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

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Answer 1 · 2017-12-09T10:36:44

Type $1$ $1$ = $P$ $P$ ( Rejecting $H_{0}$ $H_0$ | $H_{0}$ $H_0$ True)
Type $2$ $2$ = $P$ $P$ ( Accept $H_{0}$ $H_0$ | $H_{0}$ $H_0$ False )

Null Hypothesis: $H_{0} : μ = μ_{0}$ $H_0 : mu = mu_0$
Alternative Hypothesis: $H_{1} : μ <, >, \neq μ_{0}$ $H_1: mu<,>, != mu_0$

Type 1 errors in hypothesis testing is when you reject the null hypothesis $H_{0}$ $H_0$ but in reality it is true

Type 2 errors in hypothesis testing is when you Accept the null hypothesis $H_{0}$ $H_0$ but in reality it is false

We can use the idea of:

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

$P (α ∣ β) = \frac{P (α \cap β)}{P (β)}$ $P(alpha|beta) =( P(alphannbeta))/(P(beta))$

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

Type $1$ $1$ = $P$ $P$ ( Rejecting $H_{0}$ $H_0$ | $H_{0}$ $H_0$ True)
Type $2$ $2$ = $P$ $P$ ( Accept $H_{0}$ $H_0$ | $H_{0}$ $H_0$ False )