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Rookie stat question here, hoping to get a sanity check...

So lets say that I have two groups, A and B.

I run a two tailed t test in R, t.test(A, B, var.equal = TRUE).

If my t-statistic value is greater than (+/-) the t critical value, with a very low p, I can consider the result statistically significant.

What I would like to confirm is whether I can then make a statement regarding the relationship between the means of the two groups...

ex: If the t stat is (+), can I say that the mean of group A is statistically significantly larger than that of group B, and conversely if the t stat is (-), that the mean of group A is statistically significantly smaller than that of group B?

I thought this was the case, but am now second guessing myself...

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If you want to test whether the mean is greater in one particular direction, you should be using a one-sided test. So, if you think group A will have a larger mean than B (and it's is A minus B), then you reject if $t > t^{\star}$ where $t^{\star}$ is the 95th percentile of the proper $t$-distribution (assuming $\alpha=.05$).

This is in contrast, to the two-sided test, where you reject if $|t| > t^{\star}$, where $t^{\star}$ is the upper 97.5th percentile.

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