STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems...

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


Exponential distribution, =0.5 Sample size=50,50 random samples


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.


-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


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


Distribution of sample means

paXaXP

paXaP

paXP

nn

n

n

][

a=?

n1

n2


Confidence Interval

is called the confidence coefficient.


One-sided CI


Example


• 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


Remarks


• Let’s recall the procedures of determining before drawing any random sample:


Methods of Finding Confidence Intervals -The Pivotal Quantity Method


Now, if for each possible sample value


),...,( 1 nxx

211 );,,( qxxqq n ),...,()(),...,( 1211 nn xxtxxt

Remarks


skipped


Definition of Location parameter


Example of location parameter


Definition of scale parameter


Example of scale parameter


Example


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


• 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?


Review of sampling distributions


Normal distributions


Chi-square distribution


Standard normal and chi-square distributions


1~

n

n

n tnS

XT


Student’s t-distribution

Student’s t distribution with k degrees of freedom


Back to discussion on confidence intervals


Confidence interval for the mean


1~

n

n

n tnS

XT


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


(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


2

1,2

1

22

2

1,2

2 )1()1(

n

n

n

n SnSn


(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


Confidence Interval for Difference in Means


),(~

),(~

2

2

2

1

nNY

mNX

n

m

),(~)(22

21 nmNYX nm

)1,0(~)()(

22

21 N

nm

YXZ nm


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


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


confidence interval of is)%1(100 )( 21

))(,)(( 2

2,

2

2,22

PnmnmPnmnm Smn

nmtYXS

mn

nmtYX


Some pivotal functions for samples of size n


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


• However, we cannot find since p is unknown.

),(~ˆ),(~n

pqpNp

n

XnpqnpNX

)1,0(~ˆ

Nnpq

pp

n

pq


• For large n,

)1,0(~)ˆ1(ˆ

ˆN

npp

pp

1** zZzP


STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems...

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