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Section 7.4

Section 7.4. Hypothesis Testing for Proportions. Section 7.4 Objectives. Use the z -test to test a population proportion p. z -Test for a Population Proportion. z -Test for a Population Proportion A statistical test for a population proportion.

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Section 7.4

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  1. Section 7.4 Hypothesis Testing for Proportions

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

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

  4. Using a z-Test for a Proportion p Verify that np ≥ 5 and nq ≥ 5 In Words In Symbols • State the claim mathematically and verbally. Identify the null and alternative hypotheses. • Specify the level of significance. • Sketch the sampling distribution. • Determine any critical value(s). State H0 and Ha. Identify . Use Table 5 in Appendix B.

  5. Using a z-Test for a Proportion p In Words In Symbols Determine any rejection region(s). Find the standardized test statistic. Make a decision to reject or fail to reject the null hypothesis. Interpret the decision in the context of the original claim. If z is in the rejection region, reject H0. Otherwise, fail to reject H0.

  6. Example: Hypothesis Test for Proportions 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

  7. p = 0.45 p ≠ 0.45 Solution: Hypothesis Test for Proportions • H0: • Ha: •  = • Rejection Region: • Test Statistic 0.05 Fail to reject H0 • 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. 0.025 0.025 z -1.96 1.96 -1.96 0 1.96 1.14

  8. Example: Hypothesis Test for Proportions 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

  9. p ≤ 0.55 p > 0.55 Solution: Hypothesis Test for Proportions • H0: • Ha: •  = • Rejection Region: • Test Statistic 0.05 Reject H0 • 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. 0.05 1.645 z 0 1.645 2.07

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

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