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.

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These notes have been prepared by Amanda J. Shaker. The copyright for the material in these notes resides with the author named above, with the Department of Mathematics and Statistics and with La Trobe University. Copyright in this work is vested in La Trobe University including all La Trobe University branding and naming. Unless otherwise stated, material within this work is licensed under a Creative Commons Attribution-Non Commercial-Non Derivatives License CC BY-NC-ND