Increasing the Size of the Samples
In Figure 3, I have doubled the size of each sample, from 20 to 40.
Figure 3 The degrees of freedom for the t-test has increased from 38 to 78.
The situation in Figure 3 results in power for the t-test that's greater than 50%. Notice that the standard error of the difference between means, in cell I25, is 5.82. In Figures 1 and 2, the standard error is 8.34.
The critical value is found by multiplying the t value for a particular probability (here, 1.64) by the standard error of the difference between means. Because the standard error has been reduced (from 8.34 to 5.82) by doubling the sample sizes, the critical value is lowered. Just as in Figure 2, lowering the critical value increases the power of the t-test.
The power of the test shown in Figure 3 is actually 55.8%. You can see this in the chart. There, the section of the right hand curve that represents power occupies the entire right half of the curve plus a bit of its left half. Notice that the power section extends to the left of the mean of the right hand curve.
Statistical power of 56% is a major increase over the 23% that we started with, but by using a dependent group’s t-test it's possible to do better yet.