Chapter 19Chapter 19

Comparing Two ProportionsComparing Two Proportions

OutlineOutline

• Two-sample problems: proportionsTwo-sample problems: proportions

• The sampling distribution of a The sampling distribution of a differencedifference between proportions between proportions

• Large-sample confidence intervals Large-sample confidence intervals for comparing proportionsfor comparing proportions

• Significance tests for comparing Significance tests for comparing proportionsproportions

1. 1. Two-sample problems: proportionsTwo-sample problems: proportions

• In a two sample problem, we compare two In a two sample problem, we compare two populations or the responses to two treatments populations or the responses to two treatments based on two based on two independentindependent samples. samples.

• Notation:Notation:

PopulationPopulation Population Population proportionproportion

Sample sizeSample size Sample Sample proportionproportion

11

22

nn11

nn22

1p̂

2p̂1p

2p

Case Study - Case Study - Machine ReliabilityMachine Reliability

A study is performed to test of the reliability of products produced by two machines. Machine A produced 8 defective parts in a run of 140, while machine B produced 10 defective parts in a run of 200.

This is an example of when to use the two-proportion z procedures.

nn DefectsDefects

Machine AMachine A 140140 88

Machine BMachine B 200200 1111

Case Study - Case Study - Summer JobsSummer Jobs A university financial aid office polled a simple

random sample of undergraduate students to study their summer employment.

Not all students were employed the previous summer. Here are the results:

Is there evidence that the proportion of male students who had summer jobs differs from the proportion of female students who had summer jobs?

Summer StatusSummer Status MenMen WomenWomen

EmployedEmployed 718718 593593Not EmployedNot Employed 7979 139139TotalTotal 797797 732732

2. 2. The sampling distributions ofThe sampling distributions of

•When the samples are large, the When the samples are large, the distribution of distribution of is is approximate approximate normalnormal..

•The mean of The mean of is (p is (p11-p-p22)) . . ((unbiasedunbiased))

•The standard deviation of The standard deviation of is is

21 ˆˆ pp

2

22

1

11 )1()1(

n

pp

n

pp

)ˆˆ( 21 pp

)ˆˆ( 21 pp

)ˆˆ( 21 pp

3. 3. Large-sample confidence Large-sample confidence intervals intervals for comparing proportions for comparing proportions

Example19.1, 19.2 (page 493, page 495)Example19.1, 19.2 (page 493, page 495)Does preschool help?Does preschool help?

• To study the long term effects of preschool To study the long term effects of preschool programs, two groups of poor children were programs, two groups of poor children were looked at since early childhood. Group 2 looked at since early childhood. Group 2 attended preschool, but group 1 did not.attended preschool, but group 1 did not.

• One response variable of interest is the need One response variable of interest is the need for social services as adults. Let pfor social services as adults. Let p11 be the be the proportion from population 1 who need such proportion from population 1 who need such services, and similarly for pservices, and similarly for p22..

• Compute a 95% confidence interval for (pCompute a 95% confidence interval for (p11-p-p22).).

4. 4. Significance tests for comparing Significance tests for comparing proportionsproportions

• HH00: p: p11=p=p22 vs H vs Haa: one-sided or two-sided: one-sided or two-sided

• If HIf H00 is true, then the two proportions is true, then the two proportions are equal to some common value are equal to some common value pp..

• Instead of estimating Instead of estimating pp1 1 and and pp22 separately, we will combine the two separately, we will combine the two samples to estimate samples to estimate pp. (why is it . (why is it better?)better?)

Pooled Sample ProportionPooled Sample Proportion

• This combined or pooled estimate is This combined or pooled estimate is called the called the pooled sample pooled sample proportionproportion, ,

pooled sample proportion

samplesboth in nsobservatio ofnumber total

samplesboth in successes ofnumber totalˆ p

• Then the Then the Standard ErrorStandard Error of of becomes: becomes:

)ˆˆ( 21 pp

)11

)(1()1()1(

)1()1(

2121

2

22

1

11

nnpp

n

pp

n

pp

n

pp

n

ppSE

Example:Example:• Example 19.4 and 19.5: Choosing a Example 19.4 and 19.5: Choosing a

matemate– (page 500)(page 500)