statistics for business and economics: bab 8

7/24/2019 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


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

x


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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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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


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=

. . . . . .



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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!!"



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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=


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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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

. . . ( . )

&& ' *$ && ' &&

!""



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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



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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 !$ && !$

"#'+'



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End o Chapter 8

statistics for business and economics: bab 8

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