# How do you calculate Type 1 error and Type 2 error probabilities?

Dec 9, 2017

Type $1$ = $P$( Rejecting ${H}_{0}$ | ${H}_{0}$ True)
Type $2$ = $P$( Accept ${H}_{0}$ | ${H}_{0}$ False )

#### Explanation:

Null Hypothesis: ${H}_{0} : \mu = {\mu}_{0}$
Alternative Hypothesis: ${H}_{1} : \mu < , > , \ne {\mu}_{0}$

Type 1 errors in hypothesis testing is when you reject the null hypothesis ${H}_{0}$ but in reality it is true

Type 2 errors in hypothesis testing is when you Accept the null hypothesis ${H}_{0}$ but in reality it is false

We can use the idea of:

Probability of event $\alpha$ happening, given that $\beta$ has occured:

$P \left(\alpha | \beta\right) = \frac{P \left(\alpha \cap \beta\right)}{P \left(\beta\right)}$

So applying this idea to the Type 1 and Type 2 errors of hypothesis testing:

Type $1$ = $P$( Rejecting ${H}_{0}$ | ${H}_{0}$ True)
Type $2$ = $P$( Accept ${H}_{0}$ | ${H}_{0}$ False )