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Hypothesis Testing for Proportions 1 Section 7.4

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Section 7.4 Objectives 2 Use the z-test to test a population proportion p

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z-Test for a Population Proportion 3 A statistical test for a population proportion. Can be used when a binomial distribution is given such that np 5 and nq 5. The test statistic is the sample proportion. The standardized test statistic is z.

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Using a z-Test for a Proportion p 4 1.State the claim mathematically and verbally. Identify the null and alternative hypotheses. 2.Specify the level of significance. 3.Sketch the sampling distribution. 4.Determine any critical value(s). State H 0 and H a. Identify . Use Table 5 in Appendix B. Verify that np ≥ 5 and nq ≥ 5 In WordsIn Symbols

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Using a z-Test for a Proportion p 5 5.Determine any rejection region(s). 6.Find the standardized test statistic. 7.Make a decision to reject or fail to reject the null hypothesis. 8.Interpret the decision in the context of the original claim. If z is in the rejection region, reject H 0. Otherwise, fail to reject H 0. In WordsIn Symbols

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Example: Hypothesis Test for Proportions 6 Zogby International claims that 45% of people in the United States support making cigarettes illegal within the next 5 to 10 years. You decide to test this claim and ask a random sample of 200 people in the United States whether they support making cigarettes illegal within the next 5 to 10 years. Of the 200 people, 49% support this law. At α = 0.05 is there enough evidence to reject the claim? Solution: Verify that np ≥ 5 and nq ≥ 5. np = 200(0.45) = 90 and nq = 200(0.55) = 110

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Solution: Hypothesis Test for Proportions 7 H 0 : H a : = Rejection Region: p = 0.45 p ≠ 0.45 0.05 Decision: At the 5% level of significance, there is not enough evidence to reject the claim that 45% of people in the U.S. support making cigarettes illegal within the next 5 to 10 years. Test Statistic z 0-1.96 0.025 1.96 0.025 -1.961.96 1.14 Fail to reject H 0

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Example: Hypothesis Test for Proportions 8 The Pew Research Center claims that more than 55% of U.S. adults regularly watch their local television news. You decide to test this claim and ask a random sample of 425 adults in the United States whether they regularly watch their local television news. Of the 425 adults, 255 respond yes. At α = 0.05 is there enough evidence to support the claim? Solution: Verify that np ≥ 5 and nq ≥ 5. np = 425(0.55) ≈ 234 and nq = 425 (0.45) ≈ 191

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Solution: Hypothesis Test for Proportions 9 H 0 : H a : = Rejection Region: p ≤ 0.55 p > 0.55 0.05 Decision: At the 5% level of significance, there is enough evidence to support the claim that more than 55% of U.S. adults regularly watch their local television news. Test Statistic z 01.645 0.05 2.07 1.645 Reject H 0

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Section 7.4 Summary 10 Used the z-test to test a population proportion p

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