#### Power Calculation Exercises

Nationally, performance on a mathematics test for 4^{th} graders is reported to be normally distributed with a mean of 40.0 and a standard deviation of 9.0.

1. You would like to know whether the average performance of children in your school district differs from the national average. You believe that a difference as small as 3.0 points is important to detect. How many randomly selected students do you need to include in your sample to have power of 80% to detect a difference of 3.0 points using a two-tailed test with alpha = .05?

HINT: We wish to compute sample size n. We need to specify the other three components of BEAN. That is, we need to specify B, E, and A. With a two-tailed test, we have only half of alpha on each tail of the distribution.

(The answer is around 70. Compute the exact answer.)

ANSWER: The computed answer is 70.7. Round up to 71 students.

2. Your friend Bumble collected data from a sample of 25 children from a large school where the population mean is 45.0. If he uses a one-tailed test with alpha = .01, how likely is he to attain statistical significance? That is, what is his statistical power?

HINT: We wish to compute power, which is (1 – beta error). Thus, we need to specify E, A, and N: the effect size, alpha error, and sample size.

(The answer is around 65%. Compute the exact answer.)

ANSWER: Statistical power is .674 or about 67%.