Power of a test

1
Power of a test
2
The power of a test (against a specific
alternative value)

• Is the probability that the test will reject the
  null hypothesis when the alternative is true
• In practice, we carry out the test in hope of
  showing that the null hypothesis is false, so
  high power is important

3
Suppose H0 is false what if we decide to reject
it?
Suppose H0 is false what if we decide to fail
to reject it?
We correctly reject a false H0!
H0 True H0 False
Reject
Fail to reject
Suppose H0 is true what if we decide to fail to
reject it?
Type I
Correct
a
Power
Suppose H0 is true what if we decide to reject
it?
Correct
Type II
b
4
A researcher selects a random sample of size 49
from a population with standard deviation s 35
in order to test at the 1 significance level the
hypothesis H0 m 680 Ha m gt 680 What is the
probability of committing a Type I error?
a .01
5
H0 m 680 Ha m gt 680 For what values of the
sample mean would you reject the null hypothesis?
Invnorm(.99,680,5) 691.63
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H0 m 680 Ha m gt 680 If H0 is rejected,
suppose that ma is 700. What is the probability
of committing a Type II error? What is the power
of the test?
Normalcdf(-1099,691.63,700,5) .0471
Power 1 - .0471 .9529
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H0 m 680 Ha m gt 680 If H0 is rejected,
suppose that ma is 695. What is the probability
of committing a Type II error? What is the power
of the test?
Normalcdf(-1099,691.63,695,5) .2502
Power 1 - .2502 .7498
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Reject H0
Fail to Reject H0
a
m0
Power 1 - b
b
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What happens to a, b, power when the sample
size is increased?
Reject H0
Fail to Reject H0
a
m0
b
ma
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Facts

• The researcher is free to determine the value of
  a.
• The experimenter cannot control b, since it is
  dependent on the alternate value.
• The ideal situation is to have a as small as
  possible and power close to 1. (Power gt .8)
• As a increases, power increases. (But also the
  chance of a type I error has increased!)
• Best way to increase power, without increasing a,
  is to increase the sample size

11
Bottles of a popular cola are suppose to contain
300 ml of cola. A consumer group believes the
company is under-filling the bottles. (Assume s
50 with n 30) Find the power of this test
against the alternative m 296 ml.