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Understanding the Z-Test for Population Proportion in Hypothesis Testing

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akiva
10 diapositives · 276 Views · Aug 14, 2014

This resource provides a comprehensive overview of the z-test for hypothesis testing of population proportions. It outlines the necessary conditions for applying the z-test, including verifying np ≥ 5 and nq ≥ 5, as well as guidelines for formulating the null and alternative hypotheses, determining the level of significance, and sketching sampling distributions. The document also includes illustrative examples, detailing how to interpret statistical results, critical values, and the decision-making process regarding null hypotheses, ultimately aiding in real-world applications.

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Understanding the Z-Test for Population Proportion in Hypothesis Testing
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  1. Chapter 7 Hypothesis Testing with One Sample

  2. Hypothesis Testing for Proportions § 7.4

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

  4. Hypothesis Test for Proportions 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 values. State H0 and Ha. Identify . Use Table 4 in Appendix B. Continued.

  5. Hypothesis Test for Proportions Using a z-Test for a Proportion p Verify that np 5 and nq  5. In Words In Symbols • Determine any rejection regions. • 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. Hypothesis Test for Proportions Example: Statesville college claims that more than 94% of their graduates find employment within six months of graduation. In a sample of 500 randomly selected graduates, 475 of them were employed. Is there enough evidence to support the college’s claim at a 1% level of significance? Verify that the products np and nq are at least 5. np = (500)(0.94) = 470 and nq = (500)(0.06) = 30 H0:p  0.94 Ha: p > 0.94 (Claim) Continued.

  7. 2.33 0 z Test statistic Hypothesis Test for Proportions Example continued: Statesville college claims that more than 94% of their graduates find employment within six months of graduation. In a sample of 500 randomly selected graduates, 475 of them were employed. Is there enough evidence to support the college’s claim at a 1% level of significance? H0:p 0.94 Ha: p > 0.94 (Claim) Because the test is a right-tailed test and  = 0.01, the critical value is 2.33. Continued.

  8. 2.33 0 z Hypothesis Test for Proportions Example continued: Statesville college claims that more than 94% of their graduates find employment within six months of graduation. In a sample of 500 randomly selected graduates, 475 of them were employed. Is there enough evidence to support the college’s claim at a 1% level of significance? H0:p 0.94 Ha: p > 0.94 (Claim) The test statistic falls in the nonrejection region, so H0 is not rejected. At the 1% level of significance, there is not enough evidence to support the college’s claim.

  9. Hypothesis Test for Proportions Example: A cigarette manufacturer claims that one-eighth of the US adult population smokes cigarettes. In a random sample of 100 adults, 5 are cigarette smokers. Test the manufacturer's claim at  = 0.05. Verify that the products np and nq are at least 5. np = (100)(0.125) = 12.5 and nq = (100)(0.875) = 87.5 H0:p = 0.125 (Claim) Ha: p  0.125 Because the test is a two-tailed test and  = 0.05, the critical values are ± 1.96. Continued.

  10. z0 = 1.96 z0 = 1.96 0 2.27 z Hypothesis Test for Proportions Example continued: A cigarette manufacturer claims that one-eighth of the US adult population smokes cigarettes. In a random sample of 100 adults, 5 are cigarettes smokers. Test the manufacturer's claim at  = 0.05. Ha: p  0.125 H0:p = 0.125 (Claim) The test statistic is Reject H0. At the 5% level of significance, there is enough evidence to reject the claim that one-eighth of the population smokes.