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Section 7.4

Section 7.4. Hypothesis Testing for Proportions. Larson/Farber 4th ed. Section 7.4 Objectives. Use the z -test to test a population proportion p. Larson/Farber 4th ed. z -Test for a Population Proportion. z -Test for a Population Proportion

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Section 7.4

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  1. Section 7.4 Hypothesis Testing for Proportions Larson/Farber 4th ed.

  2. Section 7.4 Objectives • Use the z-test to test a population proportion p Larson/Farber 4th ed.

  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. Larson/Farber 4th ed.

  4. In Words In Symbols Using a z-Test for a Proportion p Verify that np ≥ 5 and nq ≥ 5 • 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 α. Use Table 5 in Appendix B. Larson/Farber 4th ed.

  5. In Words In Symbols Using a z-Test for a Proportion p • 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. Larson/Farber 4th ed.

  6. Example: Hypothesis Test for Proportions A research center estimates that no more than 40% of US adults eat breakfast every day. IN a random sample of 250 US adults, 41.6% say they eat breakfast every day. At α = 0.01, is there enough evidence to reject the researcher’s claim? • Solution: • Verify that np ≥ 5 and nq ≥ 5. p = 0.40 and q = 0.60 np = 250(0.40) = 100 and nq = 250(0.60) = 150 Larson/Farber 4th ed.

  7. p ≤ 0.40 (claim) p > 0.40 Solution: Hypothesis Test for Proportions • H0: • Ha: • α = • Rejection Region: • Test Statistic 0.01 Fail to reject H0 • Decision: At the 1% level of significance, there is insufficient evidence to reject the researcher’s claim that no more than 40% of US adults eat breakfast every day. 0.01 z 0 2.33 0.516 Larson/Farber 4th ed.

  8. Example: Hypothesis Test for Proportions 7.4, #12 - An environmentalist clams that more than 60% of british consumers are concerned about the use of genetic modification in food production and want to avoid genetically modified foods. You want to test this claim. You find that a random sample of 100 consumers, 65% say they are concerned about the use of genetically modified foods. At α = 0.10, can you support the environmentalist’s claim? • Solution: • Verify that np ≥ 5 and nq ≥ 5. np = 100(0.60) = 60 and nq = 100 (0.40) = 40 Larson/Farber 4th ed.

  9. p ≤ 0.60 0.10 p > 0.60 z 0 1.28 Solution: Hypothesis Test for Proportions • H0: • Ha: • α = • Rejection Region: • Test Statistic 0.10 Fail to Reject H0 • Decision: At the 10% level of significance, there is not enough evidence to support the environmentalist’s claim. 1.02 Larson/Farber 4th ed.

  10. Section 7.4 Summary • Used the z-test to test a population proportion p • HW: 5 - 15 EO Larson/Farber 4th ed.

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