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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. Except where otherwise specified, all text and images on this page are copyright InfluentialPoints, all rights reserved. Images not copyright InfluentialPoints credit their source on web-pages attached via hypertext links from those images.