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![Page 1: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/1.jpg)
STATISTICS INTERVAL ESTIMATION
Professor Ke-Sheng ChengDepartment of Bioenvironmental Systems Engineering
National Taiwan University
![Page 2: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/2.jpg)
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.
![Page 3: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/3.jpg)
Exponential distribution, =0.5 Sample size=10,50 random samples
12/6/2011 3Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 4: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/4.jpg)
Exponential distribution, =0.5 Sample size=50,50 random samples
12/6/2011 4Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 5: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/5.jpg)
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.
12/6/2011 5Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 6: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/6.jpg)
-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
12/6/2011 6Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 7: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/7.jpg)
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.
![Page 8: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/8.jpg)
Distribution of sample means
paXaXP
paXaP
paXP
nn
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n
][
a=?
n1
n2
12/6/2011 8Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 9: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/9.jpg)
Confidence Interval
is called the confidence coefficient.
12/6/2011 9Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 10: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/10.jpg)
One-sided CI
12/6/2011 10Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 11: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/11.jpg)
Example
12/6/2011 11Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 12: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/12.jpg)
• constitutes a random interval and is a confidence interval for .
),( 21 TT
]23
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[
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XP
nX
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nX
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n
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nn
12/6/2011 12Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/6/2011 13Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 14: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/14.jpg)
12/6/2011 14Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 15: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/15.jpg)
Remarks
12/6/2011 15Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/6/2011 16Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 17: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/17.jpg)
12/6/2011 17Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 18: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/18.jpg)
12/6/2011 18Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 19: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/19.jpg)
12/6/2011 19Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 20: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/20.jpg)
• 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.
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12/6/2011 21Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 22: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/22.jpg)
Methods of Finding Confidence Intervals -The Pivotal Quantity Method
12/6/2011 22Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 23: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/23.jpg)
Now, if for each possible sample value
12/6/2011 23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
),...,( 1 nxx
211 );,,( qxxqq n ),...,()(),...,( 1211 nn xxtxxt
![Page 24: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/24.jpg)
Remarks
12/6/2011 24Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 25: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/25.jpg)
skipped
12/6/2011 25Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/6/2011 26Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Definition of Location parameter
12/6/2011 27Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Example of location parameter
12/6/2011 28Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Definition of scale parameter
12/6/2011 29Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Example of scale parameter
12/6/2011 30Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Example
12/6/2011 31Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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1111
1111
111
)( nYXXnYXP
nYXnXnYXP
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n
ii
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ii
n
iin
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nnn
)()();(2
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2
1(
xIxIxf
12/6/2011 32Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 33: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/33.jpg)
12/6/2011 33Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 34: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/34.jpg)
12/6/2011 34Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 35: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/35.jpg)
• 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?
12/6/2011 35Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 36: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/36.jpg)
Review of sampling distributions
12/6/2011 36Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 37: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/37.jpg)
Normal distributions
12/6/2011 37Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 38: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/38.jpg)
12/6/2011 38Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 39: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/39.jpg)
12/6/2011 39Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 40: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/40.jpg)
12/6/2011 40Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 41: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/41.jpg)
Chi-square distribution
12/6/2011 41Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 42: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/42.jpg)
12/6/2011 42Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 43: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/43.jpg)
12/6/2011 43Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 44: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/44.jpg)
Standard normal and chi-square distributions
12/6/2011 44Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
![Page 45: STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.](https://reader035.fdocuments.net/reader035/viewer/2022081414/55147f43550346d36e8b48bb/html5/thumbnails/45.jpg)
1~
n
n
n tnS
XT
12/6/2011 45Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Student’s t-distribution
Student’s t distribution with k degrees of freedom
12/6/2011 46Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Back to discussion on confidence intervals
12/6/2011 47Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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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.
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12/6/2011 49Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/6/2011 50Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/6/2011 51Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/6/2011 52Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/6/2011 53Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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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.
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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
12/6/2011 55Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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(i) is unknown
is a pivotal quantity and has a Chi-square distribution with d.o.f. n-1.
2
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ini Sn
XXQ
1
22
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q
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Snq nnn
12/6/2011 56Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/6/2011 57Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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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.
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(ii) if is known
is a pivotal quantity and has a Chi-square distribution with d.o.f. n.
n
i
i
n
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XX
Q1
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12/6/2011 59Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/6/2011 60Laboratory 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
12/6/2011 61Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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has a t-distribution with d.o.f. (m+n-2)
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confidence interval of is)%1(100 )( 21
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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
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• However, we cannot find since p is unknown.
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• For large n,
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