A Brief Introduction to Power

1. A Brief Introduction to Power Tyrone Li ‘12 and Ariana White ‘12 Buckingham Browne & Nichols School Cambridge, MA

2. 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.

3. Why is Power Important? • For Example: • 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.

4. 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

5. 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.

6. 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: POWER False: Beta Error Ha Center

7. A Normal Curve of the Null Hypothesis: (The distribution of the old mixture) Accept HO Reject HO You reject the Null Hypothesis if the Sample Statistic falls in this Area. So you accept 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 If the null hypothesis is true: Then your solution is false… alpha error (Type 1) Alpha Level determines where to Accept and Reject the null hypothesis

8. (The distribution of the new mixture) Accept HO Reject HO So you accept the Null Hypothesis if the Sample Statistic falls in this Area. You reject the Null Hypothesis if the Sample Statistic falls in this Area. Sampling Distribution of the proportion of the new mixture converted to energy Sample Center If the null hypothesis is false: Then your solution is false… beta error (Type 2) Alpha Level determines where to Accept and Reject the null hypothesis A Normal Curve of the Alternate Hypothesis:

9. ERRORS? Type 2 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? Type I Error: When the null hypothesis is rejected when it is actually true -also known as alpha-error 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!

10. 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 efficiency between the old and the new mixtures. Conclude that there is a difference in the efficiency between the old and the new mixtures. Old Mixture Distribution: Significance level: α = .05 HO Center = .2 New Mixture Distribution: Ha Center = .5

11. 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

12. What do these calculator commands mean?? InvNorm For Example: INVN 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). Therefore we know that the probability on this side (Accepting Ho) is .95 This probability is .05 which we know because the alpha level is .05 .95 .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*

13. 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*

14. 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.

15. So… .95 .05 HO Center = .2 POWER! Ha Center = .5

16. 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.

17. Let’s Review: Decision Based on the Sample _____?____ Accept HO _____?_____ Reject HO ______ HO when HO is ______: Accept ______ HO when HO is ______: Reject True True _____-Error. Alpha ______ HO when HO is ______: Accept ______ HO when HO is ______: Reject False False _____-Error. Beta POWER!!!

18. How can I remember that? Type I Error POWER Alpha error Type II Error Correct Beta error

19. Some Things to Remember • Power is the probability that a statistical test with a fixed alpha level will reject the null hypothesis when analternate parameter is true. • Calculator commands to remember: InvNorm( and Normalcdf(