ka-fu wong © 2007 econ1003: analysis of economic data supplement8-1 additional real-life examples...
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Supplement8-1 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data
Supplement 8: Additional real-life examples Additional real-life examples (proportions)(proportions)
*The part of biasedness (including the proof) is a result of a correspondence between Dr. Ka-fu Wong and YueShen Zhou. The example was drawn from a clip sent over by Nipun Sharma. Use it at your own risks. Comments, if any, should be sent to [email protected].
Supplement8-2 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data
Mean and variance (1)
Suppose a proportion p of the population is female. An observation is randomly drawn from the population. Code x = 1 if the drawn observation is female. Code x = 0 if the drawn observation is male. What is the population mean and variance of this random variable X?
E(X) = (1)Prob(x=1) + (0)Prob(x=0) = (1)p + (0)(1-p) = p
Var(X) = (1-p)2Prob(x=1) + (0-p)2Prob(x=0) = (1-p)2p + p2(1-p) = (1-p) p [1-p + p] = (1-p)p
Supplement8-3 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data
Mean and variance (2)
Suppose a proportion p of the population is female. A sample of n observations is randomly drawn with replacement from the population. Code x = 1 if a drawn observation is female, 0 otherwise. What is the population mean and variance of m = (x1+…+xn)/n?
E(m) = E[(x1+…+xn)/n] = [E(x1) + E(x2) + … + E(xn)]/n= E(X) = p
Var(m) = Var[(x1+…+xn)/n] = Var[(x1+…+xn)]/n2 = [Var (x1)+…+Var(xn)]/n2 = Var(X)/n = (1-p)p/n
Supplement8-4 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data
Central limit theorem of proportion
Let x1,…,xn be a iid sample from a population with p proportion of success.
(Failure coded as 0 and success as 1.)
∑xi/n is simply the proportion of success and hence the simple average of the outcomes from the n trials.
∑xi/n will be approximately normal according to CLT.
n
pppNp
n
xp
n
i i
)1(,~ˆ
ˆ 1
Supplement8-5 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data
Do you think Chinese officials spent too much government money on the following?
Base on a poll of 18,000 persons.
Supplement8-6 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data
Can we construct the 95% confidence intervals for the population proportion?
Standard error= [p(1-p)/n]1/2
n p std errorlower limit upper limit
18000 0.958 0.001495103 0.95507 0.9609303
18000 0.862 0.002570733 0.856961 0.8670385
18000 0.86 0.002586289 0.854931 0.865069
18000 0.85 0.002661453 0.844784 0.8552164
18000 0.807 0.00294157 0.801235 0.8127654
18000 0.802 0.002970185 0.796179 0.8078215
18000 0.679 0.003479775 0.67218 0.6858202
18000 0.5 0.00372678 0.492696 0.5073044
Supplement8-7 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data
Why would some books used a different formula?
The population proportion is unknown. Thus, an estimate of the variance of sample proportion will be
n
pppVar
n
pppNp
n
xp
n
i i
)1()ˆ(
)1(,~ˆ
ˆ 1
?)ˆ1(ˆ
)ˆ(n
ppparEstimatedV
n
pp
n
ppE
n
pp
n
ppE
)1(
1
)ˆ1(ˆ
)1()ˆ1(ˆ
When n is large, the difference between two estimators of sample variance are negligible. This is why some books use n, some use (n-1).
Supplement8-8 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data
Why would some books used a different formula?
n
pppVar
n
pppNp
nxp
)1()ˆ(
))1(
,(~ˆ
/ˆ
n
pp
n
ppE
n
pp
n
ppE
)1(
1
)ˆ1(ˆ
)1()ˆ1(ˆ
When n is large, the difference between two estimators of sample variance are negligible. This is why some books use n, some use (n-1).
is a biased estimator for n
pp )ˆ1(ˆ n
pp )1(
is an unbiased estimator for 1
)ˆ1(ˆ
n
ppn
pp )1(
Supplement8-9 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data
Proof: A biased estimator
n
pp
n
n
n
ppn
pppnpn
npppnpn
pn
ppp
n
pEpVarpn
pEpEn
ppEnn
ppE
)1(1)1()1(
)1()1(1
)1(1
)1(1
])ˆ()ˆ([1
)]ˆ()ˆ([1
)ˆˆ(1)ˆ1(ˆ
2
2
22
2
2
2
2
Supplement8-10 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data
Proof: An unbiased estimator
n
pp
nn
ppn
pppnpnn
npppnpnn
pn
ppp
n
pEpVarpn
pEpEn
ppEnn
ppE
)1(
)1(
)1()1(
)1()1()1(
1
)1()1(
1
)1(
1
1
])ˆ()ˆ([1
1
)]ˆ()ˆ([1
1
)ˆˆ(1
1
1
)ˆ1(ˆ
2
2
2
2
2
Supplement8-11 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data
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Supplement 8: Supplement 8: Additional real-life examplesAdditional real-life examples