Directional Conclusion from Two-Tailed Tests
It is valid to infer which population mean is larger when the
null hypothesis µ
1= µ
2 is rejected
regardless of whether
one-tailed or
two-tailed tests are used. For a one-tailed test, the null hypothesis
µ
1 ≤ µ
2 is rejected allowing
the conclusion that µ
1 > µ
2. 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.