Why is $\beta=$Probability that we accept the null hypothesis, $H_0$ where $\beta$ is the probability of a type II error in a statistical test?

Question

Is it because it's the likelihood that we get a test statistic under the conditions we set of $H_0$ and $H_1$, the alternative hypothesis that causes us to accept the null hypothesis when it's false?

What is the reasoning behind this?

Thank You

Answer 1

I don't really understand what you are asking..there are two types of errors when testing, they are

1. rejecting $H_0$ given that $H_0$ is correct (type 1 error)
2. accepting $H_0$ given that $H_1$ is correct (type 2 error)

The corresponding probabilities are

\begin{align*} P(\text{reject } H_0| H_0 \text{ correct}) &= \alpha = \text{ level of test (significance)}\\ P(\text{accept } H_0| H_1 \text{ correct}) &= 1-\beta = 1-\text{ power of test} \end{align*}

So we have $$ \beta = 1- P(\text{accept } H_0| H_1 \text{ correct}) = P(\text{reject } H_0| H_1 \text{ correct}) $$

When testing, we wish to minimize level, and maximize power, since this would lead to a reduction in both types of errors, but this is in general not possible. So what we do instead is specify an acceptable level $\alpha$, say $\alpha=0.05$, and then search over all possible tests with fixed $\alpha$ and maximum $\beta$.