# COMPARING THE MEANS OF TWO SETS OF MEASUREMENTS

The most common test for comparing the means of two populations is based upon Student’s t. For Student’s t-test to provide significance levels that are exact rather than approximate, all the observations must be independent and, under the null hypothesis, all the observations must come from identical normal distributions.

Even if the distribution is not normal, the significance level of the t-test is almost exact for sample sizes greater than 12; for most of the distributions one encounters in practice,^{5} the significance level of the t-test is usually within a percent or so of the correct value for sample sizes between 6 and 12.

For testing against nonnormal alternatives, more powerful tests than the t-test exist. For example, a permutation test replacing the original observations with their normal scores is more powerful than the t-test [Lehmann, 1986, p. 321].

Permutation tests are derived by looking at the distribution of values the test statistic would take for each of the possible assignments of treatments to subjects. For example, if in an experiment two treatments were assigned at random to six subjects so that three subjects got one treatment and three the other, there would have been a total of 20 possible assignments of treatments to subjects.^{6} To determine a p-value, we compute for the data in hand each of the 20 possible values the test statistic might have taken. We then compare the actual value of the test statistic ...

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