A Brief Introduction to Power
Tyrone Li ‘12 and Ariana White ‘12
Buckingham Browne & Nichols School
Cambridge, MA
What is Power?
Power is the probability of finding that a
sample is significant when it really is
significant.
Formally Put: Power is the probability of a
test of a sample showing that the
alternate situation is true when it is in fact
true.
Why is Power Important?
• For Example:
-• Power
Say you
have
gasoline that
is is
lets
youmixed
see if aconducting
a test
more
cars
to use.
worth efficient
the timefor
and
money
required.
- 30 trials on your own car have shown that
.5 of the gasoline is transferred into energy
while the generic only transfers .2 into
energy.
• However, you need to know whether it will be
worth it to spend millions of dollars and time
to
– Produce the mixture
– Find a sample
– And conduct a test proving that it is significantly
better than the generic gasoline
• Power is the probability that your test
would show that the new mixture is more
efficient than the old when it in fact is more
efficient.
- If power is high (closer to 1, ex: .85), then
there is a higher probability that your test will
conclude that the statistic is significant.
-If power is low (closer to 0, ex: .13),
then there is a low probability that
your test will conclude that the
statistic is significant.
A Way to Look at Things…
Decision Based on The Sample
Accept HO
Reject HO
Null Hypothesis:
True:
Accept HO
False:
Alpha Error
HO
Center
Alternate Hypothesis Normal Curve:
False:
Beta
Error
POWER
Ha
Center
A Normal Curve of the Null Hypothesis:
(The distribution of the old mixture)
Accept HO
So you accept the Null
Hypothesis if the Sample
Statistic falls in this Area.
Reject HO
You reject the Null
Hypothesis if the Sample
Statistic falls in this Area.
Sampling Distribution of
the proportion of the old
gasoline converted to
energy
Sample
Center
Alpha Level determines
where to Accept and Reject
the null hypothesis
If the null hypothesis is true:
Then your solution is false…
alpha error (Type 1)
A Normal Curve of the Alternate Hypothesis:
(The distribution of the new mixture)
Accept HO
Reject HO
So you accept the Null
Hypothesis if the Sample
Statistic falls in this Area.
Sampling Distribution of
the proportion of the new
mixture converted to
energy
If the null hypothesis is false:
Then your solution is false…
beta error (Type 2)
Sample
Center
Alpha Level determines
where to Accept and Reject
the null hypothesis
You reject the Null
Hypothesis if the Sample
Statistic falls in this Area.
ERRORS?
Type I Error: When
What the
is it?null hypothesis
is rejected when it is
actually true
Type 2 Error: When
What isthe
it?null hypothesis
fails to be rejected when
the alternative hypothesis
is in fact true
-also known as alpha-error
The probability that you’ll find
that the new mixture IS more
efficient when the new mixture
IS NOT more efficient!
-also known as beta-error
The probability you’ll find that
the new mixture IS NOT more
efficient when the new mixture
IS more efficient!
What’s the Power?
• First, we should create a diagram in the context of our
particular problem.
Accept HO
Reject HO
Conclude that there is no difference in the
Conclude that there is a difference in the
efficiency between the old and the new mixtures. efficiency between the old and the new mixtures.
Old Mixture Distribution:
Significance level: α = .05
HO Center = .2
New Mixture Distribution:
Ha Center = .5
Accept HO
Reject HO
Significance level: α = .05
Old Mixture Distribution:
HO Center = .2
New Mixture Distribution:
Ha Center = .5
To find the value of the dividing
point between accepting and
rejecting the null hypothesis,
use Inverse Norm
Once you have found INVN, you
can find the percentage of
POWER (this area) using the
ncdf command on your calculator
What do these calculator commands mean??
InvNorm
For Example:
INVN means inverse normal– when you know the probability of the part of
a normal
below
Thereforecurve
we know
that a value and you’re looking for that value (when
the probability
on this
computing
power
you use the alpha level of the
hypothesis).
Thisnull
probability
is .05 which
side (Accepting Ho) is .95
.95
we know because the alpha
level is .05
.05
HO Center = .2
What do I put into my calculator?
2nd: distribution: InvNorm(
Then plug in the (% of area below center, center, standard deviation)
*don’t forget parentheses and commas*
Using the InvNorm you just found you can now find POWER!
Now you look at the
Alternate Hypothesis:
Normalcdf
InvNorm has the same
value here as we found
on the null hypothesis
POWER
Ha Center = .5
To find the probability of Power, use ncdf (normal cumulative distribution function)
Normalcdf is the probability of a value being in an area.
What do I put into my calculator?
2nd: distribution: Normalcdf(
Then plug in the (lower bound (in this case InvNorm), upper bound (as far as
possible E99, center, standard deviation) of the section you’re solving for
*don’t forget parentheses and commas*
Key Commands
• Normalcdf: To find the probability that a
particular variable will fall in an interval you
supply.
• InvNorm: To find the Z-score of a
probability you supply.
So…
InvNorm (. 95 ,.2,
(. 2)(. 8)
30
)  .32
.95

.05
HO Center = .2
POWER!
Ha Center = .5
Normalcdf

(. 32 , E 99 ,.5,
(. 5)(. 5)
30
)  .976
What Does this Mean?
• The probability of a sample test showing
that the new mixture of gasoline is better
than the original, old gasoline (when the
new mixture is in fact better) is about .976
• It is worth it to conduct the test because
you will probably conclude that your data
is significant.
Let’s Review:
Decision Based on the Sample
_____?____
Accept HO
Accept HO when
______
True
HO is ______:
_____?_____
Reject HO
Reject HO when
______
True
HO is ______:
_____-Error.
Alpha
Accept HO when
______
False
HO is ______:
_____-Error.
Beta
Reject HO when
______
False
HO is ______:
POWER!!!
How can I remember that?
Type I Error
Alpha error
Correct
POWER
Type II Error
Beta error
Some Things to Remember
• Power is the probability that a statistical
test with a fixed alpha level will reject the
null hypothesis when an alternate
parameter is true.
• Calculator commands to remember:
InvNorm( and Normalcdf(