Matakuliah : I0134/Metode Statistika
Tahun
: 2007
Pendugaan Parameter Proposi dan Beda Beda Proporsi
Pertemuan 17
Interval Estimation
of a Population Proportion
• Interval Estimate
p  z / 2
where:
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p (1  p )
n
1 - is the confidence coefficient
z/2 is the z value providing an area of
/2 in the upper tail of the standard
normal probability distribution
p is the sample proportion
Example: Political Science, Inc.
• Interval Estimation of a Population Proportion
Political Science, Inc. (PSI) specializes in voter polls and
surveys designed to keep political office seekers informed of
their position in a race. Using telephone surveys, interviewers
ask registered voters who they would vote for if the election
were held that day.
In a recent election campaign, PSI found that 220
registered voters, out of 500 contacted, favored a particular
candidate. PSI wants to develop a 95% confidence interval
estimate for the proportion of the population of registered
voters that favors the candidate.
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Example: Political Science, Inc.
• Interval Estimate of a Population Proportion
p (1  p )
p  z / 2
n
where: n = 500, p = 220/500 = .44, z/2 = 1.96
. 44(1. 44)
. 44  1. 96
500
.44 + .0435
PSI is 95% confident that the proportion of all voters
that favors the candidate is between .3965 and .4835.
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Sample Size for an Interval Estimate
of a Population Proportion
• Let E = the maximum sampling error mentioned in the precision
statement.
p(1  p)
• We have
E  z / 2
n
• Solving for n we have
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( z / 2 ) 2 p(1  p)
n
E2
Example: Political Science, Inc.
• Sample Size for an Interval Estimate of a Population Proportion
Suppose that PSI would like a .99 probability that the
sample proportion is within + .03 of the population proportion.
How large a sample size is needed to meet the required
precision?
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Example: Political Science, Inc.
• Sample Size for Interval Estimate of a Population Proportion
At 99% confidence, z.005 = 2.576.
( z / 2 ) 2 p(1  p) ( 2.576) 2 (. 44)(.56)
n

 1817
2
2
E
(. 03)
Note: We used .44 as the best estimate of p in the
above expression. If no information is available
about p, then .5 is often assumed because it provides
the highest possible sample size. If we had used
p = .5, the recommended n would have been 1843.
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Estimating the Difference between Two Proportions
•Sometimes we are interested in comparing the proportion of
“successes” in two binomial populations.
•The germination rates of untreated seeds and seeds treated
with a fungicide.
•The proportion of male and female voters who favor a
particular candidate for governor.
•To make this comparison,
A random sample of size n1 drawn from
binomial population 1 with parameter p1.
A random sample of size n2 drawn from
binomial population 2 with parameter p2 .
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Estimating the Difference between Two Means
•We compare the two proportions by making
inferences about p1-p2, the difference in the two
population proportions.
•If the two population proportions are the
same, then p1-p2 = 0.
•The best estimate of p1-p2 is the difference
in the two sample proportions,
x1 x2
pˆ 1  pˆ 2  
n1 n2
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The Sampling Distribution of
pˆ1  pˆ 2
1. The mean of pˆ 1  pˆ 2 is p1  p2 , the difference in
the population proportion s.
2. The standard deviation of pˆ 1  pˆ 2 is SE 
p1q1 p2 q2

.
n1
n2
3. If the sample sizes are large, the sampling distributi on
of pˆ 1  pˆ 2 is approximat ely normal, and SE can be estimated
as SE 
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pˆ 1qˆ1 pˆ 2 qˆ 2

.
n1
n2
Estimating p1-p2
•For large samples, point estimates and their
margin of error as well as confidence intervals
are based on the standard normal (z) distribution.
Point estimate for p1-p2 : pˆ1  pˆ 2
pˆ1qˆ1 pˆ 2 qˆ 2
Margin of Error :  1.96

n1
n2
Confidence interval for p1  p2 :
( pˆ1  pˆ 2 )  z / 2
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pˆ1qˆ1 pˆ 2 qˆ 2

n1
n2
Example
Youth Soccer
Male
Female
Sample size
80
70
Played soccer
65
39
• Compare the proportion of male and female college
students who said that they had played on a soccer team
during their K-12 years using a 99% confidence interval.
pˆ1qˆ1 pˆ 2 qˆ2
( pˆ1  pˆ 2 )  2.58

n1
n2
65 39
.81(.19) .56(.44)
 (  )  2.58

 .25  .19
80 70
80
70
or .06  p1  p2  .44.
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Example, continued
.06  p1  p2  .44
• Could you conclude, based on this confidence interval,
that there is a difference in the proportion of male and
female college students who said that they had played
on a soccer team during their K-12 years?
• The confidence interval does not contains the value
p1-p2 = 0. Therefore, it is not likely that p1= p2. You
would conclude that there is a difference in the
proportions for males and females.
A higher proportion of males than
females played soccer in their youth.
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