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t Test, Paired Samples
A paired samples t test is a hypothesis test for determining whether the population means of two dependent groups are the same. The researcher begins by selecting a sample of paired observations from the two groups. Thus, each observation in each group is paired (matched) with another observation from the other group. The researcher then calculates the difference between each of these paired observations and conducts a one-sample t test on these difference scores via the formula
where
For example, assume a researcher is interested in the effect of stress on students’ performance. She has each of her participants take part in both a “stress” and a “no stress” condition and compares how each participant performs in the two conditions. Because the researcher is not interested in the sample comparisons per se but in all students, she needs to determine whether any difference between the two conditions she finds in her sample is real or caused by chance. The paired samples t test will assist her in making this determination.
Note that the formula given above tests whether the population means are different (i.e., whether the population mean difference score equals zero). It is also possible to use the paired samples t test to examine whether the difference between the population means is greater than a constant c (rather than 0). In this situation, the numerator of the formula for t changes to (
Uses
The paired samples t test is appropriate when the researcher wants to know whether the population means of two dependent groups are the same or different. This test is similar to the independent samples t test, which also tests whether the population means of two groups are the same or different. The difference between the two tests involves whether the observations in the two groups are paired with each other.
There are multiple reasons why the observations in the two groups might be dependent (paired). The most straightforward situation is when participants take part in both conditions, which is sometimes referred to as a within-subjects design. The stress scenario discussed previously was an example of such a situation; each participant took part in both the stress and no-stress conditions. However, when the observations are paired in any manner, a paired t test is appropriate. For example, assume a political strategist is testing the effectiveness of two potential television advertisements. He takes pairs of states with similar voting records and runs one of the two advertisements in one state and the other advertisement in the other state, examining voter preference after seeing the advertisement. For example, New Jersey and Delaware might be considered one pair, Mississippi and Alabama a second pair, and so on.
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