Sample size required to test means dependent samples
The $ES$ is defined as
$$ ES=\frac{|\mu_d|}{\sigma_d} $$
where
- $\mu_d$ is the mean of the difference,
- $\sigma_d$ is the standard deviation of the difference.
The formula for determining the sample size required in each group to ensure that the test has a specified power is
$$ n =\bigg(\frac{Z_{1-\alpha/2}+Z_{1-\beta}}{ES}\bigg)^2 $$
where
- $\alpha$ is the selected level of significance,
- $1-\beta$ is the selected power and
- $ES$ is the effect size.
