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Understanding the Power of a Statistical Test

Learn the significance of test power against alternative values in hypothesis testing. Explore Type I and Type II errors to enhance statistical decision-making. Discover how sample size affects test accuracy.

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Understanding the Power of a Statistical 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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