1. Power of a Hypothesis Test

2. Power (against a specific alternative value)
P(we will correctly reject a false H0)
We want to show H0 is false, so high power is important

3. We correctly reject a false H0!
H0 True
H0 False
Reject
Fail to Reject
Type I
Correct α
Power
Type II
Correct β

4. What is the probability of committing a Type I error?
A researcher selects a random sample of size 49 from a population with standard deviation σ = 35 in order to test at the 1% significance level the hypotheses:
H0: μ = 680
Ha: μ > 680
What is the probability of committing a Type I error?
α = .01

5. H0: μ = 680
Ha: μ > 680
For what values of the sample mean would you reject the null hypothesis?
x-bar must be in top 1%
invNorm(.99, 680, 5) = or higher

6. What is the power of the test?
If H0 is false, suppose that μa is really What is the probability of committing a Type II error?
What is the power of the test?
Fail to reject H0  x-bar must be below
β = normalcdf(-1E99, , 700, 5) = .0471
Power = 1 – = .9529

7. What is the power of the test?
If H0 is false, suppose that μa is really 695. What is the probability of committing a Type II error?
What is the power of the test?
β = normalcdf(-1E99, , 695, 5) = .2502
Power = 1 – = .7498

8. Fail to Reject H0
Reject H0 α μ0 ma
Power = 1 – β β

9. As n increases, what happens to α, β, and power?
Fail to Reject H0
Reject H0 a m0 b ma

10. So what do we know? We choose α
We can't control β, since it depends on the alternate mean
Ideally: have α as small as possible and power close to 1 (we like power > .8)
As α increases, power increases… but so has the chance of a type I error!
Best way to increase power, without increasing α, is to increase sample size (n)

11. Identify the decision:
A water quality control board reports that water is unsafe for drinking if the mean nitrate concentration exceeds 30 ppm. Water specimens are taken at random from a well.
Identify the decision:
a) You decide that the water is not safe to drink when it really is safe.
Type I Error

12. Identify the decision:
A water quality control board reports that water is unsafe for drinking if the mean nitrate concentration exceeds 30 ppm. Water specimens are taken at random from a well.
Identify the decision:
a) You decide that the water is not safe to drink when it really is not safe.
Correct – Power!

13. Hamburgers from a popular fast food chain are supposed to contain 300 calories. A consumer group believes the restaurant is making burgers with too many calories. They plan to take a random sample of 30 burgers. Based on past studies, the standard deviation of burger calories at this restaurant is 50.
Find the power of this test against the alternative μ = 330 calories.

14. Reject: invNorm (.95, 300, 9.13) = 315.02 or higher
Power = normalcdf(315.02, 1E99, 330, 9.13)
Power =