007 Confidence Interval

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© 2002 Prentice-Hall, Inc. Chap 8-1

Basic Business Statistics(8th Edition

Chapter 8

Con!dence Inter"alEsti#ation



© 2001 Prentice-Hall, Inc.Chap 8-2

Esti#ation Process

Mean, , isunknown

Population $ando#Sa#ple

MeanX = 50

Sample

I am 95%

confident that

is between 40& 0!



© 2001 Prentice-Hall, Inc.Chap 8-%

Point Esti#ates

Esti#atePopulation

Para#eters &

'ith SampleStatistics

ean

Proportion

)ariance

*i+erence

µ

p

2σ

1 2 µ µ −

X

S P

2S

1 2 X X −



© 2001 Prentice-Hall, Inc.Chap 8-

Interval EstimateInterval Estimatess

Pro"ides rane o "alues /aes into consideration "ariation in sa#ple

statistics ro# sa#ple to sa#ple Is ased on oser"ation ro# one sa#ple

i"es inor#ation aout closeness to unno'npopulation para#eters

Is stated in ter#s o le"el o con!dence 3e"er 1004 certain




Con!dence Inter"alEsti#ates

Mean

σ Unknown

Confidence

Intervals

Proportion

σ Known




Con!dence Inter"al or( 7no'n

ssu#ptionsssu#ptions

Population standard de"iation is no'nPopulation standard de"iation is no'n

Population is nor#all9 distriutedPopulation is nor#all9 distriuted

I population is not nor#al, use lare sa#pleI population is not nor#al, use lare sa#ple

Con!dence inter"al esti#ate

µ σ

/ 2 / 2 X Z X Z

n nα α

σ σ µ − ≤ ≤ +


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Chap 8-:

;e"el o Con!dence;e"el o Con!dence

*enoted 9

relati"e re<uenc9 interpretation

In the lon run, o all thecon!dence inter"als that can e constructed'ill contain the unno'n para#eter

speci!c inter"al 'ill either contain or notcontain the para#eter

3o proailit9 in"ol"ed in a speci!c inter"al

( )100 1 %α −

( )100 1 %α −


http://slidepdf.com/reader/full/007-confidence-interval 8/22© 2001 Prentice-Hall, Inc.Chap 8-8

Inter"al and ;e"el oCon!dence

Confdence Interval s

Inter"als

e=tend

ro#

to

o inter"alsconstructedcontain >

donot.

_ Sa#plin *istriution o the ean

X X Z σ −

X σ

/ 2α / 2α

X X

µ µ =

1 α −

X X Z σ +

( )1 100%α −

µ

100 %α

/ 2 X Z

α µ σ +/ 2 X

Z α

µ σ −


http://slidepdf.com/reader/full/007-confidence-interval 9/22© 2001 Prentice-Hall, Inc.Chap 8-?

@actors +ectinInter"al Aidth (Precision

*ata "ariation easured 9

Sa#ple sie

;e"el o con!dence

Inte"#als $tend f"om

© 1984-1994 T/Maker Co.

X - Z σ

to X + Z σ

σ

X n

σ σ =

( )100 1 %α −



*eter#inin Sa#ple Sie (Cost

Too Big:

$e<uirestoo #an9

resources

Toosmall:

AonDtdo the o



*eter#inin Sa#ple Sie or ean*eter#inin Sa#ple Sie or ean

Ahat sa#ple sie is needed to e ?04 con!dent

o ein correct 'ithin F 5G pilot stud9

suested that the standard de"iation is 5.

$ound p

( )2 22 2

2 2

1.645 45219.2 220

Error 5

Z n

σ = = = ≅



ssu#ptions Population standard de"iation is unno'n

Population is nor#all9 distriuted I population is not nor#al, use lare sa#ple

se studentDs t distriution


Con!dence Inter"al or( nno'n

µ

σ

/ 2, 1 / 2, 1n n

S S X t X t

n nα α

µ − −

− ≤ ≤ +


http://slidepdf.com/reader/full/007-confidence-interval 13/22© 2001 Prentice-Hall, Inc.Chap 8-1%

StudentDs t *istriution

Z

t 0

t (df = 5)

t (df = 1)ell'ShapedS(mmet"ic

)*atte"+

ails

Standa"d

-o"mal



*erees o @reedo# (*erees o @reedo# (df df

Number of observations that are free to varyNumber of observations that are free to vary

after sample mean has been calculatedafter sample mean has been calculated

E=a#ple ean o % nu#ers is 2

de."ees of f"eedom= n '/

= '/

= 1

1

2

3

1 (or any number)2 (or any number)

3 (cannot vary)

X X

X

=

=

=



StudentDs t /ale

Upper !ail "rea

df .25 .10 #$5

1 1.000 3.078 6.314

% 0.817 1.886

%#&%$

3 0.765 1.638 2.353

t $ %#&%$

t Values

et! n " 3

#$ " n 1 " 2

α " .10 α/2 ".05

' % = #$5



E=a#ple

/ 2, 1 / 2, 1

& &50 2.0639 50 2.063925 25

46.69 53.30

n n

S S X t X t

n nα α

µ

µ

µ

− −− ≤ ≤ +

− ≤ ≤ +

≤ ≤

' ran#om samle o$ 25 as 50 an# &.

*et u a 95% con$+#ence +nterval est+mate $or

n X S

µ

= = =


http://slidepdf.com/reader/full/007-confidence-interval 17/22© 2001 Prentice-Hall, Inc.Chap 8-1:

Con!dence Inter"alCon!dence Inter"al

Esti#ate or ProportionEsti#ate or Proportion

ssu#ptions /'o cateorical outco#es

Population ollo's ino#ial distriution

3or#al appro=i#ation can e used i

and


5np ≥ ( )1 5n p− ≥

( ) ( )/ 2 / 2

1 1S S S S

S S

p p p p p Z p p Z

n nα α

− −− ≤ ≤ +



Chap 8-18

E=a#pleE=a#ple

rando# sa#ple o 00 "oters sho'ed %2

preerred candidate . Set up a ?54

con!dence inter"al esti#ate or p.

( ) ( )

( ) ( )

/ /

1 1

.0& 1 .0& .0& 1 .0&.0& 1.96 .0& 1.96400 400

.053 .10,

s s s s

s s

p p p p p Z p p Z

n n

p

p

α α 2 2

− −− ≤ ≤ +

− −− ≤ ≤ +

≤ ≤



Chap 8-1?

*eter#inin Sa#ple Sie or

Proportion

Jut o a population o 1,000, 'e rando#l9

selected 100, o 'hich %0 'ere deecti"e.

Ahat sa#ple sie is needed to e 'ithin

F 54 'ith ?04 con!denceG

$ound

( ) ( ) ( )2 2

2 2

1 1.645 0.3 0.,

Error 0.05

22,.3 22&

Z p p

n

−= =

= ≅



Chap 8-20

Condence Interval for Population TotalCondence Interval for Population Total

Amount Amount

Point esti#ate


NX

( ) ( )( )

/ 2, 11

n N nS NX N t N n

α −−±−



Chap 8-21

Con!dence Inter"al or Population /otalCon!dence Inter"al or Population /otal

E=a#pleE=a#ple

n auditor is aced 'ith apopulation o 1000"ouchers and 'ants toesti#ate the total "alue

o the population. sa#ple o 50 "ouchers isselected 'ith a"erae"oucher a#ount o

K10:6.%?, standardde"iation o K2:%.62.Set up the ?54con!dence inter"alesti#ate o the total

a#ount or the



Chap 8 22

E=a#ple SolutionE=a#ple Solution

( ) ( )

( )

( ) ( ) ( ) ( )

/ 2, 1

1000 50 -106.39 -23.62

1

23.62 1000 501000 106.39 1000 2.0096

1000 1100

1,06,390 5,&30.&5

n

N n X S

N nS NX N t

N nα −

= = = =

−±

−

−= ±

−

= ±

/he ?54 con!dence inter"al or the populationtotal a#ount o the "ouchers is et'een1,000,55?.15, and 1,152,220.85

007 Confidence Interval

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