When to Use This Version

This version should be used to test the equality of two sample means when the two samples can be regarded as dependent or paired, meaning that each individual in the first sample is matched with exactly one individual in the second sample in such a way that the two individuals can be expected to respond in similar fashion to the experimental condition.

Examples where a paired sample approach can be used:

• Before and after treatment studies on the same individual
• Same store sales in business
• Twin studies
• Any study where each individual in sample 1 is carefully matched to a single individual in sample 2.

This version of the two-sample test actually looks at the difference in the measured values between matched pairs of subjects.

In a sense, it reduces the two sample problem to a single sample problem where the measures are the differences

$$d_i\;=\;x_i-y_i$$

between matched pairs of subjects. We test the hypothesis that the population mean of the differences $\mu_d$ is zero.

Assumptions The following assumptions are made:

1) The sample is obtained using simple random sampling

2) The data are matched pairs

3) At least one of the following statements is true:

• The difference between matched subjects is normally distributed
• The sample size $n$ is $30$ or more (the central limit theorem applies)