The paired sample t-test, sometimes called the dependent sample t-test, is a statistical procedure used to determine whether the mean difference between two sets of observations is zero. In a paired sample t-test, each subject or entity is measured twice, resulting in pairs of observations. Common applications of the paired sample t-test include case-control studies or repeated-measures designs. Suppose you are interested in evaluating the effectiveness of a company training program. One approach you might consider would be to measure the performance of a sample of employees before and after completing the program, and analyze the differences using a paired sample t-test.
Like many statistical procedures, the paired sample t-test has two competing hypotheses, the null hypothesis and the alternative hypothesis. The null hypothesis assumes that the true mean difference between the paired samples is zero. Under this model, all observable differences are explained by random variation. Conversely, the alternative hypothesis assumes that the true mean difference between the paired samples is not equal to zero. The alternative hypothesis can take one of several forms depending on the expected outcome. If the direction of the difference does not matter, a two-tailed hypothesis is used. Otherwise, an upper-tailed or lower-tailed hypothesis can be used to increase the power of the test. The null hypothesis remains the same for each type of alternative hypothesis. The paired sample t-test hypotheses are formally defined below:
• The null hypothesis (\(H_0\)) assumes that the true mean difference (\(\mu_d\)) is equal to zero.
• The two-tailed alternative hypothesis (\(H_1\)) assumes that \(\mu_d\) is not equal to zero.
• The upper-tailed alternative hypothesis (\(H_1\)) assumes that \(\mu_d\) is greater than zero.
• The lower-tailed alternative hypothesis (\(H_1\)) assumes that \(\mu_d\) is less than zero.
The mathematical representations of the null and alternative hypotheses are defined below:\(H_0:\ \mu_d\ =\ 0\)
Note. It is important to remember that hypotheses are never about data, they are about the processes which produce the data. In the formulas above, the value of \(\mu_d\) is unknown. The goal of hypothesis testing is to determine the hypothesis (null or alternative) with which the data are more consistent.
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