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Lesson 10 - 7 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 = μ0 ± zα · 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 xl and xu 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 Ho : μ = 6.40 (our preservative is same as the current) Ha: μ = 7.40 (new is significantly better than the current) TYPE I: TYPE II: Example 1 • Our hypotheses H0: μ = 6.40 (our preservative is the same as the current one) H1: μ = 7.40 (our preservative is significantly better than the current one) H0 H1 Example 1 • Type I error – Assumes that H0 is true (that NEW is no better) – Our experiment leads us to reject H0 H1 H0 Critical Value This area is the Type I error Example 1 • Type I errors – Assumes that H0 is true (that NEW is no better) – Our experiment leads us to reject H0 – 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 H1 is true (that NEW is better) – Our experiment leads us to not reject H0 H0 This area is the Type II error H1 Critical Value (the same as before) Example 1 • Type II errors – Assumes that H1 is true (that NEW is better) – Our experiment leads us to not reject H0 – 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 3.2 ----- = 0.41 60 ● The critical value (for a right-tailed test) is 6.40 + 1.645 0.41 = 7.08 Example 1 • The Type II error, β, is the probability of not rejecting H0 when H1 is true – H1 is that the true mean is 7.40 – The area where H0 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 7.08 – 7.40 β = P( z < ------------------) = P(z < -0.77) = 0.22 0.41 • Thus β, the Type II error, is 0.22 • Power of the test is 1 – β = 0.78 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