One of the most common tests of significance is the test for paired data. This test can be used to analyze data collected on individuals before and after a treatment, or data from experiments where the subjects are paired. For example, Mitchell, Burroughs, and Beadles (1936) collected data on ten pairs of rats, with one rat from each pair assigned to a raw peanut diet and the other member of the pair to a roasted peanut diet. Usually, in the analysis of paired data, the experimenter will compute the differences between the pairs and the analysis will be based solely on these differences. These differences are assumed to have a symmetric distribution. Under the null hypothesis the distribution will be symmetric about zero.

To demonstrate that the distribution of differences will be symmetric about zero, suppose that we are trying to evaluate the effectiveness of a medication to lower diastolic blood pressure and we observe a blood pressure of 85 (mm Hg) before the drug is taken and 80 after the drug is taken. Under the null hypothesis, we are just as likely to obtain 80 as the first reading and 85 as the second reading. If we define the difference as the posttreatment blood pressure minus the pretreatment blood pressure we are just as likely to record a +5 as a -5. Because the same argument can be made for all measurements made under the null hypothesis, it can be seen that the distribution of the differences is symmetric ...

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