sample distribution statistics

8/11/2019 sample distribution statistics

1/17

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


3/17

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


5/17

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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7/17

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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8/17

, 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


9/17

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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10/17

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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11/17

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


12/17

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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13/17

Snedecor-Fishers F-distribution

For

Let

Then

with pdf given by

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14/17

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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15/17

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

15

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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sample distribution statistics