Transcript Power of a test - Academic Magnet High School

Power of a test
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
Suppose H0 is false
Suppose
H
is
false
We
correctly
reject
0
– what if we decide
– what
if we
a reject
false
Hit?
! H0
0decide
to
to fail to reject
it?
True
H0
False
Suppose H0
Reject Type I Correct
is true –
what if we
a
Power
decide
to
Fail to Correct Type II
Suppose H0 is true
fail to
b
–reject
what if we decide reject
it?
to reject it?
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 > 680
What is the probability of committing
a Type I error?
a = .01
H0: m = 680
Ha: m > 680
For what values of the sample mean
would you reject the null hypothesis?
Invnorm(.99,680,5) =691.63
H0: m = 680
Ha: m > 680
If H0 is rejected, suppose that ma is
700. What is the probability of
committing a Type II error?
Normalcdf(-10^99,691.63,700,5) =.0471
What is the power of the test?
Power = 1 - .0471 = .9529
H0: m = 680
Ha: m > 680
If H0 is rejected, suppose that ma is
695. What is the probability of
committing a Type II error?
Normalcdf(-10^99,691.63,695,5) =.2502
What is the power of the test?
Power = 1 - .2502 = .7498
Fail to Reject H0
Reject H0
a
m0
Power = 1 - b
b
ma
What happens to a, b, & power
when the sample size is increased?
Fail to Reject H0
Reject H0
a
m0
b
ma
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 > .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
A water quality control board reports
that water is unsafe for drinking if the
mean nitrate concentration exceeds 30
ppm. Water specimens are taken from a
well.
Identify the decision:
Type I
Error
a) You decide that the water is not safe
to drink when in fact it is safe.
A water quality control board reports
that water is unsafe for drinking if the
mean nitrate concentration exceeds 30
ppm. Water specimens are taken from a
well.
Identify the decision:
Correct
– Power!!
b) You decide that the water is not safe
to drink when in fact it is not safe.
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.