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.

Men Women

Examples

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

Men Women

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.

Men Women

Rep

Dem

Rep

Demp1 p2

p1 > p2

Examples

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

Democrats Republicans

Women

Men

Women

Men

Examples

The proportion of patients who recovered, given treatment A vs. the proportion of patients who recovered, given treatment B.p1 = recovery rate under treatment A.

p2 = recovery rate under treatment B.

Comparing proportions

To estimate the difference between population proportions p1 and p2, we need the sample proportions p1

^ and p2^.

The sample difference p1^ – p2

^ is an estimator of the true difference p1 – p2.

Case Study 13

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 p1 = proportion of men who believe that Mayor Wilder is doing a good or excellent job.

Let p2 = 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 (p1^ = 0.41).

37% of men rated Wilder’s performance good or excellent (p2

^ = 0.37).

Hypothesis Testing

The hypotheses.H0: p1 – p2 = 0 (i.e., p1 = p2)

H1: p1 – p2 > 0 (i.e., p1 > p2)

The significance level is = 0.05. What is the test statistic? That depends on the sampling distribution

of p1^ – p2

^.

The Sampling Distribution of p1

^ – p2^

If the sample sizes are large enough, then p1

^ is N(p1, 1), where

and p2^ is N(p2, 2), where

1

111

1

n

pp

2

222

1

n

pp

The Sampling Distribution of p1

^ – p2^

The sample sizes will be large enough ifn1p1 5, and n1(1 – p1) 5, and

n2p2 5, and n2(1 – p2) 5.

Statistical Fact #1

For any two random variables X and Y,

YXYX

222YXYX

22YXYX

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

22, is YXYXNYX

The Sampling Distribution of p1

^ – p2^

Therefore,

where

21 ˆˆ2121 , is ˆˆ ppppNpp

2

22

1

11ˆˆ

)1()1(21 n

pp

n

pppp

The Test Statistic

Therefore, the test statistic would be

if we knew the values of p1 and p2. We will approximate them with p1

^ and p2^.

2

22

1

11

21

11

0ˆˆ

npp

npp

ppZ