# Power of a test. power The power of a test (against a specific alternative value) Is the probability that the test will reject the null hypothesis when.

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Power of a test

power 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

H 0 True H 0 False Reject Fail to reject Type I Correct Type II Power Suppose H 0 is true – what if we decide to fail to reject it? Suppose H 0 is false – what if we decide to reject it? Suppose H 0 is true – what if we decide to reject it? Suppose H 0 is false – what if we decide to fail to reject it? We correctly reject a false H 0 !

A researcher selects a random sample of size 49 from a population with standard deviation = 35 in order to test at the 1% significance level the hypothesis: H 0 : = 680 H a : > 680 What is the probability of committing a Type I error? =.01

H 0 : = 680 H a : > 680 For what values of the sample mean would you reject the null hypothesis? Invnorm(.99,680,5) =691.63

H 0 : = 680 H a : > 680 If H 0 is rejected, suppose that a 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

H 0 : = 680 H a : > 680 If H 0 is rejected, suppose that a 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

Reject H 0 Fail to Reject H 0 Power = 1 - 0 a

What happens to,, & power when the sample size is increased? Reject H 0 Fail to Reject H 0 0 a

Facts: The researcher is free to determine the value of. The experimenter cannot control, since it is dependent on the alternate value. The ideal situation is to have as small as possible and power close to 1. (Power >.8) powerAs increases, power increases. (But also the chance of a type I error has increased!) sample sizeBest way to increase power, without increasing, is to increase the sample size

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 = 50 with n = 30) Find the power of this test against the alternative = 296 ml.

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