Using a sample mean to make decisions

We assume initially that a population is normally distributed with known standard deviation, σ, and that we want a test for the hypotheses:

H0 :   μ  =  μ0
HA :   μ  >  μ0

Large values of throw doubt on H0, so our decision should be of the form:

Data Decision
< k    accept H0
is k or higher    reject H0   

The probabilities of Type I and Type II errors are shown in the red cells of the table below:

Decision
  accept H0     reject H0  
Truth H0 is true     
HA (H0 is false)      

Example: Test for the hypotheses:

H0 :   μ = 10
HA :   μ > 10

If it is known that σ = 4, then the mean of a random sample of n = 16 values is approximately normal with mean µ and standard deviation 1. If the decision rule rejects H0 when the sample mean is less than k, the diagram below illustrates the probabilities of Type I and Type II errors.

Increasing k reduces P(Type I error) but increases P(Type II error). The choice of k for the decision rule is a trade-off between the acceptable sizes of the two types of error.