Using a sample mean to make decisions
We now introduce the idea of decision rules with a test about whether a population mean is a particular value, µ0, or greater. We assume initially that the population is normally distributed and that its standard deviation, σ, is known.
H0 : μ = μ0
HA : μ > μ0
The decision about whether to accept or reject H0
should depend on the value of the sample mean, .
Large values throw doubt on H0.
Data | Decision |
---|---|
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accept H0 |
![]() |
reject H0 |
We want to choose the value k to make the probability of errors low. This is however complicated because of the two different types of error.
Decision | |||
---|---|---|---|
accept H0 | reject H0 | ||
Truth | H0 is true | ![]() |
|
HA (H0 is false) | ![]() |
Increasing the value of k to make the Type I error probability small (top right) also increases the Type II error probability (bottom left) so the choice of k for the decision rule is a trade-off between the acceptable sizes of the two types of error.
Illustration
The diagram below relates to a normal population whose standard deviation is known to be σ = 4. We will test the hypotheses
H0 : μ = 10
HA : μ > 10
The test is based on the sample mean of n = 16 values from this distribution. The sample mean has a normal distribution,
![]() |
~ normal (μ, | ![]() |
= 1) |
This normal distributions can be used to calculate the probabilities of the two types of error. The diagram below illustrates how the probabilities of the two types of error depend on the critical value for the test, k.
Drag the slider at the top of the diagram to adjust k. Observe that making k large reduces the probability of a Type I error, but makes a Type II error more likely. It is impossible to simultaneously make both probabilities small with only n = 16 observations.
Note also that there is not a single value for the probability of a Type II error — the probability depends on how far above 10 the mean µ lies. Drag the slider on the row for the alternative hypothesis to observe that:
The probability of a Type II error is always high if µ is close to 10, but is lower if µ is far above 10.
This is as should be expected — the further above 10 the population mean, the more likely we are to detect that it is higher than 10 from the sample mean.