# Making one-sided conclusions from two-sided tests

I'm reading Montgomery's Design and Analysis of Experiments. On page 39, he rejected a two-sided $$t$$-test against the null hypothesis that modified formulation of some cement mortar doesn't change its bond strength, and then, presumably by noticing that samples from the modified formulation has lower mean, he added

One can conclude that the modified formulation reduces the bond strength (just because we conducted a two-sided test, this does not preclude drawing a one-sided conclusion when the null hypothesis is rejected).

I don't get it. How can you make a one-sided conclusion from a two-sided test?

Or more generally, can you make any conclusion beyond the alternative on rejecting the null? If so, can I conclude that the expectation of (un)modified cement mortar is exactly the corresponding sample mean?

• You have concluded there's a difference. You can see which direction it was in (which tail your statistic fell into). I am pretty sure this is already answered somewhere on site. this one hits some of the issues but that's not what I am thinking of – Glen_b Sep 24 at 5:32
• @Glen_b I can get the intuition, but let's suppose instead of two directions, we have a million possible directions. Are you still sure that the direction suggested by the sample mean is the direction where the difference truly lies in? – nalzok Sep 24 at 5:39
• You only have two directions in the conditions described in the question. If you want to discuss some other situation, ask specifically about it in the question – Glen_b Sep 24 at 5:42
• @Glen_b Will do, but the one-million-direction case just serves as an example. I mean, just like the number of random variables, univariate analysis is very different from bivariate analysis, but the latter is not so much different from other multivariate analysis? So I don't think two directions is fundamentally different from one million directions. – nalzok Sep 24 at 5:49
• I can't tell if it's different until I understand what you want to know. I imagine it's probably going to be sufficiently clear (enough to be able to generalize) from the bivariate case – Glen_b Sep 24 at 5:51