AP STATISTICS LESSON 10 – 4 ( DAY 1 ) INFERENCE AS DECISION ESSENTIAL QUESTION: What are the two types of errors made when making decisions using inference? Objectives: • To identify type I and type II errors. • To calculate type I and type II errors. Inference as Decision Tests of significance assess the strength of evidence against the null hypothesis. The alternative hypothesis ( the statement we seek evidence for) enters the test only to help us see what outcomes count against the null hypothesis. Using significance tests with fixed α, however, suggests another way of thinking. A level of significance α chosen in advance points to the outcome of the test as a decision. The transformation from measuring the strength of evidence to making decisions is not a small step. Acceptance Sampling There are circumstances that call for a decision or action as the end result of inference. Acceptance sampling is one such circumstance. We will use acceptance sampling to show how a different concept – inference as decision – changes the reasoning used in tests of significance. Type I and Type II Errors There are simply two hypotheses, and we must accept one and reject the other. It is convenient to continue to call the two hypotheses Ho and Ha , but Ho no longer has the special status (the statement we try to find evidence against) that it had in tests of significance. In the acceptance sampling problem, we must decide between Ho and Ha. Type I and Type II Errors (continued…) • If we reject Ho ( accept Ha) when in fact Ho is true, this is a Type I error. • If we accept (reject Ha ) H0 when in fact Ha is true, this is a Type II error. The Two Types of Error in Testing Hypotheses Error Probabilities We assess any rule for making decisions by looking at the probabilities of the two types of errors. This is in keeping with the idea that statistical inference is based on asking, “ What would happen if I used this procedure many times?” Significance tests with fixed level α give a rule for making decisions, because the test either rejects Ho or fails to reject it. We then describe the performance of a test by the probabilities of type I and type II errors. Example 10.21 Page 595 Are These Potato Chips Too Salty? Steps for finding type I and type II errors: Step 1: To find the type I error you use the α or significance level = type I error. Finding type II error: Step 2: find Z values for α level: Ho : μ = 2 Ha : μ ≠ 2 Example 10.21 (continued…) z < -1.96 or z > 1.96 Step1: write the rule in terms of x. 2 – 1.96σ/√ n ≤ x ≤ 2 + 1.96σ/√ n 1.9723 ≤ x ≤ 2.027 Step 2: find the probability of accepting Ho assuming that the alternative is true. Take μ = 2.05 and standardize to find the probability. Figure 10.20 Page 596 • The light shaded area is a Type 1 error (the probability of rejecting Ho: μ = 2 when in fact μ = 2. • The probability of a Type II error (dark shaded area) is the probability of accepting Ho when in fact μ = 2.05. Significance and Type I Error The significance level α of any fixed level test is the probability of a Type I error. That is, α is the probability that the test will reject the null hypothesis Ho when Ho is in fact true. Interpreting Type II Error The probability of a type II error in example 10.22 is .0571. This tells us that this test will lead us to fail to reject Ho : μ = 2 for about 6% of all batches of chips with a μ = 2.05. In other words, we will accept 6% of batches of potato chips so bad that their mean salt content is 2.05 mg.