Statistics for Business and Economics: bab 8
Transcript of Statistics for Business and Economics: bab 8
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Slides Prepared by
JOHN S. LOUCKSSt. Edwards University
2002 South-Western /Thomson Learning
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Chapter 8Interval Estimation
Interval Estimation o a !opulation "ean#
Large-Sample Case
Interval Estimation o a !opulation "ean#
Small-Sample Case
$etermining the Sample Si%e Interval Estimation o a !opulation !roportion
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Interval Estimation o a !opulation "ean#Large-Sample Case
Sampling Error
!ro'a'ilit( Statements a'out the SamplingError
Constru)ting an Interval Estimate#
Large-Sample Case *ith +no*n
Cal)ulating an Interval Estimate#
Large-Sample Case *ith ,nno*n
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Sampling Error
The a'solute value o the dieren)e 'et*een
an un'iased point estimate and the populationparameter it estimates is )alled the samplingerror
or the )ase o a sample mean estimating a
population mean the sampling error isSampling Error 3
| |x
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!ro'a'ilit( Statements5'out the Sampling Error
+no*ledge o the sampling distri'ution o
ena'les us to mae pro'a'ilit( statementsa'out the sampling error even though thepopulation mean is not no*n
5 pro'a'ilit( statement a'out the sampling
error is a pre)ision statement
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!re)ision Statement
There is a 1 - pro'a'ilit( that the value oa sample mean *ill provide a samplingerror o or less
!ro'a'ilit( Statements5'out the Sampling Error
/2/2 /2/21 -
o all values
1 - o all valuesx
z x /2
Sampling
distri'ution o
Sampling
distri'ution o x
x
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Interval Estimate o a !opulation "ean#Large-Sample Case n9 &0:
With +no*n
*here# is the sample mean
1 - is the )on;den)e )oe
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Interval Estimate o a !opulation "ean#Large-Sample Case n9 &0:
With ,nno*n
In most appli)ations the value o thepopulation standard deviation is unno*n Wesimpl( use the value o the sample standarddeviation s as the point estimate o the
population standard deviationx z
s
n
/2
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>ational $is)ount has 260 retail outlets
throughout the ,nited States >ationalevaluates ea)h potential lo)ation or a ne*retail outlet in part on the mean annualin)ome o the individuals in the mareting area
o the ne* lo)ationSampling )an 'e used to develop an
interval estimate o the mean annual in)omeor individuals in a potential mareting area or
>ational $is)ount5 sample o si%e n3 &6 *as taen The
sample mean is ?21100 and the samplestandard deviation s is ?.400 We *ill use =4 as the )on;den)e )oeational $is)ount In)
x
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!re)ision Statement
There is a =4 pro'a'ilit( that the value o asample mean or >ational $is)ount *ill providea sampling error o ?1.70 or lessAAdetermined as ollo*s#
=4B o the sample means that )an 'eo'served are *ithin 1=6 o thepopulation mean
I then 1=6 3 1.70
x
!"#$
!""%&===
n
sx
x
E@ample# >ational $is)ount In)
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E@ample# >ational $is)ount In)
Interval Estimate o the !opulation "ean#
,nno*n
Interval Estimate o is#
?21100 ?1.70
or ?1=6&0 to ?22470
We are =4B )on;dent that the interval )ontainsthe
population mean
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Interval Estimation o a !opulation "ean#Small-Sample Case nD &0:
!opulation is >ot >ormall( $istri'uted
The onl( option is to in)rease the sample si%e to
n9 &0 and use the large-sample interval-estimation
pro)edures
!opulation is >ormall( $istri'uted and is+no*n
The large-sample interval-estimation pro)edure
)an
'e used
!opulation is >ormall( $istri'uted and is,nno*n
The appropriate interval estimate is 'ased on a
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t $istri'ution
The t distri'ution is a amil( o similar
pro'a'ilit( distri'utions 5 spe)i;) t distri'ution depends on a
parameter no*n as the degrees o reedom 5s the num'er o degrees o reedom
in)reases the dieren)e 'et*een the tdistri'ution and the standard normalpro'a'ilit( distri'ution 'e)omes smaller andsmaller
5 t distri'ution *ith more degrees o reedom
has less dispersion The mean o the t distri'ution is %ero
Inter al Estimation o a !opulation "ean
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Interval Estimation o a !opulation "ean#Small-Sample Case nD &0: *ith
,nno*n Interval Estimate
*here 1 - 3 the )on;den)e )oe
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E@ample# 5partment ents
Interval Estimation o a !opulation "ean#
Small-Sample Case nD &0: *ith ,nno*n
5 reporter or a student ne*spaper is*riting an
arti)le on the )ost o o-)ampus housing 5sample o 10 one-'edroom units *ithin a hal-mile o )ampus resulted in a sample mean o?440 per month and a sample standard
deviation o ?60Let us provide a =4B )on;den)e interval
estimate o the mean rent per month or thepopulation o one-'edroom units *ithin a hal-
mile o )ampus WeFll assume this populationto 'e normall distri'uted
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tGalue
5t =4B )on;den)e 1 - 3 =4 3 04 and /23 024
t024is 'ased on n - 1 3 10 - 1 3 = degrees o
reedom
In the t distri'ution ta'le *e see that t0243
2262
Degrees Area in Upper Tailof Freedom .10 .05 .025 .01 .005
. . . . . .
7 1.14 18=4 2&64 2==8 &.==
8 1&=7 1860 2&06 28=6 &&44
9 1&8& 18&& 2262 2821 &240
10 1&72 1812 2228 276. &16=
. . . . . .
E@ample# 5partment ents
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Interval Estimation o a !opulation "ean#
Small-Sample Case nD &0: *ith ,nno*n
440 .2=2
or ?40708 to ?4=2=2
We are =4B )on;dent that the mean rent permonth or the population o one-'edroom units*ithin a hal-mile o )ampus is 'et*een?40708 and ?4=2=2
x ts
n
."2!
'"
$"
2$2.2!!"
E@ample# 5partment ents
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Let E3 the ma@imum sampling error
mentioned in the pre)ision statement Eis the amount added to and su'tra)ted rom
the point estimate to o'tain an intervalestimate
Eis oten reerred to as the margin o error We have
Solving or n*e have
Sample Si%e or an Interval Estimateo a !opulation "ean
E zn
=
/2
n z
E=
( )/ 22 2
2
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1=1=
E@ample# >ational $is)ount In)
Sample Si%e or an Interval Estimate o a
!opulation "ean
Suppose that >ationalFs management team*ants an estimate o the population meansu)h that there is a =4 pro'a'ilit( that the
sampling error is ?400 or lessHo* large a sample si%e is needed to meet
the reuired pre)isionJ
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E@ample# >ational $is)ount In)
Sample Si%e or Interval Estimate o a !opulation
"ean
5t =4B )on;den)e %0243 1=6
e)all that 3 .400
Solving or n*e have
We need to sample &12 to rea)h a desiredpre)ision o
?400 at =4B )on;den)e
zn
/2 !""=
2 2
2
1=6: .400:&1117
400:n= =
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Interval Estimationo a !opulation !roportion
Interval Estimate
*here# 1 - is the )on;den)e )oe
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E@ample# !oliti)al S)ien)e In)
Interval Estimation o a !opulation !roportion
!oliti)al S)ien)e In) !SI: spe)iali%es invoter polls and surve(s designed to eeppoliti)al o
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Interval Estimate o a !opulation !roportion
*here# n3 400 3 220/400 3 .. z/2 =
1=6
.. 0.&4
!SI is =4B )on;dent that the proportion o allvoters
that avors the )andidate is 'et*een &=64and .8&4
p z p p
n
/
( )2
'
p
. . . ( . )
&& ' *$ && ' &&
!""
E@ample# !oliti)al S)ien)e In)
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2.2.
Let E3 the ma@imum sampling error
mentioned in the pre)ision statement We have
Solving or n*e have
Sample Si%e or an Interval Estimateo a !opulation !roportion
E z p p
n=
/
( )2
'
n z p p
E=
( ) ( )/ 22
2
'
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Sample Si%e or an Interval Estimate o a
!opulation !roportionSuppose that !SI *ould lie a ==
pro'a'ilit( that the sample proportion is *ithin 0& o the population proportion
Ho* large a sample si%e is needed to meetthe reuired pre)isionJ
E@ample# !oliti)al S)ien)e In)
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Sample Si%e or Interval Estimate o a
!opulation !roportion 5t ==B )on;den)e %0043 2476
>ote# We used .. as the 'est estimate opinthe
a'ove e@pression I no inormation isavaila'le
a'outp then 4 is oten assumed 'e)ause itprovides
the highest possi'le sample si%e I *e hadused
p3 4 the re)ommended n*ould have 'een18.&
n z p p
E=
=
( ) ( ) ( . ) (. )(. )
(. )
/ 22
2
2
2
' 2 !$ && !$
"#'+'
E@ample# !oliti)al S)ien)e In)
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End o Chapter 8