Testing The Difference Between Proportions

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8.4 Testing the Difference

Between Proportions

Statistics

Mrs. SpitzSpring 2009


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Objectives/Assignment

How to perform a z-test for the

difference between two population

proportions p1and p2.

ssignment! pp. "0"-"0# $1-12


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Two Sample !Test for the Difference

Between Proportions

%n this section& 'ou will learn how to use a z-

test to test the difference between two

population proportions p1and p2using a

sample proportion from each population. (orinstance& suppose 'ou want to determine

whether the proportion of female college

students who earn a bachelor)s degree in

four 'ears is different from the proportion ofmale college students who earn a bachelor)s

degree in four 'ears.


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Between Proportions

*o use a z-test to test such a difference& the

following conditions are necessar'!

1. *he samples must be independent.

2. *he samples must be large enough to use a

normal sampling distribution. *hat is!

n1p1+ ,& n11+ ,& n2p2+ ,& n22+ ,


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Between Proportions

%f these conditions are met& then the

sampling distribution for the

difference between the sample

proportions& is a normal distribution

with mean!

, 21 pp

21 21pp

pp =

nd a standard error

2

22

1

11 21

n

qp

n

qp

pp +=


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Two sample !test for "ifference

between proportions

otice that 'ou need to /now the

population proportions to calculate the

standard error. ecause a h'pothesis

test for p1- p2is based on the conditionof eualit' that p1 p2& 'ou can

calculate a weighted estimate of p

using!

21

21

nn

xxp

+

+= 111

pnx = and 222 pnx =

where


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Two!Sample !Test for the Difference

Between Proportions

sing the weighted estimate & the

standard error of the sampling

distribution for is!, 21 pp

)11

(21

21nn

qppp

+=

where

pq =1

lso& when determining whether the z-test can be used

for the difference between proportions& 'ou should use

in place of p1and p2and in place of 1and 2q

p

p


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Two Sample !Test for the

Difference Between Proportions

%f the sampling distribution for is normal&

'ou can use a two-sample z-test to test the difference

between two population proportions& and

, 21 pp

1p 2p


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#f the n$ll h%pothesis states that p&' p()

then the e*pression) p&! p(is e+$al to , in

the prece"ing test.


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-*. & A Two!Sample !Test for the

Difference Between Proportions

%n a stud' of 200 adult female and 2,0adult male %nternet users& 304 of thefemales and 354 of the males said that

the' plan to shop online at least onceduring the ne6t month. t 0.10& testthe claim that there is a difference in theproportion of female %nternet users whoplan to shop online and the proportionof male %nternet users who plan to shoponline.


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Sol$tion

7ou want to determine whether there is adifference in the proportions. So& the null andalternati8e h'potheses are!

Ho! p1 p2 and p1p2claim:

ecause the test is two-tailed and the le8el ofsignificance is 0.10& the critical 8alues are1.#",. *he re;ection regions are z < -1.#",and z = 1.#",.


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Sol$tion

*he weighted estimate of the populationproportion is!

21

21

nn

xxp++= 344.0

450155

2502009560 =

+

+=p

nd .ecause 2000.3"":& 2000.#,#:&

2,00.3"": and 2,0:0.#,# are at least ,&

'ou can use the two-sample z-test.

.656.0344.011 === pq


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Sol$tion

*he standardized test statistic is!

)11(

)()(

21

2121

nnqp

ppppz

+

=

775.1)

250

1

200

1)(656.0)(344.0(

)0()38.030.0(

+

=z


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Sol$tion

*he graph shows thelocation of there;ection regions andthe standardized teststatistic. ecause z

is in the re;ectionregion& 'ou shoulddecide to re;ect thenull h'pothesis. 7ouha8e enoughe8idence at the 104

le8el to concludethere is a difference inthe proportion offemale and theproportion of male

%nternet users whoplan to shop online.


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-*. ( Two!Sample !Test for the Difference

Between Proportions

medical research team conducted a stud'

to test the effect of a cholesterol-reducing

medication. t the end of the stud'& the

researchers found that of the ">00 sub;ectswho too/ the medication& 301 died of heart

disease. ?f the "300 sub;ects who too/ a

placebo& 3,> died of heart disease. t

0.01& can 'ou conclude that the death rate islower for those who too/ the medication than

for those who too/ the placebo@


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Sol$tion

7ou want to determine whether there is a

difference in the proportions. So& the null and

alternati8e h'potheses are!

Ho! p1+ p2 and p1< p2claim:

ecause the test is left-tailed and the le8el of

significance is 0.01& the critical 8alue is

-2.33. *he re;ection region is z < -2.33


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Sol$tion

*he weighted estimate of the populationproportion is!

21

21

nn

xxp+

+= 073.0

9000658

43004700357301 =

+

+=p

nd .ecause ">000.0>3:& ">000.92>:&

"3000.0>3: and "300:0.92> are at least,& 'ou can use the two-sample z-test.

.927.0073.011 === pq


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Sol$tion

*he standardized test statistic is!

)11(

)()(

21

2121

nnqp

ppppz

+

=

461.3)

4300

1

4700

1)(927.0)(073.0(

)0()083.0064.0(

+

=z


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Sol$tion

*he graph shows thelocation of there;ection region andthe standardized teststatistic. ecause z

is in the re;ectionregion& 'ou shoulddecide to re;ect thenull h'pothesis. tthe 14 le8el& there isenough e8idence toconclude that thedeath rate is lower forthose who too/ themedication than forthose who too/ the

placebo.

Testing The Difference Between Proportions

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