Power of a test

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

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

What is power? Power is the probability of rejecting the null hypothesis when in fact it is false. Power is the probability of making a correct decision (to reject the null hypothesis) when the null hypothesis is false. Power is the probability that a test of significance will pick up on an effect that is present. Power is the probability that a test of significance will detect a deviation from the null hypothesis, should such a deviation exist. Power is the probability of avoiding a Type II error.

What is power? Power is the probability of rejecting the null hypothesis when in fact it is false. Power is the probability of making a correct decision (to reject the null hypothesis) when the null hypothesis is false. Power is the probability that a test of significance will pick up on an effect that is present. Power is the probability that a test of significance will detect a deviation from the null hypothesis, should such a deviation exist. Power is the probability of avoiding a Type II error.

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

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

What is the power of the test? H0: m = 680 Ha: m > 680 If H0 is rejected, suppose that ma is 700. 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 = 1 - .0471 = .9529

What is the power of the test? H0: m = 680 Ha: m > 680 If H0 is rejected, suppose that ma 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

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

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

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 n Decreasing  will increase power, but this is usually not feasible. Another way to increase power is to consider an alternative farther from µ0

For the AP (and my) Exam An understanding of error and power is what’s important, not the mindless calculations that most students don’t understand anyway. STUDY the types of error, make sure you can state them in words. Be sure to state what the power of the test IS and how to increase it! ap central - on power

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.