Lesson Probability of a Type-II Error and the Power of the Test

Objectives Determine the probability of making a Type II error Compute the power of the test

Vocabulary Power of the test – value of 1 – β Power curve – a graph that shows the power of the test against values of the population mean that make the null hypothesis false.

Probability of Type II Error Determine the sample mean that separates the rejection region from the non-rejection region x-bar = μ 0 ± z α · σ/√n Draw a normal curve whose mean is a particular value from the alternative hypothesis, with the sample mean(s) found in step 1 labeled. The area described below represents β, the probability of not rejecting the null hypothesis when the alternative hypothesis is true. a. Left-tailed Test: Find the area under the normal curve drawn in step 2 to the right of x-bar b. Two-tailed Test: Find the area under the normal curve drawn in step 2 between x l and x u c. Right-tailed Test: Find the area under the normal curve drawn in step 2 to the left of x-bar

Example 1 The current wood preservative (CUR) preserves the wood for 6.40 years under certain conditions. We have a new preservative (NEW) that we believe is better, that it will in fact work for 7.40 years H o : H a : TYPE I: TYPE II: μ = 6.40 (our preservative is same as the current) μ = 7.40 (new is significantly better than the current)

Example 1 Our hypotheses  H 0 : μ = 6.40 (our preservative is the same as the current one) H0H0 H1H1  H 1 : μ = 7.40 (our preservative is significantly better than the current one)

Example 1 Type I error –Assumes that H 0 is true (that NEW is no better) –Our experiment leads us to reject H 0 Critical Value H0H0 H1H1 This area is the Type I error

Example 1 Type I errors –Assumes that H 0 is true (that NEW is no better) –Our experiment leads us to reject H 0 –Result – we conclude that NEW is significantly better, when it actually isn’t –This will lead to unrealistic expectations from our customers that our product actually works better

Example 1 Type II errors –Assumes that H 1 is true (that NEW is better) –Our experiment leads us to not reject H 0 Critical Value (the same as before) H0H0 H1H1 This area is the Type II error

Example 1 Type II errors –Assumes that H 1 is true (that NEW is better) –Our experiment leads us to not reject H 0 –Result – we conclude that NEW is not significantly better, when it actually is –This will lead to customers not getting a better treatment because we didn’t realize that it was better

Example 1 ●Test Details  We test our product on n = 60 wood planks  We have a known standard deviation σ = 3.2  We use a significance level of α = 0.05 ●The standard error of the mean is ●The critical value (for a right-tailed test) is  0.41 = = 0.41  60

Example 1 The Type II error, β, is the probability of not rejecting H 0 when H 1 is true –H 1 is that the true mean is 7.40 –The area where H 0 is not rejected is where the sample mean is 7.08 or less The probability that the sample mean is 7.08 or less, given that it’s mean is 7.40, is Thus β, the Type II error, is 0.22 Power of the test is 1 – β = – 7.40 β = P( z < ) = P(z < -0.77) =

Summary and Homework Summary –The Type II error, β, is the probability of not rejecting the null hypothesis when the alternative hypothesis was actually true –Type II errors can be computed only when the alternative hypothesis is also an equality –The power of a test, 1 – β, measures how well the test distinguishes between the two hypotheses Homework –pg 565 – 567; 3, 4, 7, 9