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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 Suppose H0 is false Suppose H is false We correctly reject 0 – what if we decide – what if we a reject false Hit? ! H0 0decide to to fail to reject it? True H0 False Suppose H0 Reject Type I Correct is true – what if we a Power decide to Fail to Correct Type II Suppose H0 is true fail to b –reject what if we decide reject it? to reject it? 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 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? Normalcdf(-10^99,691.63,700,5) =.0471 What is the power of the test? Power = 1 - .0471 = .9529 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? Normalcdf(-10^99,691.63,695,5) =.2502 What is the power of the test? Power = 1 - .2502 = .7498 Fail to Reject H0 Reject H0 a m0 Power = 1 - b b ma 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 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: Type I Error a) You decide that the water is not safe to drink when in fact it is safe. 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: Correct – Power!! b) You decide that the water is not safe to drink when in fact it is not safe. 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.