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Pendugaan Parameter Proposi dan Beda Beda Proporsi Pertemuan 17. Matakuliah: I0134/Metode Statistika Tahun: 2007. Interval Estimation of a Population Proportion. Interval Estimate where: 1 - is the confidence coefficient z /2 is the z value providing an area of - PowerPoint PPT Presentation
Pendugaan Parameter Proposi dan Beda Beda Proporsi Pertemuan 17
Matakuliah : I0134/Metode StatistikaTahun : 2007
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Interval Estimationof a Population Proportion
• Interval Estimate
where: 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 is the sample proportion
p zp pn
/
( )2
1p z
p pn
/
( )2
1
pp
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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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• Interval Estimate of a Population Proportion
where: n = 500, = 220/500 = .44, z/2 = 1.96
.44 + .0435PSI is 95% confident that the proportion of all votersthat favors the candidate is between .3965 and .4835.
p zp pn
/( )
21
p zp pn
/( )
21
pp
. .. ( . )
44 1 9644 1 44
500
. .
. ( . )44 1 96
44 1 44500
Example: Political Science, Inc.
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• Let E = the maximum sampling error mentioned in the precision statement.
• We have
• Solving for n we have
Sample Size for an Interval Estimateof a Population Proportion
E zp pn
/
( )2
1E z
p pn
/
( )2
1
nz p p
E
( ) ( )/ 22
2
1n
z p p
E
( ) ( )/ 22
2
1
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• 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?
Example: Political Science, Inc.
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• Sample Size for Interval Estimate of a Population Proportion At 99% confidence, z.005 = 2.576.
Note: We used .44 as the best estimate of p in theabove expression. If no information is availableabout p, then .5 is often assumed because it providesthe highest possible sample size. If we had usedp = .5, the recommended n would have been 1843.
nz p p
E
( ) ( ) ( . ) (. )(. )
(. )/ 2
2
2
2
2
1 2 576 44 56
031817n
z p p
E
( ) ( ) ( . ) (. )(. )
(. )/ 2
2
2
2
2
1 2 576 44 56
031817
Example: Political Science, Inc.
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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,
.parameter with 1 population binomial
fromdrawn size of sample randomA
1
1
p
n.parameter with 1 population binomial
fromdrawn size of sample randomA
1
1
p
n
.parameter with 2 population binomial
fromdrawn size of sample randomA
2
2
p
n.parameter with 2 population binomial
fromdrawn size of sample randomA
2
2
p
n
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Estimating the Difference between Two Means
•We compare the two proportions by making inferences about p-p, the difference in the two population proportions.
•If the two population proportions are the same, then p1-p= 0.•The best estimate of p1-pis the difference in the two sample proportions,
2
2
1
121 ˆˆ
n
x
n
xpp
2
2
1
121 ˆˆ
n
x
n
xpp
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The Sampling Distribution of 21 ˆˆ pp
.ˆˆˆˆ
SE as
estimated becan SE and normal,ely approximat is ˆˆ of
ondistributi sampling thelarge, are sizes sample theIf .3
.SE is ˆˆ ofdeviation standard The 2.
s.proportion population the
in difference the, is ˆˆ ofmean The 1.
2
22
1
11
21
2
22
1
1121
2121
n
qp
n
qp
pp
n
qp
n
qppp
pppp
.ˆˆˆˆ
SE as
estimated becan SE and normal,ely approximat is ˆˆ of
ondistributi sampling thelarge, are sizes sample theIf .3
.SE is ˆˆ ofdeviation standard The 2.
s.proportion population the
in difference the, is ˆˆ ofmean The 1.
2
22
1
11
21
2
22
1
1121
2121
n
qp
n
qp
pp
n
qp
n
qppp
pppp
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Estimating p1-p
•For large samples, point estimates and their margin of error as well as confidence intervals are based on the standard normal (z) distribution.
2
22
1
11
2121
ˆˆˆˆ1.96 :Error ofMargin
ˆˆ :for estimatePoint
n
qp
n
qp
pp-pp
2
22
1
11
2121
ˆˆˆˆ1.96 :Error ofMargin
ˆˆ :for estimatePoint
n
qp
n
qp
pp-pp
2
22
1
112/21
21
ˆˆˆˆ)ˆˆ(
:for interval Confidence
n
qp
n
qpzpp
pp
2
22
1
112/21
21
ˆˆˆˆ)ˆˆ(
:for interval Confidence
n
qp
n
qpzpp
pp
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Example
• 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.
2
22
1
1121
ˆˆˆˆ58.2)ˆˆ(
n
qp
n
qppp
70
)44(.56.
80
)19(.81.58.2)
70
39
80
65( 19.52.
.44. .06or 21 pp
Youth Soccer Male Female
Sample size 80 70
Played soccer 65 39
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Example, continued
• 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 pp11--pp= 0= 0.. Therefore, it is not likely that pp11= = ppYou would conclude that there is a difference in the proportions for males and females.
44. .06 21 pp 44. .06 21 pp
A higher proportion of males than females played soccer in their youth.