1. Power of a test

2. The power of a test (against a specific alternative value)
Is the probability that the test will reject the null hypothesis when the alternative is true
In practice, we carry out the test in hope of showing that the null hypothesis is false, so high power is important

3. H0 True H0 False Reject Fail to reject
Suppose H0 is false – what if we decide to reject it?
Suppose H0 is false – what if we decide to fail to reject it?
We correctly reject a false H0!
H0 True
H0 False
Reject
Fail to reject
Suppose H0 is true – what if we decide to fail to reject it?
Type I
Correct a
Power
Suppose H0 is true – what if we decide to reject it?
Correct
Type II b

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 s = 35 in order to test at the 1% significance level the hypothesis:
H0: m = 680
Ha: m > 680
What is the probability of committing a Type I error?
a = .01

5. H0: m = 680
Ha: m > 680
For what values of the sample mean would you reject the null hypothesis?
Invnorm(.99,680,5) =691.63

6. What is the power of the test?
H0: m = 680
Ha: m > 680
If H0 is rejected, suppose that ma is What is the probability of committing a Type II error?
What is the power of the test?
Normalcdf(-10^99,691.63,700,5) =.0471
Power = = .9529

7. What is the power of the test?
H0: m = 680
Ha: m > 680
If H0 is rejected, suppose that ma is What is the probability of committing a Type II error?
What is the power of the test?
Normalcdf(-10^99,691.63,695,5) =.2502
Power = = .7498

8. Fail to Reject H0
Reject H0 a ma m0
Power = 1 - b b

9. What happens to a, b, & power when the sample size is increased?
Fail to Reject H0
Reject H0 a m0 b ma

10. Facts: The researcher is free to determine the value of a.
The experimenter cannot control b, since it is dependent on the alternate value.
The ideal situation is to have a as small as possible and power close to 1. (Power > .8)
As a increases, power increases. (But also the chance of a type I error has increased!)
Best way to increase power, without increasing a, is to increase the sample size

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 from a well.
Identify the decision:
a) You decide that the water is not safe to drink when in fact it 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 from a well.
Identify the decision:
b) You decide that the water is not safe to drink when in fact it is not safe.
Correct – Power!!

13. Bottles of a popular cola are suppose to contain 300 ml of cola
Bottles of a popular cola are suppose to contain 300 ml of cola. A consumer group believes the company is under-filling the bottles. (Assume s = 50 with n = 30)
Find the power of this test against the alternative m = 296 ml.