statistics interval estimation professor ke-sheng cheng department of bioenvironmental systems...
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STATISTICS INTERVAL ESTIMATION
Professor Ke-Sheng ChengDepartment of Bioenvironmental Systems Engineering
National Taiwan University
Exponential distribution (negative exponential distribution)
.0)();( ),0[ ,xIexf x
X
/1][ XE
t for t
tm
XVar
X )(
/1][ 2
Mean rate of occurrence in a Poisson process.
12/6/2011 2Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Exponential distribution, =0.5 Sample size=10,50 random samples
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Exponential distribution, =0.5 Sample size=50,50 random samples
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Variation of sample means• 6 random samples (sample size n= 50) of a
normal density with unit variance and an unknown mean [N( , 1)].
• 6 random samples (sample size n= 500) of a normal density with unit variance and the same unknown mean [N( , 1)].
• Examine the variation of sample means.
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-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0 1 2 3 4 5 6
n=50
n=500
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Estimating the mean
• Point estimation – – Increasing the sample size reduces the standard
deviation of our estimate.– If the sample size is given (we are given a random
sample), how do we see (interpret) our sample estimate? How do we (or in what sense) judge whether our estimate is close to the population mean?
• Interval estimation
12/6/2011 7Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Distribution of sample means
paXaXP
paXaP
paXP
nn
n
n
][
a=?
n1
n2
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Confidence Interval
is called the confidence coefficient.
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One-sided CI
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Example
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• constitutes a random interval and is a confidence interval for .
),( 21 TT
]23
2[
]66
[
]66
[
n
XP
nX
nP
nX
nXP
n
n
nn
12/6/2011 12Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 13Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 14Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Remarks
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12/6/2011 16Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 17Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 18Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 19Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• Let’s recall the procedures of determining before drawing any random sample:
12/6/2011 20Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 21Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Methods of Finding Confidence Intervals -The Pivotal Quantity Method
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Now, if for each possible sample value
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),...,( 1 nxx
211 );,,( qxxqq n ),...,()(),...,( 1211 nn xxtxxt
Remarks
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skipped
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Definition of Location parameter
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Example of location parameter
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Definition of scale parameter
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Example of scale parameter
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Example
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1111
1111
111
)( nYXXnYXP
nYXnXnYXP
nYnnYPYYPYYP
n
ii
n
iin
n
ii
n
ii
n
iin
n
ii
nnn
)()();(2
1,21
)2
1,
2
1(
xIxIxf
12/6/2011 32Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 33Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 34Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• In the above example, the confidence coefficient is determined after the random sample is obtained. It is not pre-determined (out of our control).
• What if we want to find the confidence interval of Θ with a pre-determined confidence coefficient?
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Review of sampling distributions
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Normal distributions
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12/6/2011 38Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 39Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 40Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Chi-square distribution
12/6/2011 41Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 42Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 43Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Standard normal and chi-square distributions
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1~
n
n
n tnS
XT
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Student’s t-distribution
Student’s t distribution with k degrees of freedom
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Back to discussion on confidence intervals
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Confidence interval for the mean
12/6/2011 48Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 49Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 50Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 51Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 52Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 53Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
1~
n
n
n tnS
XT
12/6/2011 54Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Confidence interval for the variance
• Given a random sample from a normal distribution with mean and variance . We want to determine the confidence interval of .
nXX ,....,1
22
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(i) is unknown
is a pivotal quantity and has a Chi-square distribution with d.o.f. n-1.
2
2
21
2
)1()(
n
n
ini Sn
XXQ
1
22
2
2
22
2
1)1()1()1(
q
Sn
q
Snq
Snq nnn
12/6/2011 56Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 57Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
2
1,2
1
22
2
1,2
2 )1()1(
n
n
n
n SnSn
12/6/2011 58Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
(ii) if is known
is a pivotal quantity and has a Chi-square distribution with d.o.f. n.
n
i
i
n
ii
XX
Q1
22
1
2
)(
)(
1
1
2
2
2
1
2
221
2
121
)()(
)(
q
X
q
X
qX
qqQq
n
ii
n
ii
n
ii
12/6/2011 59Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 60Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Confidence Interval for Difference in Means
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),(~
),(~
2
2
2
1
nNY
mNX
n
m
),(~)(22
21 nmNYX nm
)1,0(~)()(
22
21 N
nm
YXZ nm
12/6/2011 62Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
12/6/2011 63Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
has a t-distribution with d.o.f. (m+n-2)
)2(])()([
)()(
21 1
22
21
nmYYXXmn
nm
YX
nmU
ZT
m
i
n
inimi
nm
2
21 )()(
P
nm
Smn
nm
YX
)2(])()([1 1
222
nmYYXXSm
i
n
inimiP
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1][2,2,
22nmnm
tTtP
2,2
212,
22
)()(
nm
P
nmnm
t
Smn
nm
YXt
2
2,212
2,22
)()( PnmnmPnmnm Smn
nmtYXS
mn
nmtYX
12/6/2011 65Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
confidence interval of is)%1(100 )( 21
))(,)(( 2
2,
2
2,22
PnmnmPnmnm Smn
nmtYXS
mn
nmtYX
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Some pivotal functions for samples of size n
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Confidence interval for a population proportion, p
• Let X be a random variable with binomial density Binom(n, p). A random number of X, say x, is given, and we want to find a 95% confidence interval for p.
• As n approaches infinity, X can be approximated by a normal distribution with mean np and variance npq, i.e.,
nxp /ˆ
12/6/2011 68Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• However, we cannot find since p is unknown.
),(~ˆ),(~n
pqpNp
n
XnpqnpNX
)1,0(~ˆ
Nnpq
pp
n
pq
12/6/2011 69Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• For large n,
)1,0(~)ˆ1(ˆ
ˆN
npp
pp
1** zZzP
12/6/2011 70Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.