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Section 10.4.2 Power AP Statistics March 11, 2008 CASA What is Power? Power is a test of sensitivity. Your statistical test may be able to detect differences, but how well does it detect difference of a pre-determined nature? The Power procedure allows to state the probability of our procedure to catch the differences. AP Statistics, Section 8.2.1 2 Power Procedure Begin by stating your H0 and Ha as usual. Find the z* or t* that would allow you to reject H0. Find the x-bar that matches up with the z* or t*. Assuming that you have a particular true mean, what is the probability that you would be to still reject the H0? AP Statistics, Section 8.2.1 3 Power Example: Example 10.23 Can a 6-hour study program increase your score on SAT? A team of researchers is planning as study to examine this question. Based on the result of a previous study, they are willing to assume that the change has σ=50. Research would like significance at the .05 level. AP Statistics, Section 8.2.1 4 Power Example: Example 10.23 A change of 50 points would be considered important, and the researchers would like to have a reasonable chance of detecting a change is this large or larger. Is 25 subjects a large enough sample for this project? AP Statistics, Section 8.2.1 5 Step 1: State your hypothesis H0: µ=0 Ha: µ>0 Where µ represents the change is in the SAT score. AP Statistics, Section 8.2.1 6 Step 2: Find the z* value and find the data value x We'll set α=.05, z* invNorm(.95) gives us a / n z*=1.645. What is the lowest x-bar x 0 would show significance? 1.645 Summary: If we had a 50 / 25 study with n=25 and xbar>16.45, we would have significance. 1.645 50 / 25 x 16.45 x AP Statistics, Section 8.2.1 7 Step 3: Chance at importance We stated that gains of 50 points would be considered "important". We state this as the alternative µ=50. The power against the alternative µ=50 increase is the probability that H0 is rejected when µ=50. Restated: What the area from 16.45 to ∞ under a normal curve centered at µ=50. AP Statistics, Section 8.2.1 8 Step 3 normalcdf(16.45,1E99,50,50/√(25))=.9996 Summary: because the power is so high, there is a great chance of finding a significance when the real increase is 50. AP Statistics, Section 8.2.1 9 Increase Power by… increase alpha increase sample size AP Statistics, Section 8.2.1 10 Exercises 10.71-10.77 odd, 10.79-10.89 AP Statistics, Section 8.2.1 11

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