Download §9.4--Type II Error Probabilities Power

§9.4–Type II Error Probabilities
Power
Tom Lewis
Fall Term 2009
Tom Lewis ()
§9.4–Type II Error Probabilities Power
Fall Term 2009
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Fall Term 2009
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Outline
1
Calculating type II probabilities
2
Power
3
Power curves
Tom Lewis ()
§9.4–Type II Error Probabilities Power
Calculating type II probabilities
Problem
16 random samples are drawn from a normal population with σ = 5. The
hypothesis test
H0 : µ = 116
Ha : µ 6= 116
is to be conducted at the 5% significance level.
Find the rejection and
√ non-rejection regions for
z = (x − 116)/(5/ 16).
Interpret the rejection and non-rejection regions for x.
If the true mean of the population is 120, then what is the probability
of a type II error?
Repeat these parts for n = 100 random samples. What happens to
the probability of a type II error?
Tom Lewis ()
§9.4–Type II Error Probabilities Power
Fall Term 2009
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Power
The power of a test
The power of a hypothesis test is the probability of not making a
type II error.
Since a type II error occurs when a false null hypothesis is not
rejected, the power of a test is the probability of rejecting a false null
hypothesis.
We have the formula:
Power = 1 − β.
Problem
What effect does increasing the sample size have on the power of a test?
Tom Lewis ()
§9.4–Type II Error Probabilities Power
Fall Term 2009
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Power curves
Problem
Power curves For the hypothesis test on the first slide, calculate β and the
power of the test for each of the following assumed true values of the
population mean, µ:
120, 119.5, 119, 118.5, 118, . . . , 116, 115.5, 115
Make a graph of the power of the test versus the assumed value of the
mean, µ.
Tom Lewis ()
§9.4–Type II Error Probabilities Power
Fall Term 2009
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Power curves
A power curve
Here is the power curve for the hypothesis test with n = 16 samples.
0.8
0.6
0.4
0.2
114
Tom Lewis ()
116
§9.4–Type II Error Probabilities Power
118
120
Fall Term 2009
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Power curves
A power curve
Here is the power curve for the hypothesis test with n = 100 samples.
1.0
0.8
0.6
0.4
0.2
114
Tom Lewis ()
116
118
§9.4–Type II Error Probabilities Power
120
Fall Term 2009
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Power curves
A power curve
Here is are the two power curves superimposed.
1.0
0.8
0.6
0.4
0.2
114
Tom Lewis ()
116
§9.4–Type II Error Probabilities Power
118
120
Fall Term 2009
8/8