Independent Independent Samples: Samples:

Comparing Comparing ProportionsProportionsLecture 41Lecture 41

Section 11.5Section 11.5

Fri, Apr 14, 2006Fri, Apr 14, 2006

Comparing ProportionsComparing Proportions

We now wish to compare We now wish to compare proportions between two proportions between two populations.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.

ExamplesExamples

The “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.

ExamplesExamples

The 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 proportions

To 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 Testing

See 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 Testing

The 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

1

111

1

n

pp

2

222

1

n

pp

The Sampling Distribution The Sampling Distribution of of

pp11^̂ – – pp22

^̂

The sample sizes will be large The sample sizes will be large enough ifenough if nn11pp11 5, and 5, and nn11(1 – (1 – pp11) ) 5, and 5, and

nn22pp22 5, and 5, and nn22(1 – (1 – pp22) ) 5. 5.

The Sampling The Sampling Distribution of Distribution of

pp11^̂ – – pp22

^̂ Therefore, Therefore,

wherewhere

2

22

12121 , is ˆˆ ppNpp

2

22

1

1122

21

2

22

1

1122

21

)1()1(

)1()1(

n

pp

n

pp

n

pp

n

pp

The Test StatisticThe Test Statistic

Therefore, the test statistic Therefore, the test statistic would bewould be

if 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…

2

22

1

11

21

11

0ˆˆ

npp

npp

ppZ

Pooled Estimate of Pooled Estimate of pp

In hypothesis testing for the In hypothesis testing for the difference between proportions, difference between proportions, typically the null hypothesis istypically the null hypothesis is

HH00: : pp11 = = pp22

Under 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 pp

Rather 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)

21

2211

21

21

2222

22

1111

11

ˆˆˆ

ˆˆ

ˆˆ

nn

pnpn

nn

xxp

pnxn

xp

pnxn

xp

The 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^̂..

21

22

21

21

22

21

11)ˆ1(ˆ

11)ˆ1(ˆ

nnpp

nnpp

CautionCaution

If the null hypothesis does If the null hypothesis does notnot say say

HH00: : pp11 = = pp22

then we should then we should notnot use the pooled use the pooled estimate estimate pp^̂, but should use the , but should use the unpooledunpooled estimate estimate

2

22

1

11ˆˆ

ˆ1ˆˆ1ˆ21 n

pp

n

pppp

The Test StatisticThe Test Statistic

So the test statistic isSo the test statistic is

21

21

11ˆ1ˆ

0ˆˆ

nnpp

ppZ

The Value of the Test The Value of the Test StatisticStatistic

Compute Compute pp^̂::

Now compute Now compute zz::

.2741.03564

977

25471017

611366ˆ

p

.253.701655.0

12.0

25471

10171

)7259.0)(2741.0(

2399.03599.0

Z

The The pp-value, etc.-value, etc.

Compute the Compute the pp-value: -value:

PP((ZZ > 7.253) = 2.059 > 7.253) = 2.059 10 10-13-13.. Reject Reject HH00.. The data indicate that a greater The data indicate that a greater

proportion of Canadians than French proportion of Canadians than French have a positive feeling about the have a positive feeling about the U.S.U.S.

Exercise 11.34Exercise 11.34

Sample #1: 361 “Wallace” cars Sample #1: 361 “Wallace” cars reveal that 270 have the sticker.reveal that 270 have the sticker.

Sample #2: 178 “Humphrey” cars Sample #2: 178 “Humphrey” cars reveal that 154 have the sticker.reveal that 154 have the sticker.

Do these data indicate that Do these data indicate that pp11 pp22??

ExampleExample

State the hypotheses.State the hypotheses. HH00: : pp11 = = pp22

HH11: : pp11 pp22

State the level of significance.State the level of significance. = 0.05.= 0.05.

ExampleExample

Write the test statistic.Write the test statistic.

21

21

11ˆ1ˆ

0ˆˆ

nnpp

ppz

ExampleExample

Compute Compute pp11^̂, , pp22

^̂, and , and pp^̂..

7866.0539

424

178361

154270ˆ

p

7479.0361

270ˆ1 p

8652.0178

154ˆ 2 p

ExampleExample

Now we can compute Now we can compute zz..

126.303752.0

1173.0

1781

3611

)7866.01)(7866.0(

8652.07479.0

Z

ExampleExample

Compute the Compute the pp-value.-value. pp-value = 2 -value = 2 normalcdf(-E99, -3.126) normalcdf(-E99, -3.126)

= 2(0.0008861)= 2(0.0008861)= 0.001772.= 0.001772.

The decision is to reject The decision is to reject HH00.. State the conclusion.State the conclusion.

The data indicate that the proportion of The data indicate that the proportion of Wallace cars that have the sticker is Wallace cars that have the sticker is different from the proportion of different from the proportion of Humphrey cars that have the sticker.Humphrey cars that have the sticker.

TI-83 – Testing Hypotheses TI-83 – Testing Hypotheses Concerning Concerning pp11

^̂ – – pp22^̂

Press STAT > TESTS > 2-Press STAT > TESTS > 2-PropZTest...PropZTest...

EnterEnter xx11, , nn11

xx22, , nn22

Choose the correct alternative Choose the correct alternative hypothesis.hypothesis.

Select Calculate and press ENTER.Select Calculate and press ENTER.

TI-83 – Testing Hypotheses TI-83 – Testing Hypotheses Concerning Concerning pp11

^̂ – – pp22^̂

In the window the following appear.In the window the following appear. The title.The title. The alternative hypothesis.The alternative hypothesis. The value of the test statistic The value of the test statistic zz.. The The pp-value.-value. pp11^.^. pp22^.^. The pooled estimate The pooled estimate pp^.^. nn11.. nn22..

ExampleExample

Work Exercise 11.34 using the TI-83.Work Exercise 11.34 using the TI-83.

Confidence Intervals for Confidence Intervals for pp11

^̂ – – pp22^̂

The formula for a confidence interval The formula for a confidence interval for for pp11

^̂ – – pp22^̂ is is

Caution: Note that we Caution: Note that we do notdo not use the use the pooled estimate for pooled estimate for pp^̂..

.

11ˆˆ

2

22

1

1121 n

pp

n

ppzpp

TI-83 – Confidence Intervals TI-83 – Confidence Intervals for for

pp11^̂ – – pp22

^̂

Press STAT > TESTS > 2-PropZInt…Press STAT > TESTS > 2-PropZInt… EnterEnter

xx11, , nn11

xx22, , nn22

The confidence level.The confidence level. Select Calculate and press ENTER.Select Calculate and press ENTER.

TI-83 – Confidence Intervals TI-83 – Confidence Intervals for for

pp11^̂ – – pp22

^̂

In the window the following appear.In the window the following appear. The title.The title. The confidence interval.The confidence interval. pp11^.^.

pp22^.^.

nn11..

nn22..

ExampleExample

Find a 95% confidence interval using Find a 95% confidence interval using the data in Exercise 11.34.the data in Exercise 11.34.