenly into the two tails. It is a more conservative test by design. The answer to your question is emphatically—no. You should not conduct a one-sided test after the discovery of a significant finding under a two-sided alternative. Your indifference to any one particular side, a priori, should be the deciding factor.

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  • $\begingroup$ For a constructive counterpoint to your initial assertion about "not good practice," please see "A five-decision testing procedure" at tandfonline.com/doi/full/10.1080/00031305.2018.1437075. It might cause you to rethink your emphatic "no." $\endgroup$ – whuber Aug 19 at 16:56
  • $\begingroup$ @whuber I will give it a read. Do you recommend I revise that statement to one that is less categorical? $\endgroup$ – Thomas Bilach Aug 19 at 17:02
  • $\begingroup$ It's a good read. I am interested in hearing your reaction once you have gone through it. $\endgroup$ – whuber Aug 19 at 17:03

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