Planning Test Size

We would like to minimize Type 2 Error to hold to our Hypothesis. We thus would like to know how many samples we should take to get below a certain threshold.

One-Sided Tests

For One-Sided Gauß Test

$\beta (\mu )=\Phi \left(z_{1-\alpha }-\frac{|\mu -{\mu }_0|}{\sigma /\sqrt{n}}\right)$

Thus by applying the inverse we get: $n\ge n_0:=\frac{{\left(z_1-\alpha -z_{\beta }\right)}^2{\sigma }^2}{{|\mu -{\mu }_0|}^2}$

Two-Sided Test

For Two-Sided Gauß Test

$\beta (\mu )=\Phi \left(z_{1-\alpha /2}-\frac{|\mu -{\mu }_0|}{\sigma /\sqrt{n}}\right)+\Phi \left(z_{1-\alpha /2}+\frac{|\mu -{\mu }_0|}{\sigma /\sqrt{n}}\right)-1$

Here we have to solve for $n_0$ numerically. For example by plotting and reading the value.