### Chapter 8. Testing for Differences in Mean and Proportion: Paired Samples t-test

### Paired Samples t-test: Purpose, Hypotheses, and Assumptions

In this chapter, we will consider research designs in which a continuous variable is measured twice on a simple random sample of #n# subjects. These two measurements of the same variable will be denoted by #X# and #Y#. Together, these two scores form a *matched pair* for each subject in the sample.

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Paired Data Research Designs

Typical research designs that produce **paired** **data** are:

- Subjects are given a
*pre-test*whose score is #X#, then some treatment is given, then they are given a*post-test*whose score is #Y#. - Subjects are measured under two different conditions at two different times. Under such circumstances, it is important that the order of the conditions is randomized to prevent an
*order effect*from occurring. Then #X# is the measurement under one condition and #Y# is the measurement under the other. - Subjects are not individual people by
*dyads*(pairs of people), such as twins, or couples in a relationship. Then #X# is measured on one member of the couple and #Y# is measured on the other. - Measurements are taken on two different parts of the body, such as the left and right arm, or the left and right eye. Then #X# is measured on one part and #Y# on the other.

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In such cases, we are not necessarily interested in making inferences about or #X# or #Y#. Rather, we want to draw conclusions about the difference #D# between them.

Whether you define the difference as #D=X-Y# or #D=Y-X# does not matter for the outcome of the statistical test, as long as you remain consistent in your choice of how the difference is computed.

Let #\mu_D# denote the unknown *mean difference *for the matched pairs if #D# was measured on the entire population, and #\sigma_D# the unknown standard deviation. To conduct inferences about #\mu_D#, a *paired samples #t#-test *should be used.

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Paired samples t-test: Purpose and Hypotheses

The **paired samples **#\boldsymbol{t}#**-test **is used to test hypotheses about the *mean difference* #\mu_D# between two paired samples.

Specifically, the test is used to determine whether or not it is plausible that #\mu_D# differs from some value #\Delta#. In most situations #\Delta=0#, so we will only present this specific setting.

The hypotheses of a *two-tailed* paired samples #t#-test are fairly straightforward:

\[H_0: \mu_D = 0\]

\[H_a: \mu_D \neq 0\]

The hypotheses for *one-tailed* paired samples t-tests are a bit trickier to formulate, however, as they depend on the definition of the difference score #D# and the expectations of the researcher.

If you define #D# in such a way that the mean difference #\mu_D# is expected to be *positive*, a *right-tailed *test should be used:

\[H_0: \mu_D \leq 0\]

\[H_a: \mu_D \gt 0\]

If, on the other hand, you define #D# in such a way that the mean difference #\mu_D# is expected to be *negative, *a *left-tailed *test should be used:

\[H_0: \mu_D \geq 0\]

\[H_a: \mu_D \lt 0\]

Assumptions of the Paired Samples t-test

The following assumptions are required to hold in order for a *paired samples** t-test* to produce valid results:

**Random sampling**is used to draw the samples.- The two samples are
**related**. - The
*sampling**distribution of the sample mean difference*is approximately**normally distributed**. This condition of normality is met under the following circumstances:- If the sample is
*small*#(n \lt 30)#, it is required that the difference scores are normally distributed:

\[D\sim N(\mu_D, \sigma_D)\] - If the sample is sufficiently
*large*#(n \geq 30)#, the*Central Limit Theorem*can be invoked and this requirement is not needed.

- If the sample is

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