424 Chapter 9 Hypothesis Testing SUMMARY ∙ A confidence interval contains all the values that are plausible at a particular level. ∙ A hypothesis test tells us precisely how plausible a particular value is. Check Your Understanding 17. A 95% confidence interval for �� is computed to be (1.75, 3.25). For each of the following hypotheses, state whether H0 will be rejected at the 0.05 level. a. H0 : �� = 3 versus H1: �� ≠ 3 No b. H0 : �� = 4 versus H1: �� ≠ 4 Yes c. H0 : �� = 1.7 versus H1: �� ≠ 1.7 Yes d. H0 : �� = 3.5 versus H1: �� ≠ 3.5 Yes 18. You want to test H0 : �� = 4 versus H1: �� ≠ 4, so you compute a 95% confidence interval for ��. The 95% confidence interval is 5.1 < �� < 7.2. a. Do you reject H0 at the �� = 0.05 level? Yes b. Your friend thinks that �� = 0.01 is a more appropriate significance level. Can you tell from the confidence interval whether to reject at this level? No Answers are on page 433. Objective 4 Describe the relationship between �� and the probability of error The Relationship Between �� and the Probability of an Error Recall that a Type I error occurs if we reject H0 when it is true, and a Type II error occurs if we do not reject H0 when it is false (see Table 9.2). When designing a hypothesis test, we would like to make the probabilities of these two errors small. In order to do this, we need to know how to calculate the probabilities of these errors. It is straightforward to find the probability of a Type I error: It is equal to the significance level. So, for example, if we perform a test at a significance level of �� = 0.05, the probability of a Type I error is 0.05. Table 9.2 H0 true H0 false Reject H0 Type I error Correct Don’t reject H Correct Type II error 0 SUMMARY When a test is performed with a significance level ��, the probability of a Type I error is ��. The probability of a Type II error is denoted by the letter ��. Computing the probability of a Type II error is more difficult than finding the probability of a Type I error. A Type II error occurs when H0 is false, and a decision is made not to reject. The probability of a Type II error depends on the true value of the parameter being tested. We will learn how to compute these probabilities in Section 9.7. Because �� is the probability of a Type I error, why don’t we always choose a very small value for ��? The reason is that the smaller a value we choose for ��, the larger the value of ��, the probability of making a Type II error, becomes (unless we increase the sample size). SUMMARY The smaller a value we choose for the significance level ��: ∙ The smaller the probability of a Type I error becomes. ∙ The larger the probability of a Type II error becomes. In general, making a Type I error is more serious than making a Type II error. When a Type I error is much more serious, a smaller value of �� is appropriate. When a Type I error is only slightly more serious, a larger value of �� can be justified.
navidi_monk_elementary_statistics_2e_ch7-9
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