One- and two-tailed tests
Sometimes we want to test whether two group means are the same or different — a two-tailed test. In other situations, the alternative hypothesis is that one specific group mean is larger than the other; this is a one-tailed test.
Test statistic and p-value
The p-value for a one-tailed test about two means is found in a similar way to that for a two-tailed test. It is based on the same test statistic with the same t-distribution when the null hypothesis holds. The only difference is that the p-value is the tail area on only one side of the distribution.
Examples
We now show a few examples of hypothesis tests and the conclusions that can be drawn from them.
Properties of p-values
We again stress that a statistical hypothesis test cannot provide a definitive answer. The randomness of sample data means that:
A simulation demonstrates that when the underlying population means are the same in two groups, a one-tailed test results in p-values that are equally likely to be anywhere between 0 and 1.
There is a (small) probability of getting random data that misleadingly suggest that the second group mean is higher.