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 |
---|---|
![]() |
accept H0 |
![]() |
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.