- Nondirectional Hypotheses
- Making a Directional Hypothesis
- Increasing the Size of the Samples
- The Dependent Groups t-Test
- A Recap
Making a Directional Hypothesis
You can increase the power by making a directional hypothesis instead of a nondirectional hypothesis. See Figure 2.
Figure 2 The alpha level is no longer split, but occupies solely the wedge in the right tail of the left hand curve.
Figure 1 assumes a nondirectional alternative hypothesis: that the treatment group mean is different from the control group mean. Therefore, we must allow for two possibilities: that the treatment mean is larger, and that it is smaller than the control group mean. In that case, some of the alpha rate must be in each tail of the distribution that represents the null hypothesis.
But if we exclude the possibility that the treatment mean could be smaller than the control mean, we can put the entire alpha into the right tail of the left curve. That is what is shown in Figure 2. Notice that alpha is no longer labeled as "Alpha / 2" but simply as "Alpha." The entire 5% of the distribution has been placed in the right tail of the distribution.
The effect of doing that is to lower the critical value. Notice that alpha is cut off from the rest of the left curve at 14 on the horizontal axis. Compare that to Figure 1, where the critical value is almost 17.
That means with a directional hypothesis, you don’t have to get a mean difference as large as you do with a nondirectional hypothesis in order to reject the null hypothesis. That's another way of saying that the power of the t-test is greater when you use a directional hypothesis.
In this case, the T.DIST.RT() function returns 0.34 or 34%: more than 10% greater than with a nondirectional hypothesis (23%). The reason for this increase in power is the shift of the critical value down the horizontal axis, increasing the area under the right hand curve that lies to the right of the critical value.
That's a useful increase, but 34% power still isn't very good. Another method of increasing power is to increase the sample size, discussed next.