# lesson 12 - 2 tests about a population parameter

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Lesson 12 - 2 Tests about a Population Parameter Slide 2 Knowledge Objectives Explain why p 0, rather than p-hat, is used when computing the standard error of p-hat in a significance test for a population proportion. Explain why the correspondence between a two- tailed significance test and a confidence interval for a population proportion is not as exact as when testing for a population mean. Explain why the test for a population proportion is sometimes called a large sample test. Discuss how significance tests and confidence intervals can be used together to help draw conclusions about a population proportion. Slide 3 Construction Objectives Conduct a significance test for a population proportion using the Inference Toolbox. Slide 4 Vocabulary Statistical Inference Slide 5 Requirements to test, population proportion Simple random sample Normality: np 0 10 and n(1-p 0 ) 10 [for normal approximation of binomial] Independence: n 0.10N [to keep binomial vs hypergeometric] Unlike with confidence intervals where we used p-hat in all calculations, in this test with use p 0, the hypothesized value (assumed to be correct in H 0 ) Slide 6 One-Proportion z-Test Slide 7 zz -z /2 z /2 -z Critical Region Reject null hypothesis, if P-value < Left-TailedTwo-TailedRight-Tailed z 0 < - z z 0 < - z /2 or z 0 > z /2 z 0 > z P-Value is the area highlighted |z 0 |-|z 0 | z0z0 z0z0 p p 0 Test Statistic: z 0 = -------------------- p 0 (1 p 0 ) n Slide 8 Reject null hypothesis, if p 0 is not in the confidence interval Confidence Interval Approach Lower Bound Upper Bound p0p0 P-value associated with lower bound must be doubled! Confidence Interval: p z /2 (p(1-p)/n p + z /2 (p(1-p)/n Example 2 Nexium is a drug that can be used to reduce the acid produced by the body and heal damage to the esophagus due to acid reflux. Suppose the manufacturer of Nexium claims that more than 94% of patients taking Nexium are healed within 8 weeks. In clinical trials, 213 of 224 patients suffering from acid reflux disease were healed after 8 weeks. Test the manufacturers claim at the =0.01 level of significance. H 0 : % healed =.94 H a : % healed >.94 One-sided test n 10000 in US!!) np(1-p) > 10 checked 224(.94)(.06) = 12.63 Slide 13 Example 2 p p 0 Test Statistic: z 0 = -------------------- p 0 (1 p 0 ) n 0.950893 0.94 Test Statistic: z 0 = ------------------------- = 0.6865 0.94(0.06)/224 = 0.01 so one-sided test yields Z = 2.33 Since Z 0 < Z , we fail to reject H 0 therefore there is insufficient evidence to support manufacturers claim Slide 14 Example 3 According to USDA, 48.9% of males between 20 and 39 years of age consume the minimum daily requirement of calcium. After an aggressive Got Milk campaign, the USDA conducts a survey of 35 randomly selected males between 20 and 39 and find that 21 of them consume the min daily requirement of calcium. At the = 0.1 level of significance, is there evidence to conclude that the percentage consuming the min daily requirement has increased? H 0 : % min daily = 0.489 H a : % min daily > 0.489 One-sided test n 700 in US!!) np(1-p) > 10 failed 35(.489)(.511) = 8.75 Slide 15 Example 3 Since the sample size is too small to estimate the binomial with a z-distribution, we must fall back to the binomial distribution and calculate the probability of getting this increase purely by chance. P-value = P(x 21) = 1 P(x < 21) = 1 P(x 20) (since its discrete) 1 P(x 20) is 1 binomcdf(35, 0.489, 20) (n, p, x) P-value = 0.1261 which is greater than , so we fail to reject the null hypothesis (H 0 ) insufficient evidence to conclude that the percentage has increased Slide 16 Using Your Calculator Press STAT Tab over to TESTS Select 1-PropZTest and ENTER Entry p 0, x, and n from given data Highlight test type (two-sided, left, or right) Highlight Calculate and ENTER Read z-critical and p-value off screen From first problem: z 0 = 0.686 and p-value = 0.2462 Since p > , then we fail to reject H 0 insufficient evidence to support manufacturers claim. Slide 17 Comments about Proportion Tests Changing our definition of success or failure (swapping the percentages) only changes the sign of the z-test statistic. The p-value remains the same. If the sample is sufficiently large, we will have sufficient power to detect a very small difference On the other hand, if a sample size is very small, we may be unable to detect differences that could be important Standard error used with confidence intervals is estimated from the sample, whereas in this test it uses p 0, the hypothesized value (assumed to be correct in H 0 ) Slide 18 Summary and Homework Summary We can perform hypothesis tests of proportions in similar ways as hypothesis tests of means Two-tailed, left-tailed, and right-tailed tests Normal distribution or binomial distribution should be used to compute the critical values for this test Confidence intervals provide additional information that significance tests do not namely a range of plausible values for the true population parameter Homework pg 771 12-23 to 12-12.27

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