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Power of a test

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.

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Power of a test

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  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. 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 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. 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

  7. 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

  8. ma Fail to Reject H0 Reject H0 a 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) • 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

  11. 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.

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