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# Sample size for testing a difference between paired means

We introduced the topic of sample size in Unit 5. There we were estimating the required sample size for comparing a sample mean with a known parametric mean using the Z-test.

The formula we use for the paired t-test is identical, albeit we accept it is an approximation because the standard deviation is not known but estimated:

Algebraically speaking:

For a one-tailed test:

 n = (zα + zβ)2 σd2 δ2
where

• n is the required number of pairs,
• zα is obtained from your probability calculator or tables given that P(Z < zα) = 1 − α and α is the significance level,
• zβ is obtained from your probability calculator or tables, given that P(Z < zβ) = 1 − β and 1 − β is the power,
• δ is the difference that one wishes to be able to detect,
• σd is the known standard deviation of the difference. In practice we only have an estimate of this.

For a two-tailed test, we use zα/2 in place of zα. This is an approximation since it ignores the possibility of a type III error. However, for large treatment effects, it will not usually introduce any serious error.