Download 9.1d Power

Chapter 9: Power
Target Goal:
I can make really good decisions .
9.1d
h.w: pg 548: 23, 25
Hand draw Type I/II Curves
• The probability of a Type II Error tells us
the probability of accepting the null
hypothesis when it is actually false.
• The complement of this would be the
probability of not accepting (in other words
rejecting) the null hypothesis when it is
actually false. (good decision!)
Power
• To calculate the probability of rejecting the
null hypothesis when it is actually false,
compute
Power = 1 – P(Type II Error),
Or, (1 – b).
This is called the power of a significance
test.
• A P-value describes what would happen
supposing that the null hypothesis is true.
• Power describes what would happen
supposing that a particular alternative
hypothesis is true.
Ex: Exercise is good
Can a six month exercise program
increase the total body mineral
content (TBBMC) of young women?
A team of researchers is planning a
study to examine this question.
Type II error  calculated to be = .20.
Power = 1 – β = 0.80
Interpret results:
This significance test correctly rejects the null
hypothesis that exercise has no effect on
TBBMC 80% of the time if the true effect
of exercise is a 1% increase in TBBMC.
Power: the probability of rejecting the
null hypothesis when it is actually false.
We must use the shifted curve!
How Does Power Change?
Suppose
Suppose that s = $85 and n = 100.
H0: µ = 500
We would reject H0 for x > 513.98.
Ha: μ > 500
Power is the probability of correctly rejecting H0.
What is the power of the test if µ = 520?
Power = 1 - .239 = .761
 = .239
Rejection Region
Notice that power is in the SAME curve as 
Power = 1 – 
H0: μ = 500
Suppose that s = $85 and n = 100.
Ha: µ > 500
We would reject H0 for x > 513.98.
Notice that, as the distance between the null hypothesized value for m
and our alternative value for m increases,  decreases AND power
Find b and power.
increases.
If we reject H0, then m > 500.
What if m = 530?
b = .03
Power = .97 vs,
Rejection Region
530
520
.
H0: m = 500
Suppose that s = $85 and n = 100.
Ha: m > 500
We would reject H0 for x > 513.98.
Find  and power.
 = .03
 = .68
If the null hypothesis is false, then m > 500.
that, as the
distance
What
if m = 510? between
Notice
the null hypothesized value for μ and
our alternative value for μ decreases, β
power = .97 vs.
increases AND power decreases.
power = .32
Rejection Region
Suppose that s = $85 and n = 100.
H0: m = 500
Ha: m > 500
What happens if we use
= .01? and power will
willaincrease
decrease.
Rejection Region

Rejection Region
Power

Power
What happens to a, , & power when
standard deviation
the sample size isTheincreased?
Thedecrease
significance
level (a)
will
making
remains
thetaller
same –and
so theand
the
curve
β
decreases
Reject
H0where the rejection
value
skinnier.
power increases!
Fail to Reject H0
region begins must move.
a
m0

Power
ma
If power is too small, increase the
Power
1. Increase α.
2. Consider an
alternative that is
further away from
μo.
3. Increase the sample
size.
4. Decrease σ.
What does this look like?
Increase α.
Slides reference point to the left.
Consider an alternative that
is further away from
μo.
Shift “new” curve to the right.
Increase the sample size.
Less spread and narrower curve.
Decrease σ.
Less spread and narrower curve.
Read 542 - 544
• Explore on your own applet on page 543.