Essential Statistics

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Chapter 14: Hypothesis Tests – Small Samples

14-4 Small Sample Hypothesis Tests for the DifferenceBetween

Two PopulationMeansUsing Dependent Samples

In this section we will use the

test when the samples are

dependent

Samples are considered to be dependent when they are paired or matched in

some way. For example, an instructor may give a test at the beginning of the

semester to determine the basic math skill level of the students in a course.

At the end of the semester, the instructor will give the same test to determine

the basic math skill level of the students again. Although we have two

different sets of data, it was obtained from the same set of students

(assuming all students remain in the course). Thus, we say that the data are

dependent since the same experimental units (students in this case) were

used. Another example in which we may have dependent samples is when

patients are matched or paired according to some variable of interest.

Patients may then be assigned to two different groups. For instance, patients

may be paired according to their age, blood pressure, etc. That is, two

patients with the same age will be paired and then one will be assigned to

one sample group and the other to another sample group. Caution should be

taken when matching experimental units. In this example we matched by

age, but this does not eliminate the influence of other variables.

In performing hypothesis tests for dependent data, we use the difference of

the values of the before and after or the difference of the values of the

matched pairs to account for the dependency. By doing this, we will have a

single sample of differences, denoted by

, with which to perform the test.

Thus, the hypothesis test for the difference between two means for two

dependent samples will be reduced to a simple one-sample

test.

If we let

be the mean for the sample of differences,

be the standard

deviation for the sample of differences,

be the mean for the potential

population of differences, and

be the sample size, then the statistic