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AP Statistics: Section 9.2 We now turn out attention to conducting a significance test about a population proportion. Recall the conditions necessary to construct a confidence interval for a population proportion, because the conditions for conducting a significance test about a population proportion are the same. SRS np 10 and n 1 - p 10 Normality for distribution: ____________________ Independence Our general form for our test statistic has always been sample statistic - hypothesized value Test statistic = -------------------------------------standard error (Also called a large - sample test for a proportion) pˆ p0 p0 (1 p0 ) n np0 n(1 p0 ) z distribution Example: An experiment on the side effects of pain relievers randomly assigned arthritis patients to one of several over-the-counter pain medications. Of the 440 patients who took pain reliever A, 23 suffered some “adverse effect”. a) Does this experiment provide strong evidence that fewer than 10% of the people who take this medication have adverse effects? Hypothesis: The population of interest is arthritis sufferers who take pain reliever A. H 0 : p .1 H a : p .1 p the proportion of people taking pain reliever A who have adverse effects. Conditions: SRS : An experiment often uses volunteers. If the sample is not an SRS results may not generalize to the population. Normality of p̂ : 440(.1) 44 10 and 440(.9) 396 10 Independence : The randomization in an experiment helps with independence. And, since sampling w/o replacement N 10(440) or 4440 Calculations: z 23 .1 440 3.34 (.1)(.9) 440 TI83/84 : STAT TESTS 5 : 1 - PropZTest Interpretation: If we assume the population proportion is .1, there is a .04% chance of getting a sample with a proportion of 23 or smaller. This is 440 strong evidence that the true proportion of arthritis sufferers taking pain Reliever A and having adverse effects is less than .1. b) Describe a Type I error and a Type II in this situation and give consequences of each. Type I : Assume the proportion of arthritis sufferers taking pain reliever A and have some adverse effects is less than .1 when it is not. Consequence would be believing the drug is safer than it is Type II : Assume the proportion of arthritis sufferers taking pain reliever A and having some adverse effects is not less than .1 when it actually is. Consequence might be reduced sales or having the FDA pull the drug. Note I: The standard error for the confidence interval is computed using p̂ , while the denominator for the test statistic is computed using the value in the null hypothesis p0. Consequently, the correspondence between a two-tailed significance test and a confidence interval for a population proportion is no longer exact. However, they are still very close. Note II: Confidence Intervals provide information that significant tests do not – namely, a range of plausible values for the true population proportion.