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McGraw-Hill/Irwin Copyright 2010 by The McGraw-Hill Companies, Inc. All rights reserved.

Sampling Methods andthe Central Limit Theorem

Chapter 8(LECTURE 3)

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8-2

Central Limit Theorem

If the poplation follo!s a normal pro"a"ilit# distri"tion$ then for

an# sample si%e the sampling distri"tion of the sample mean !ill

also "e normal&

If the poplation distri"tion is s#mmetri'al ("t not normal)$ the

normal shape of the distri"tion of the sample mean emerge !ithsamples as small as &

If a distri"tion that is s*e!ed or has thi'* tails$ it ma# re+ire

samples of 3 or more to o"ser,e the normalit# featre&

The mean of the sampling distri"tion e+al to and the ,arian'e

e+al to .2/n.

CENTRAL LIMIT THEOREM If all samples of a parti'lar si%e are

sele'ted from an# poplation$ the sampling distri"tion of the sample

mean is appro0imatel# a normal distri"tion& This appro0imation

impro,es !ith larger samples&

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8-3

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8-1

Standard Error of the Mean

nX

=

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Using the Sampling

Distribtion of the Sample Mean !Sigma "no#n$

If a poplation follo!s the normal distri"tion$ the sampling

distri"tion of the sample mean !ill also follo! the normal

distri"tion&

If the shape is *no!n to "e nonnormal$ "t the sample 'ontainsat least 3 o"ser,ations$ the 'entral limit theorem garantees the

sampling distri"tion of the mean follo!s a normal distri"tion&

To determine the pro"a"ilit# a sample mean falls !ithin a

parti'lar region$ se

n

Xz

=

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8-4

The 5alit# 6ssran'e 7epartment for Cola$ In'&$ maintainsre'ords regarding the amont of 'ola in its m"o "ottle& Thea'tal amont of 'ola in ea'h "ottle is 'riti'al$ "t ,aries asmall amont from one "ottle to the ne0t& Cola$ In'&$ does not

!ish to nderfill the "ottles& 9n the other hand$ it 'annot o,erfillea'h "ottle& Its re'ords indi'ate that the amont of 'ola follo!sthe normal pro"a"ilit# distri"tion& The mean amont per "ottleis 3&2 on'es and the poplation standard de,iation is &1on'es&

6t 8 6&M& toda# the +alit# te'hni'ian randoml# sele'ted 4 "ottlesfrom the filling line& The mean amont of 'ola 'ontained in the"ottles is 3&38 on'es&

Is this an nli*el# reslt: Is it li*el# the pro'ess is ptting too m'hsoda in the "ottles: To pt it another !a#$ is the sampling errorof &8 on'es nsal:

Using the Sampling Distribtion of the Sample Mean

!Sigma "no#n$ % E&ample

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Step

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Step 2

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?hat do !e 'on'lde:

It is nli*el#$ less than a 1 per'ent 'han'e$ !e'old sele't a sample of 4 o"ser,ationsfrom a normal poplation !ith a mean of 3&2on'es and a poplation standard de,iationof &1 on'es and find the sample mean

e+al to or greater than 3&38 on'es&?e 'on'lde the pro'ess is ptting too m'h

'ola in the "ottles&

Using the Sampling Distribtion of the Sample Mean

!Sigma "no#n$ % E&ample