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Daniel S. Yates The Practice of Statistics Third Edition Chapter 11.4 Using Inference to Make Decisions Copyright © 2008 by W. H. Freeman & Company Potential Wrong Decisions • When we make decisions based on a significance test we hope to make the right decision - - but we could be wrong. • Two types of errors: – We reject H0 when H0 is in fact true. – We fail the reject H0 when H0 is false. • Each of these errors have consequences. Type I and Type II Errors Sampling Distribution Based on the Truth of a Hypothesis Reject H0 Fail to reject H0 Sampling Distribution when H0 is false. Sampling Distribution when H0 is true. α α is the probability of a type I error. β Critical μ = 6.4 Value of μ = 6.7 (Ha) (H0) x β is the probability of a type II error. α 1-β 1-α β Which type of error is more serious • Depends – which error has the more serious consequence. (Which error does more harm.) • Requires a realistic interpretation of what constitutes a type I error and what constitutes a type II. Example 11.52 • An outbreak of the mountain pine beetle has affected several types of trees in British Columbia. The beetle leaves behind a fungus that produces blue-colored stains in the wood. Some customers might worry that lumber obtained from the blue-stained trees is weaker as the result of the fungus. A Canadian company performed a test on the breaking strength of bluestained wood. The measured mean breaking strength of a sample of 100 pieces of blue-stained pine. The target breaking strength of lumber made from healthy pine trees is 10,000 pounds per square inch. 1. State the appropriate null and alternative hypothesis for the company test. H 0 : 10,000 psi H a : 10,000 psi Example 11.52 Continued 2. Describe a Type I and a Type II error, and give the consequences of each. • Type I error (Reject H0, when in fact H0 is true.) A type I error is committed by telling the consumer that the wood with blue stain is weaker when in fact it is not. • Type I consequence is that you must find more wood without the stain. • Type II error (Fail to reject H0 when in fact H0 is False.) A type II error is committed by not telling the consumer that the blue-stained wood is weaker when in fact it is. • Type II consequence is that you may lose loyal customers because they will lose their trust in the company. Example 11.52 Continued 3. Which type of error is more serious? Why? • While Type I error will cause the company to spend more money to find wood that does not contain blue stains, the Type II error is more serious since the company can not afford to lose a portion of its customer base because of perceived dishonesty. Class Practice Power of a Test 1-β 1 -α Power α β Definition of Power • Power measures the sensitivity of the test to find a significant effect when the truth of the alternative is far from the null hypothesis. • High power is desirable. Applet over Adjusting Power • Look at Activity 11C on page 729. • Power Applet. Ways to Increase The Power • Increase the sample size. • Increase α. • Consider an alternative hypothesis further away from the null hypothesis. • Decrease σ. Advise: “…choose as high an α level (Type I error probability) as your are willing to risk and as large a sample size as you can afford.” Standard • Evolving standard: – Sample size large enough for power (around 80%) – 95% confidence interval – Significance level for test, α = 0.05