Rejecting a true hypothesis,
Not rejecting a false hypothesis, Pr[Type II error] = Pr[T∈A|H
Now, let's consider the cases in which we make the correct decision:
Not rejecting a true hypothesis, Pr[Not(Type I error)] = Pr[T∈A|H
Rejecting a false hypothesis,
The complement of β is called the power of the test of the null hypothesis H
vs. the alternative H
. The power of a test is used, for example, to determine
a minimum sample size to restrict errors.
Selecting values of α and β
A typical value of the level of significance (or probability of Type I error) is α
= 0.05, (i.e., incorrect rejection once in 20 times on the average). If the
consequences of a Type I error are more serious, choose smaller values of α,
say 0.01 or even 0.001.
The value of β, i.e., the probability of making an error of Type II, depends on
α, the sample size n, and on the true value of the parameter tested. Thus, the
value of β is determined after the hypothesis testing is performed.
customary to draw graphs showing β, or the power of the test (1- β), as a
function of the true value of the parameter tested. These graphs are called
operating characteristic curves or power function curves, respectively.
Inferences concerning one mean
The problem consists in testing the null hypothesis H
alternative hypothesis, H
significance level α, using a sample of size n with a mean x and a standard
deviation s. This test is referred to as a two-sided or two-tailed test.
procedure for the test is as follows:
Pr[Type I error] = Pr[T∈R|H
Pr[Not(Type II error)] = Pr [T∈R|H
: µ≠ µ
at a level of confidence (1-α)100%, or
] = α
] = β
] = 1 - α
] = 1 - β
: µ = µ
, against the