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5.4 Inference as Decision Definition. A consumer may accept or reject a package of commodities on the basis of the quality of a sample of the package. This is called acceptance sampling. Type I and Type II Errors Definition. If we reject H0 (accept Ha ) when in fact H0 is true, this is a Type I error. If we accept H0 (reject Ha ) when in fact Ha is true, this is a Type II error. See Figure 5.17 (and TM-94). Error Probabilities Example 5.19. The mean diameter of a type of bearing is supposed to be 2.000 centimeters (cm). The bearing diameters vary normally with standard deviation σ = .010 cm. When a lot of the bearings arrives, the consumer takes an SRS of 5 bearings from the lot and measures their diameters. The consumer rejects the bearings if the sample mean diameter is significantly different from 2 at the 5% level. This is a test of the hypotheses: H0 : µ = 2 Ha : µ = 2. 1 To carry out the test, the consumer computes the z statistic z= x−2 √ .01/ 5 and rejects H0 if z > 1.96 A Type I error is to reject H0 when in fact µ = 2. What about Type II errors? Because there are many values of µ in Ha , we will concentrate on one value. The producer and the consumer agree that a lot of bearings with mean diameter 2.015 cm should be rejected. So a particular Type II error is to accept H0 when in fact µ = 2.015 Figure 5.18 (and TM-95) shows how the two probabilities of error are obtained from the two sampling distributions of x, for µ = 2 and for µ = 2.015. When µ = 2, H0 is true and to reject H0 is a Type I error. When µ = 2.015, Ha is true and to accept H0 is a Type II error. Definition. 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 H0 when H0 is in fact true. Example 5.20. Let’s calculate the probability of a Type II error for the previous example. Step 1. Write the rule for accepting H0 in terms of x. This occurs when x−2 √ ≤ 1.96. .01/ 5 or solving for x when 1.9912 ≤ x ≤ 2.0088. −1.96 ≤ Step 2. Find the probability of accepting H0 assuming that the alternative is true. Take µ = 2.015 and standardize to find the probability: P ( Type II error ) = P (1.9912 ≤ x ≤ 2.0088) 2 1.9912 − 2.015 x − 2.015 √ √ ≤ .01/ 5 .01/ 5 2.0088 − 2.015 √ ≤ .01 5 = P (−5.32 ≤ Z ≤ −1.39) = .0823. = P Figure 5.19 (and TM-96) show the relevant regions. Power Definition. The probability that a fixed level α significance test will reject H0 when a particular alternative value of the parameter is true is called the power of the test against that alternative. The power of a test against any alternative is 1 minus the probability of a Type II error for the alternative. Example. The power of the test performed in the previous example is 1 − .0823 = .9177. Different Views of Statistical Tests Note. The way of thinking about statistical tests called testing hypotheses involves: 1. State H0 and Ha just as in a test of significance. In particular, we are seeking evidence against H0 . 2. Think of the problem as a decision problem, so that the probabilities 3 of Type I and Type II errors are relevant. 3. Type I errors are more serious. So choose an α (significance level) and consider only tests with probability of Type I error no greater than α. 4. Among the tests, select one that makes the probability of a Type II error as small as possible (that is, power as large as possible). If this probability is too large, you will have to take a larger sample to reduce the chance of error. 4