lec 3 (SII)
Transcript of lec 3 (SII)
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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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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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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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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&le
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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&le