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.