390 CHAPTER 7 Hypothesis Testing with One Sample Recall from Section 6.3 that when the sample proportion is not given, you can find it using the formula pn = x n Sample proportion where x is the number of successes in the sample and n is the sample size. Hypothesis Test for a Proportion A researcher claims that 26% of U.S. adults ages 22 to 59 who do not have a parent with a bachelor’s degree have completed a bachelor’s degree themselves. In a random sample of 7400 adults ages 22 to 59 who do not have a parent with a bachelor’s degree, 1984 say they have completed a bachelor’s degree themselves. At a = 0.10, is there enough evidence to reject the researcher’s claim? (Adapted from Pew Research Center) SOLUTION The products np = 740010.262 = 1924 and nq = 740010.742 = 5476 are both greater than 5. So, you can use a z@test. The claim is “26% of U.S. adults ages 22 to 59 who do not have a parent with a bachelor’s degree have completed a bachelor’s degree themselves.” So, the null and alternative hypotheses are H0: p = 0.26 (Claim) and Ha: p ≠ 0.26. Because the test is a two-tailed test and the level of significance is a = 0.10, the critical values are -z0 = -1.645 and z0 = 1.645. The rejection regions are z 6 -1.645 and z 7 1.645. Because the number of successes is x = 1984 and n = 7400, the sample proportion is np = x n = 1984 7400 . The standardized test statistic is z = np - p 2 pq n Because np Ú 5 and nq Ú 5, you can use the z-test. = 11984 74002 - 0.26 2 10.26210.742 7400 Assume p = 0.26. ≈ 1.59. Round to two decimal places. The figure at the left shows the location of the rejection regions and the standardized test statistic z. Because z is not in the rejection region, you fail to reject the null hypothesis. Interpretation There is not enough evidence at the 10% level of significance to reject the claim that 26% of U.S. adults ages 22 to 59 who do not have a parent with a bachelor’s degree have completed a bachelor’s degree themselves. TRY IT YOURSELF 2 A researcher claims that 70% of U.S. adults ages 22 to 59 who have at least one parent with a bachelor’s degree or beyond have completed a bachelor’s degree themselves. In a random sample of 7400 adults ages 22 to 59 who have at least one parent with a bachelor’s degree or beyond, 5110 say they have completed a bachelor’s degree themselves. At a = 0.10, is there enough evidence to reject the researcher’s claim? (Adapted from Pew Research Center) Answer: Page A41 10% Level of Significance z −3 −4 −2 −1 0 1 2 3 4 −z0 = −1.645 z0 = 1.645 z ≈ 1.59 Picturing the World According to a survey, 33% of employees working from home say they have encountered at least one type of technical failure. To test this claim, you randomly select 300 employees working from home. In the sample, you find that 93 of them say they have encountered at least one type of technical failure. (Adapted from Deloitte Global) At A = 0.05, is there enough evidence to reject the claim? See Minitab steps on page 414. EXAMPLE 2
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