Hypothesis Testing for Proportions 7.4 388 CHAPTER 7 Hypothesis Testing with One Sample What You Should Learn How to use the z@test to test a population proportion p HypothesisTest for Proportions Hypothesis Test for Proportions In Sections 7.2 and 7.3, you learned how to perform a hypothesis test for a population mean m. In this section, you will learn how to test a population proportion p. Hypothesis tests for proportions can be used when politicians want to know the proportion of their constituents who favor a certain bill or when quality assurance engineers test the proportion of parts that are defective. If np Ú 5 and nq Ú 5 for a binomial distribution, then the sampling distribution for np is approximately normal with a mean of mnp = p and a standard error of snp = 2pq n. The z@test for a proportion p is a statistical test for a population proportion. The z@test can be used when a binomial distribution is given such that np Ú 5 and nq Ú 5. The test statistic is the sample proportion np and the standardized test statistic is z = np - mnp snp = np - p 2 pq n . Standardized test statistic for p z-Test for a Proportion p Using a z@Test for a Proportion p In Words In Symbols 1. Verify that the sampling distribution np Ú 5, nq Ú 5 of np can be approximated by a normal distribution. 2. State the claim mathematically State H0 and Ha. and verbally. Identify the null and alternative hypotheses. 3. Specify the level of significance. Identify a. 4. Determine the critical value(s). Use Table 4 in Appendix B. 5. Determine the rejection region(s). 6. Find the standardized test statistic z = np - p 2 pq n . and sketch the sampling distribution. 7. Make a decision to reject or fail to If z is in the rejection region, reject the null hypothesis. then reject H0. Otherwise, fail to reject H0. 8. Interpret the decision in the context of the original claim. GUIDELINES In Step 7 of the guidelines, the decision rule uses rejection regions. You can also test a claim using P-values, as described in the Study Tip at the left. Study Tip A hypothesis test for a proportion p can also be performed using P@values. Use the guidelines on page 365 for using P@values for a z@test for a mean m, but in Step 4 find the standardized test statistic by using the formula z = np - p 2 pq n . The other steps in the test are the same.
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