## Sample size required to test means independent samples

The $ES$ is defined as
```
$$
ES=\frac{|\mu_1-\mu_2|}{\sigma}
$$
```

where

- $\mu_1$ is the mean of the first group,
- $\mu_2$ is the mean of the second group,
- $\sigma$ is the standard deviation.

The formula for determining the sample size required in each group to ensure that the test has a specified power is

```
$$
n =2\bigg(\frac{Z_{1-\alpha/2}+Z_{1-\beta}}{ES}\bigg)^2
$$
```

where

- $\alpha$ is the selected level of significance,
- $1-\beta$ is the selected power and
- $ES$ is the effect size.