2.4 Link between paired and one-sample \(t\)-tests

You may be surprised to know that the paired and one-sample \(t\)-tests are actually equivalent! Recall the hypotheses of the one-sample \(t\)-test:

\[H_0:\mu = \mu_0\;\;\text{versus}\;\;H_1:\mu \neq \mu_0,\]

where:

\(\mu_0\) denotes the mean under the null hypothesis.

Compare this to the paired \(t\)-test hypotheses:

\[H_0:\mu_D = 0\;\;\text{versus}\;\;H_1:\mu_D \neq 0,\]

and you may be able to see the similarity. Here, our \(\mu_0\) value is 0, and our variable of interest is the paired differences. What the paired \(t\)-test really does, is take each individual difference, and then test to see whether the average of these differences is different from zero.

Below are the results of a one-sample \(t\)-test for the anorexia example, testing whether the paired differences are different from 0:


    One Sample t-test

data:  anorexia$paired.differences
t = -2.9376, df = 71, p-value = 0.004458
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -4.6399424 -0.8878354
sample estimates:
mean of x 
-2.763889

As we can see, results of this one-sample \(t\)-test are identical to those we saw in the previous section for the paired \(t\)-test. In the computer lab, you will have a chance to try this for yourself.