Independent Samples: Comparing ProportionsComparing ProportionsExamplesSlide 4Comparing proportionsHypothesis TestingSlide 7The Sampling Distribution of p1^ – p2^The Sampling Distribution of p1^ – p2^Slide 10The Test StatisticPooled Estimate of pSlide 13Pooled Estimate of p (The “Batting-Average” Formula)The Standard Deviation of p1^ – p2^CautionSlide 17The Value of the Test StatisticThe p-value, etc.Exercise 11.34ExampleSlide 22Slide 23Slide 24Slide 25TI-83 – Testing Hypotheses Concerning p1^ – p2^Slide 27Slide 28Confidence Intervals for p1^ – p2^TI-83 – Confidence Intervals for p1^ – p2^Slide 31Slide 32Independent Independent Samples: Samples: Comparing Comparing ProportionsProportionsLecture 41Lecture 41Section 11.5Section 11.5Fri, Apr 14, 2006Fri, Apr 14, 2006Comparing ProportionsComparing ProportionsWe now wish to compare proportions We now wish to compare proportions between two populations.between two populations.Normally, we would be measuring Normally, we would be measuring proportions for the same attribute.proportions for the same attribute.For example, we could measure the For example, we could measure the proportion of NC residents living proportion of NC residents living below the poverty level and the below the poverty level and the proportion of VA residents living proportion of VA residents living below the poverty level.below the poverty level.ExamplesExamplesThe “gender gap” – the proportion of The “gender gap” – the proportion of men who vote Republican vs. the men who vote Republican vs. the proportion of women who vote proportion of women who vote Republican.Republican.The proportion of teenagers who The proportion of teenagers who smoked marijuana in 1995 vs. the smoked marijuana in 1995 vs. the proportion of teenagers who smoked proportion of teenagers who smoked marijuana in 2000.marijuana in 2000.ExamplesExamplesThe proportion of patients who The proportion of patients who recovered, given treatment A vs. the recovered, given treatment A vs. the proportion of patients who proportion of patients who recovered, given treatment B.recovered, given treatment B.Treatment A could be a placebo.Treatment A could be a placebo.Comparing proportionsComparing proportionsTo estimate the difference between To estimate the difference between population proportions population proportions pp11 and and pp22, we , we need the sample proportions need the sample proportions pp11^^ and and pp22^^..The difference The difference pp11^^ – – pp22^^ is an is an estimator of the difference estimator of the difference pp11 – – pp22..Hypothesis TestingHypothesis TestingSee Example 11.8, p. 721 – See Example 11.8, p. 721 – Perceptions of the U.S.: Canadian Perceptions of the U.S.: Canadian versus French.versus French.pp11 = proportion of Canadians who feel = proportion of Canadians who feel positive about the U.S..positive about the U.S..pp22 = proportion of French who feel = proportion of French who feel positive about the U.S..positive about the U.S..Hypothesis TestingHypothesis TestingThe hypotheses.The hypotheses.HH0: 0: pp11 – – pp22 = 0 (i.e., = 0 (i.e., pp11 = = pp22))HH1: 1: pp11 – – pp22 > 0 (i.e., > 0 (i.e., pp11 > > pp22))The significance level is The significance level is = 0.05. = 0.05.What is the test statistic?What is the test statistic?That depends on the sampling That depends on the sampling distribution of distribution of pp11^^ – – pp22^^..The Sampling The Sampling Distribution of Distribution of pp11^^ – – pp22^^ If the sample sizes are large enough, If the sample sizes are large enough, then then pp11^^ is is NN((pp11, , 11), where ), where Similarly, Similarly, pp22^^ is is NN((pp22, , 22), where ), where 11111npp 22221npp The Sampling Distribution The Sampling Distribution of of pp11^^ – – pp22^^The sample sizes will be large The sample sizes will be large enough ifenough ifnn11pp11 5, and 5, and nn11(1 – (1 – pp11) ) 5, and 5, andnn22pp22 5, and 5, and nn22(1 – (1 – pp22) ) 5. 5.The Sampling The Sampling Distribution of Distribution of pp11^^ – – pp22^^ Therefore, Therefore, wherewhere22212121, is ˆˆppNpp22211122212221112221)1()1()1()1(nppnppnppnppThe Test StatisticThe Test StatisticTherefore, the test statistic Therefore, the test statistic would bewould beif we knew the values of if we knew the values of pp11 and and pp22..We could estimate them with We could estimate them with pp11^^ and and pp22^^..But there may be a better way…But there may be a better way… 22211121110ˆˆnppnppppZPooled Estimate of Pooled Estimate of ppIn hypothesis testing for the In hypothesis testing for the difference between proportions, difference between proportions, typically the null hypothesis istypically the null hypothesis isHH00: : pp11 = = pp22Under that assumption, Under that assumption, pp11^^ and and pp22^^ are both estimators of a common are both estimators of a common value (call it value (call it pp).).Pooled Estimate of Pooled Estimate of ppRather than use either Rather than use either pp11^^ or or pp22^^ alone to estimate alone to estimate pp, we will use a , we will use a ““pooled” estimatepooled” estimate..The pooled estimate is the The pooled estimate is the proportion that we would get if we proportion that we would get if we pooledpooled the two samples together. the two samples together.Pooled Estimate of Pooled Estimate of pp(The “Batting-Average” (The “Batting-Average” Formula)Formula)2122112121222222111111ˆˆˆˆˆˆˆnnpnpnnnxxppnxnxppnxnxpThe Standard Deviation of The Standard Deviation of pp11^^ – – pp22^^This leads to a better estimator of This leads to a better estimator of the standard deviation of the standard deviation of pp11^^ – – pp22^^..21222121222111)ˆ1(ˆ11)ˆ1(ˆnnppnnppCautionCautionIf the null hypothesis does If the null hypothesis does notnot say sayHH00: : pp11 = = pp22then we should then we should notnot use the pooled use the pooled estimate estimate pp^^, but should use the , but should
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