Section 9.1 Basic Principles of Hypothesis Testing 407 EXAMPLE 9.5 State a conclusion when the null hypothesis is not rejected Boxes of a certain kind of cereal are labeled as containing 20 ounces. An inspector thinks that the mean weight may be less than this, so he performs a test of H0 : �� = 20 versus H1: �� < 20. He does not reject the null hypothesis. State an appropriate conclusion. Solution The null hypothesis is not rejected, so we do not have sufficient evidence to conclude that the alternate hypothesis is true. We can express this as follows: ‘‘There is not enough evidence to conclude that the mean weight of cereal boxes is less than 20 ounces.’’ Another way to state this is: ‘‘The mean weight of cereal boxes may be equal to 20 ounces.’’ Objective 3 Distinguish between Type I and Type II errors Type I and Type II Errors Whenever a decision is made, there is a possibility that it is the wrong decision. There are two ways to make a wrong decision with a hypothesis test. First, if H0 is true, we might mistakenly reject it. Second, if H0 is false, we might mistakenly decide not to reject it. These two types of errors have names. Rejecting H0 when it is true is called a Type I error. Failing to reject H0 when it is false is called a Type II error. We summarize the possibilities in the following table. Explain It Again Type I and Type II errors in a trial: In a trial, the null hypothesis is that the defendant is innocent. A Type I error occurs if an innocent defendant is found guilty. A Type II error occurs if a guilty defendant is found not guilty. Reality Decision H0 True H0 False Reject H0 Type I error Correct decision Don’t reject H0 Correct decision Type II error EXAMPLE 9.6 Determining which type of error has been made The dean of a business school wants to determine whether the mean starting salary of graduates of her school is greater than $50,000. She will perform a hypothesis test with the following null and alternate hypotheses: H0 : �� = $50,000 H1: �� > $50,000 a. Suppose that the true mean is �� = $50,000, and the dean rejects H0. Is this a Type I error, a Type II error, or a correct decision? b. Suppose that the true mean is �� = $55,000, and the dean rejects H0. Is this a Type I error, a Type II error, or a correct decision? c. Suppose that the true mean is �� = $55,000, and the dean does not reject H0. Is this a Type I error, a Type II error, or a correct decision? Solution a. The true mean is �� = $50,000, so H0 is true. Because the dean rejects H0, this is a Type I error. b. The true mean is �� = $55,000, so H0 is false. Because the dean rejects H0, this is a correct decision. c. The true mean is �� = $55,000, so H0 is false. Because the dean does not reject H0, this is a Type II error. Check Your Understanding 4. A test is made of H0 : �� = 100 versus H1: �� ≠ 100. The true value of �� is 150, and H0 is rejected. Is this a Type I error, a Type II error, or a correct decision? Correct decision 5. A test is made of H0 : �� = 18 versus H1: �� > 18. The true value of �� is 20, and H0 is not rejected. Is this a Type I error, a Type II error, or a correct decision? Type II error
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