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:
For a one-tailed test:
||(zα + zβ)2 σd2
- 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.