Directional Conclusion from Two-Tailed Tests

It is valid to infer which population mean is larger when the null hypothesis µ1= µ2 is rejected regardless of whether one-tailed or two-tailed tests are used. For a one-tailed test, the null hypothesis µ1 ≤ µ2 is rejected allowing the conclusion that µ1 > µ2. A two-tailed test at the 0.05 level can be thought of as two one-tailed tests, each at the 0.025 level. If the null hypothesis is true, then the probability that one or the other of the two one-tailed tests will be significant is 0.05. This is because P(A or B) = P(A) + P(B) - P(A and B) = .025 + .025 - 0.0. The last term is zero since it is impossible for both one-tailed tests to be significant simultaneously. Therefore, even for two-tailed tests, it is valid to infer which population mean is higher when the null hypothesis is rejected. The problem can also be approached in terms of the relationship between confidence intervals and significance tests. If a test is significant, then the confidence interval contains values on only one side of zero allowing the direction of the effect to be inferred.