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Chapter 22

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University of Alberta

Biology (Biological Sciences)

BIOL499A

Blaine Mullins

Winter

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Chapter 22: Comparing Two Proportions
In this chapter, we will talk about:
Hypothesis test for comparing two proportions.
Large-sample confidence intervals for comparing two proportions,
Suppose a sample of 300 urban adult residents of Alberta revealed 63 who
favored increasing the highway speed limit from 110 km/h to 120 km/h,
whereas a sample of 250 rural residents yielded 75 who favored the increase.
Is there enough evidence to conclude a difference between proportion of
adult residents of Alberta who favored increasing the highway speed
between urban and rural?
Interested parameter1 2p
Estimate:pp1 2 0 21 0.30 0.09
The test statistics for tes0 12 is:
Estimate0 p1 2
TS eis.. ) p ˆˆp
1 2 p11 p2 2
Var p 1 2p ˆ1Var pˆ2 Var p
n1 2
p qp p p p p (1 ) (1 )
( 1 2 p 11 2 2 1 1 2 2
n1 2 n1 n2
ˆˆˆ ˆ
E ( ) p p1 1 22 ) (1 )
1 2 n1 2
If Hp0 12 p1 2 is true, then
p1 1 2 2 (1 )
SD ( )1 2p
n1 2
p(1p ) (1 ) p(1 ) 1 1
n n n n
1 2 2 1
Therefore,
11
SE(. pˆ1 2 1( p) p
2
Estimate 0 p1 2
TS
eiE.. 11
p(1
n1 2 Inference for Two Population Proportions, p1 p 2:
For testingH 0: p1 p 2 0, when n is large, the test statistic is
p p 0
TS 1 2 ~ Standard normal distribution if H is true, where
1 1 0
p (1 p)( )
n1 n 2
yy pnn ˆ
pˆ 1 2 1122
n1 2 n1 2
*
ForHp:a0 1 2 , we reject H 0ifTS z
*
For Hpap0 1 2 , we reject H 0if TS z
ForH : 0 , we reject H ifTS z * or TS z *
a 1 2 0
A 100(1)% approximate confidence interval for difference of two population
proportions, p1 p 2 , is given by:
ˆ ˆ ˆ ˆ
p p z * p 11 p 1 p 21 p )2
1 2 n n
1 2 Example 22.1: A sample of 100 females was collected from ethnic group A
and a sample of 100 from ethn

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