Confidence Intervals and Two Proportions

Presentation 9.4

Two Proportions

• When examining two proportions, we are interested in the difference between the two.

• For example,– Is the proportion of males who vote different

than the proportion of females who vote?– Statistically, we would estimate the difference

between the proportions– That is look at pmales - pfemales

From 1 proportion to 2

• For one proportion, we had the formula

• For two proportions, we look at the difference (so we subtract the proportions and use rules of variances for the standard error)

n

ppzp

ˆ1ˆ*ˆ

2

22

1

1121

ˆ1ˆˆ1ˆ*ˆˆ

n

pp

n

ppzpp

Assumptions

• The formula is appropriate when:– The samples are independently selected

random samples– In order to use the standard error formula,

make sure: population>10n for both samples– In order to approximate with the normal

distribution, make sure:

n1p1 10 n2p2 10

n1(1- p1) 10 n2(1- p2) 10

Example: Gender and Voting

• In a survey done in 2000 about voting and gender, it was found that 106 of 200 males voted in a recent election while 124 of 220 females voted. What is the difference between the proportion of females who vote and the proportion of males who vote? That is, construct a 95% confidence interval for the difference.

Example: Gender and Voting• First, find the proportions:

pfemales = 124/220 = 0.5636 call this proportion p1

pmales = 106/200 = 0.53 call this proportion p2

• Now check assumptions:– Check to use standard error.

• Each of the populations are greater than 10n– There are more than 2200 females.– There are more than 2000 males.

– Check for normal approximation.n1p1 10 n2p2 10220(.5636) 10 200(.53) 10124 10 106 10

n1(1- p1) 10 n2(1- p2) 10220(.4364) 10 200(.47) 1096 10 94 10

Notice that these are essentially counts of yeses or nos to the potential question (Do you vote?)

Example: Gender and Voting

• Once the formula is justified, just plug in and go to work.

• Notice the difference, at the 95% level, includes 0.

1289,.0617.

0486.96.10336.

200

53.153.

220

5636.15636.96.1535636.

ˆ1ˆˆ1ˆ*ˆˆ

2

22

1

1121

n

pp

n

ppzpp

Example: Gender and Voting

• Constructing the interval on the calculator.

• Choose 2-Proportion Z-interval

• Enter your statistics• Calculate to obtain

results.

Confidence Intervals

• The crucial value to look for in a 2-proportion interval of p1-p2 is zero.– If the interval is all positive, it implies that proportion

p1 is greater than p2.

– If the interval is all negative, it implies that proportion p1 is less than p2.

– If the interval contains zero, it means these is no significant difference between the proportions.• This result is rather important to conducting two sided

significant tests for two proportions.

Confidence Intervals and Two Proportions

This concludes this presentation.