Power of a Hypothesis Test

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Presentation transcript:

Power of a Hypothesis Test

Power (against a specific alternative value) P(we will correctly reject a false H0) We want to show H0 is false, so high power is important

We correctly reject a false H0! H0 True H0 False Reject Fail to Reject Type I Correct α Power Type II Correct β

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 σ = 35 in order to test at the 1% significance level the hypotheses: H0: μ = 680 Ha: μ > 680 What is the probability of committing a Type I error? α = .01

H0: μ = 680 Ha: μ > 680 For what values of the sample mean would you reject the null hypothesis? x-bar must be in top 1% invNorm(.99, 680, 5) = 691.63 or higher

What is the power of the test? If H0 is false, suppose that μa is really 700. What is the probability of committing a Type II error? What is the power of the test? Fail to reject H0  x-bar must be below 691.63 β = normalcdf(-1E99, 691.63, 700, 5) = .0471 Power = 1 – .0471 = .9529

What is the power of the test? If H0 is false, suppose that μa is really 695. What is the probability of committing a Type II error? What is the power of the test? β = normalcdf(-1E99, 691.63, 695, 5) = .2502 Power = 1 – .2502 = .7498

Fail to Reject H0 Reject H0 α μ0 ma Power = 1 – β β

As n increases, what happens to α, β, and power? Fail to Reject H0 Reject H0 a m0 b ma

So what do we know? We choose α We can't control β, since it depends on the alternate mean Ideally: have α as small as possible and power close to 1 (we like power > .8) As α increases, power increases… but so has the chance of a type I error! Best way to increase power, without increasing α, is to increase sample size (n)

Identify the decision: A water quality control board reports that water is unsafe for drinking if the mean nitrate concentration exceeds 30 ppm. Water specimens are taken at random from a well. Identify the decision: a) You decide that the water is not safe to drink when it really is safe. Type I Error

Identify the decision: A water quality control board reports that water is unsafe for drinking if the mean nitrate concentration exceeds 30 ppm. Water specimens are taken at random from a well. Identify the decision: a) You decide that the water is not safe to drink when it really is not safe. Correct – Power!

Hamburgers from a popular fast food chain are supposed to contain 300 calories. A consumer group believes the restaurant is making burgers with too many calories. They plan to take a random sample of 30 burgers. Based on past studies, the standard deviation of burger calories at this restaurant is 50. Find the power of this test against the alternative μ = 330 calories.

Reject: invNorm (.95, 300, 9.13) = 315.02 or higher Power = normalcdf(315.02, 1E99, 330, 9.13) Power = 0.9496