1. Power of a Hypothesis test

2. H 0 True H 0 False Reject Fail to reject Type I Correct Type II Power   Suppose H 0 is true – what if we decide to fail to reject it? Suppose H 0 is false – what if we decide to reject it? Suppose H 0 is true – what if we decide to reject it? Suppose H 0 is false – what if we decide to fail to reject it? We correctly reject a false H 0 !

3. power 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.

4. A researcher selects a random sample of size 49 from a population with standard deviation  = 35 in order to test at the 1% significance level the hypothesis: H 0 :  = 680 H a :  > 680 What is the probability of committing a Type I error?  =.01

5. H 0 :  = 680 H a :  > 680 For what values of the sample mean would you reject the null hypothesis? Invnorm(.99,680,5) =691.63

6. Normalcdf(-10^99,691.63,700,5) =.0471 Power = 1 -.0471 =.9529 Go to 2 nd VARS, enter normalcdf Lower: -100,000 Upper: 691.63 Mean: 700 Standard deviation: 5

7. H 0 :  = 680 H a :  > 680 If H 0 is rejected, suppose that  a is 695. 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 = 1 -.2502 =.7498

8. Reject H 0 Fail to Reject H 0 Power = 1 -  00  aa 

9. What happens to , , & power when the sample size is increased? Reject H 0 Fail to Reject H 0 00  aa 

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

11. 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. 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!!