I doing some stats exercises, in which i have to calculate the Type II error for an normal distribution.
I understand the concept of it, that you accept $H_0$ even though it's not the true means, but i am having a hard time understanding what part of the normal distribution i have to calculate.
I think i need some visual representation to understand why $$\beta = P(z < \frac{\bar{x_l} -µ_{true}}{\frac{\sigma}{\sqrt{n}}}) - (1-P(z < \frac{\bar{x_h} -µ_{true}}{\frac{\sigma}{\sqrt{n}}})) $$
is given by this formula.
Lets do a simple Z-test of a single mean:
Picture a bell curve with the usual 1.96 rejection region around the null hypothesis (usually $\mu=0$). Now, if the null hypothesis is incorrect, then the true distribution is assumed to be a normal distribution with a different value of $\mu'$.
So, lets say that $\mu'=1$: what is the probability that this distribution will "fool" your test? Its exactly the probability that this shifted normal variable will fall in your "do not reject" region.
Visually: A type I error looks at the probability contained outside the "do not reject" region, as calculated with the null distribution.
The Type II error is exactly the opposite: you are calculating the probability of falling in the "do not reject" region but with a $non-null$ distribution.