A Brief Introduction to Power

Tyrone Li ‘12 and Ariana White ‘12Buckingham 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?

• Power lets you see if conducting a test is worth the time and money required.

- Say you have mixed a gasoline that is more efficient for cars to use.

- 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.

• For Example:

• 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.

HO Center

Accept HO Reject HO

Alternate Hypothesis Normal Curve:

Null Hypothesis:

Ha Center

False: Beta Error

POWER

False: Alpha Error

True: Accept HO

A Way to Look at Things…Decision Based on The Sample

Sample Center

So you accept the Null Hypothesis if the Sample Statistic falls in this Area.

Accept HO Reject HO

You reject the Null Hypothesis if the Sample Statistic falls in this Area.

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)

(The distribution of the old mixture)

Sampling Distribution of the proportion of the old gasoline converted to

energy

A Normal Curve of the Null Hypothesis:

A Normal Curve of the Alternate Hypothesis:

Sample Center

So you accept the Null Hypothesis if the Sample Statistic falls in this Area.

Accept HO Reject HO

You reject the Null Hypothesis if the Sample Statistic falls in this Area.

Alpha Level determines where to Accept and Reject the null hypothesis

If the null hypothesis is false: Then your solution is false… beta error (Type 2)

(The distribution of the new mixture)

Sampling Distribution of the proportion of the new

mixture converted to energy

ERRORS?Type I Error: Type 2 Error:When the null hypothesis

is rejected when it is actually true

-also known as alpha-error

When the null hypothesis fails to be rejected when the alternative hypothesis is in fact true

-also known as beta-error

What is it? What is it?

The probability that you’ll find that the new mixture IS more efficient when the new mixture IS NOT more efficient!

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

Ha Center = .5

HO Center = .2

Conclude that there is a difference in the efficiency between the old and the new mixtures.

Old Mixture Distribution:

New Mixture Distribution:

Significance level: α = .05

Conclude that there is no difference in the efficiency between the old and the new mixtures.

Accept HO Reject HO

Ha Center = .5

HO Center = .2

Old Mixture Distribution:

New Mixture Distribution:

Significance level: α = .05

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??

This probability is .05 which we know because the alpha level is .05

Therefore we know that the probability on this side (Accepting Ho) is .95 .95

.05

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*

HO Center = .2

InvNormINVN means inverse normal– when you know the probability of the part of a normal curve below a value and you’re looking for that value (when computing power you use the alpha level of the null hypothesis).

For Example:

Using the InvNorm you just found you can now find POWER!Now you look at the Alternate Hypothesis:

Ha Center = .5

InvNorm has the same value here as we found on the null hypothesis POWER

To find the probability of Power, use ncdf (normal cumulative distribution function)Normalcdf is the probability of a value being in an area.

Normalcdf

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*

What do I put into my calculator?

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…

.95.05

HO Center = .2

€

InvNorm(.95,.2,(.2)(.8)

30) = .32

€

Normalcdf (.32,E99,.5,(.5)(.5)

30) = .976

Ha Center = .5

POWER!

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.

_____?____ _____?_____Accept HO Reject HO

______ HO when

HO is ______:

______ HO when HO is ______:

______ HO when HO is ______:

______ HO when HO is ______:

Decision Based on the Sample

TrueAccept

Accept

TrueReject

RejectFalse False

_____-Error.

_____-Error.

Alpha

BetaPOWER!!!

Let’s Review:

Type I ErrorPOWER

Type II ErrorCorrect

Alpha error

Beta error

How can I remember that?

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(