Independent Samples: Comparing Proportions Lecture 37 Section 11.5 Tue, Nov 6, 2007

Comparing Proportions We wish to compare proportions between two populations. We should compare proportions for the same attribute in order for it to make sense.  For example, we could measure the proportion of NC residents living below the poverty level and the proportion of VA residents living below the poverty level.

Examples The “gender gap” refers to the difference between the proportion of men who vote Republican and the proportion of women who vote Republican. MenWomen

Examples The “gender gap” refers to the difference between the proportion of men who vote Republican and the proportion of women who vote Republican. MenWomen Rep Dem Rep Dem

Examples The “gender gap” refers to the difference between the proportion of men who vote Republican and the proportion of women who vote Republican. MenWomen Rep Dem Rep Dem p1p1 p2p2 p 1 > p 2

Examples Of course, the “gender gap” could be expressed in terms of the population of Democrats vs. the population of Republicans. DemocratsRepublicans Women Men Women Men

Examples The proportion of patients who recovered, given treatment A vs. the proportion of patients who recovered, given treatment B.  p 1 = recovery rate under treatment A.  p 2 = recovery rate under treatment B.

Comparing proportions To estimate the difference between population proportions p 1 and p 2, we need the sample proportions p 1 ^ and p 2 ^. The sample difference p 1 ^ – p 2 ^ is an estimator of the true difference p 1 – p 2.

Case Study 13 City Hall turmoil: Richmond Times- Dispatch poll. City Hall turmoil: Richmond Times- Dispatch poll Test the hypothesis that a higher proportion of men than women believe that Mayor Wilder is doing a good or excellent job as mayor of Richmond.

Case Study 13 Let p 1 = proportion of men who believe that Mayor Wilder is doing a good or excellent job. Let p 2 = proportion of women who believe that Mayor Wilder is doing a good or excellent job.

Case Study 13 What is the data?  500 people surveyed.  48% were male; 52% were female.  41% of men rated Wilder’s performance good or excellent (p 1 ^ = 0.41).  37% of men rated Wilder’s performance good or excellent (p 2 ^ = 0.37).

Hypothesis Testing The hypotheses.  H 0 : p 1 – p 2 = 0 (i.e., p 1 = p 2 )  H 1 : p 1 – p 2 > 0 (i.e., p 1 > p 2 ) The significance level is  = What is the test statistic? That depends on the sampling distribution of p 1 ^ – p 2 ^.

The Sampling Distribution of p 1 ^ – p 2 ^ If the sample sizes are large enough, then p 1 ^ is N(p 1,  1 ), where and p 2 ^ is N(p 2,  2 ), where

The Sampling Distribution of p 1 ^ – p 2 ^ The sample sizes will be large enough if  n 1 p 1  5, and n 1 (1 – p 1 )  5, and  n 2 p 2  5, and n 2 (1 – p 2 )  5.

Statistical Fact #1 For any two random variables X and Y,

Statistical Fact #2 Furthermore, if X and Y are both normal, then X – Y is normal. That is, if X is N(  X,  X ) and Y is N(  Y,  Y ), then

The Sampling Distribution of p 1 ^ – p 2 ^ Therefore, where

The Test Statistic Therefore, the test statistic would be if we knew the values of p 1 and p 2. We will approximate them with p 1 ^ and p 2 ^.