sample distribution statistics
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Chapter 5 Sampling Distributions of
Statistics
A sample statistic is an estimateof a population
parameter A sample estimate is subject to sampling error
Sampling distributioncaptures the variation of a
sample estimate around the true parameter value if
repeated samples were drawn from the population
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Sampling distribution of the sample mean
A random sample,, , is drawn from apopulation with mean ()and variance ().The sample mean = is a common estimate for the
population mean . What is the sampling distribution of?
It depends on the population distribution of For~i.i.d.Bern p , = ~ , , then
=
(1 )
For~i.i.d.N , ,~(, ) For ~i.i.d.Exp , = ~(,),
then~(,)2
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Central Limit Theorem
For any arbitrary population distribution
with ()and
(), as sample size , the sampling distribution ofconvergesto a (,), i.e.
/ ~(0,1)
Note for any n, and .
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Empirical distributions of sample mean
x
Density
0 2 4 6 8 10
0.0
0
0.0
5
0.1
0
0.1
5
0.2
0
0.2
5
0.3
0
0.3
5
x
Density
0 2 4 6 8 10
0.0
0.1
0.2
0.3
0.4
0.5
0.6
x
Density
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
x
Density
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
For a (2,1)population
1 5
10 100 4
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Normal approximation to Binomial distribution
If~(, ), then when n is large
(, 1 )
X~Bin(5,0.5)
Density
0 1 2 3 4 5
0.0
0
0.0
5
0.1
0
0.1
5
0.2
0
0.2
5
0.3
0
X~Bin(20,0.5)
Density
5 10 15
0.0
0
0.0
5
0.1
0
0.1
5
X~Bin(10,0.1)
Density
0 1 2 3 4 5
0.0
0.1
0.2
0.3
0.4
X~Bin(100,0.1)
Density
0 5 10 15 20
0.0
0
0.0
2
0.0
4
0.0
6
0.0
8
0.1
0
0.1
2
5, 0.5 20, 0.5
10, 0.1 100, 0.1 5
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A general rule of thumb
10and (1 ) 10 Continuity Correction
0.5 (1)
1 0.5 (1)
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Sampling distribution of the sample variance
The sample variance ( )= is oftenused to estimate the population variance . The samplingdistribution of is also very dependent on the populationdistribution of.
When~(,
), it can be shown that 1
~
~
( 1)
and 4and are statistically independent.
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, denotes the upper critical point of adistribution (Table A.5)
Particularly, ~, and / ,Ex: Find (1) and ;
(2) a, b such that
upper tail area
= 2.5%lower tail area
= 2.5%
, ,
8
2
10,0.05 2
10,0.95
2
8 )a 0.9( 5P b
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Ex: The variance among the repeat measurements is used
to quantify the precision of an instrument. Suppose the
advertised claim for the precision of one kind of
thermometer is 0.01 . If the observed samplevariance of 10 repeat measurements for a thermometer of
this kind is significantly larger than the claimed variance
0.01, then this casts doubt on the advertised claim.What is the threshold value of sample variance so that the
probability of observing a value no less than the cut-off
value is no more than 5%?
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Students t-distribution
For~ , , it is known that ()
/
~ 0,1 . Then, ( )/ ~
denotes a students t-distribution with degrees offreedom, whose p.d.f. is given by
12 2
1
+
, < <
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Students t-distribution
Symmetric around zero
Bell-shaped
as Upper
critical value:
,
, , (Table A.4)
,,11
Ex: Find (1) ,., ,.(2) a such that
< 0.95
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Ex: A soft drink company uses a filling machine to fill cans.
Each 12 oz. can is to contain 355milliliters ofbeverage. The actually filling amount follows a normal
distribution with mean and variance .(1) If is known to be 0.5ml, then what is the probability
that mean content of a six-pack of cans is less than
354.8ml?
(2) If is unknown, the sample variance of the contents ofa six-pack of cans is measured to be 0.6ml, what is theminimum deviation of the sample mean of a six-packfrom such that the probability of observing a samplemean at least distant away from is no more than5%?
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Snedecor-Fishers F-distribution
For
Let
Then
with pdf given by
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F-distribution
,~ , Upper critical value:,, (Table A.6)
Lower critical value:,, ,,
// ~, and ,/ ,,Ex: Find (1),,. , ,,.,,,.,,,.(2) a & b such that < 8,< 0.9
,,,,
Lower tail
area =
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Ex: A company tests samples of a certain product made by
two different suppliers to determine whether the variability
in their products are different. Two samples of 9 and 13 units are drawn from the products of the twosuppliers. A decision rule for declaring the true variance ofthe two suppliers are different is defined as
for some < 1and > 1. Determine the decision rulesuch that
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2 2
1 11 22 2
2 2
ors s
c c
s s
2 22 2 2 21 1
1 1 2 2 1 22 22 2
| | 0.05
s scP Pc
s s
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Sampling distribution of order statistics
Data:,, , ~...()continuous distributionOrdered data:()< ()< < Consider sampling distributions of
() and()
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Sampling distribution of the r-th order statistic
Let ~...[0,1], then its pdf is
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