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The Difference Between ProportionsQuestions and Answers
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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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Two Sample !Test for the Difference
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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Two Sample !Test for the Difference
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.