Chapter 21 Power of a test

1. Chapter 21Power of a test

2. The power of a test (against a specific alternative value) • Is the probability that the test will correctly 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. 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! 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. A researcher selects a random sample of size 50 from a population in order to test at the 1% significance level the hypothesis: H0: p = .25 Ha: p > .25 What is the probability of committing a Type I error? a = .01

5. H0: p = .25 Ha: p > .25 For what values of the sample mean would you reject the null hypothesis? Invnorm(.99) =2.326 SD=.0612 2.326= p-hat - .25 .0612 When p-hat is at least .392

6. H0: p = .25 Ha: p > .25 If H0 is rejected, suppose that the true p-hat is .4. What is the probability of committing a Type II error? What is the power of the test? SD = .0693 z=-.1154 Normalcdf(-999,-.1154) =.454 Power = 1 - .454 = .546

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

8. What happens to a, b, & power when the sample size is increased? Fail to Reject H0 Reject H0 What happens to the standard deviation as n increases? How will that change the shape of the distribution? a m0 b ma

9. 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. • Asa 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