Tests. With the advent of high-speed computing, new procedures have been developed that permit statistical inferences to be performed under less restrictive conditions than those of classical procedures. Permutation tests constitute one such collection of new procedures. To perform a permutation test to compare two population means using paired samples, we proceed as follows.

1. For each pair, switch or don’t switch the two observations with probability 0.5. This procedure yields a new paired sample.

2. Compute the mean of the new paired differences.

3. Repeat steps 1 and 2 a large number (hundreds or thousands) of times.

4. The distribution of the resulting paired-difference means provides an estimate of the sampling distribution of the sample mean of paired differences when the null hypothesis of equal population means is true. This estimate is called a permutation distribution.

5. The (estimated) P-value of the hypothesis test equals the proportion of values of the permutation distribution that are as extreme as or more extreme than the observed mean of the paired differences. Refer to Example 10.11. Use the technology of your choice to conduct a permutation test and compare your results with those found by using the paired t-test. Discuss any discrepancy that you encounter.

Example 10.11

We now return to the hypothesis test posed in Example10.10. A random sample of 10 married couples gave the data on ages, in years, shown in the second and third columns of Table 10.10 on. At the 5% significance level, do the data provide sufficient evidence to conclude that the mean age of married men differs from the mean age of married women?