350 CHAPTER 7 Hypothesis Testing with One Sample p 0.65 0.63 0.57 0.59 0.61 Ha H0 Ha μ 19 17 18 11 12 13 14 15 16 H0 Ha μ 22 20 21 14 15 16 17 18 19 H0 Ha Stating the Null and Alternative Hypotheses Write each claim as a mathematical statement. State the null and alternative hypotheses, and identify which represents the claim. 1. A school publicizes that the proportion of its students who are involved in at least one extracurricular activity is 61%. 2. A car dealership announces that the mean time for an oil change is less than 15 minutes. 3. A company advertises that the mean life of its furnaces is more than 18 years. SOLUTION 1. The claim “the proportion . . . is 61%” can be written as p = 0.61. Its complement is p ≠ 0.61, as shown in the figure at the left. Because p = 0.61 contains the statement of equality, it becomes the null hypothesis. In this case, the null hypothesis represents the claim. You can write the null and alternative hypotheses as shown. H0: p = 0.61 (Claim) Ha: p ≠0.61 2. The claim “the mean . . . is less than 15 minutes” can be written as m 6 15. Its complement is m Ú 15, as shown in the figure at the left. Because m Ú 15 contains the statement of equality, it becomes the null hypothesis. In this case, the alternative hypothesis represents the claim. You can write the null and alternative hypotheses as shown. H0: m Ú 15 minutes Ha: m 6 15 minutes (Claim) 3. The claim “the mean . . . is more than 18 years” can be written as m 7 18. Its complement is m … 18, as shown in the figure at the left. Because m … 18 contains the statement of equality, it becomes the null hypothesis. In this case, the alternative hypothesis represents the claim. You can write the null and alternative hypotheses as shown. H0: m … 18 years Ha: m 7 18 years (Claim) In the three figures at the left, notice that each point on the number line is in either H0 or Ha, but no point is in both. TRY IT YOURSELF 1 Write each claim as a mathematical statement. State the null and alternative hypotheses, and identify which represents the claim. 1. A consumer analyst reports that the mean life of a certain type of automobile battery is not 74 months. 2. An electronics manufacturer publishes that the variance of the life of its home theater systems is less than or equal to 2.7. 3. A realtor publicizes that the proportion of homeowners who feel their house is too small for their family is more than 24%. Answer: Page A40 In Example 1, notice that the claim is represented by either the null hypothesis or the alternative hypothesis. EXAMPLE 1
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