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
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? [uncertainties]
• Interval estimation
04/20/23 2Laboratory 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
][
Given a particular random sample, 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
04/20/23 7Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 8Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 9Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Remarks
04/20/23 10Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 11Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 12Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 13Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 14Laboratory 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:
04/20/23 15Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 16Laboratory 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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04/20/23 21Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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
04/20/23 27Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 28Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 29Laboratory 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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04/20/23 33Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 34Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 35Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Chi-square distribution
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04/20/23 37Laboratory 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
04/20/23 39Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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
04/20/23 42Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 43Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 44Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 45Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 46Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 47Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
1~
n
n
n tnS
XT
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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
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04/20/23 51Laboratory 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
04/20/23 52Laboratory 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
04/20/23 53Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 54Laboratory 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
04/20/23 56Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
04/20/23 57Laboratory 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
04/20/23 58Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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
04/20/23 59Laboratory 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
04/20/23 61Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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
04/20/23 62Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
• Two-sided confidence interval of the population proportion p is given by
),(~ˆ),(~n
pqpNp
n
XnpqnpNX
)1,0(~ˆ
Nnpq
pp
04/20/23 63Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
)/(1
4//)ˆ1(ˆ)2/(ˆ2
2/1
222/12/1
22/1
nz
nznppznzp
)%1(100
04/20/23 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
64
)/(1
4//)ˆ1(ˆ)2/(ˆ
)/(1
/ˆ)(ˆ))2/(()2/(ˆ2ˆ)2/(ˆ
)/(1
/ˆ)())2/(ˆ()2/(ˆ
)(2
ˆ)(4)ˆ2()ˆ2(
0ˆ)1ˆ2
()1(
0ˆ)ˆ2()(,ˆˆ2
)ˆ(/)1(,ˆ/)1(
/)1(
ˆ
2
222
2
22222222
2
22222
2
22222
222
22
222222222
22
2/1
nz
nznppznzp
nz
npzpnznzppnzp
nz
npznnzpnzpp
zn
pnznzpnzpnp
pz
np
z
pnp
z
n
pnpzpnpznpzpzpnpnpnp
ppnppzppnppz
zznpp
pp Let
• For large n,
)1,0(~)ˆ1(ˆ
ˆN
npp
pp
1** zZzP
04/20/23 65Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
Determining required sample size for a test with desired margin of errors
• In estimation of the population proportion, if we like the margin of error to be no greater than , how large should the sample size be?
• The margin of error is . However, we do not know in advance.
04/20/23 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
66
nppz )ˆ1(ˆ2/1
p̂
nz
nzpLHS
nppz
2
22/1
2/1
2/1
4
)(
)4/(1.5.0ˆ whenmaximum its has
/)ˆ1(ˆ
Confidence interval for difference in means
04/20/23 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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04/20/23 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Confidence interval for difference in means when the population variances are different and unknown.