Statistics for Business and Economics: bab 12

7/21/2019 Statistics for Business and Economics: bab 12

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1

Slides Prepared by

JOHN S. LOUCKSSt. Edward’s University

© 2002 South-Western/Thomson Learning™





) )

Goodness of Fit Test!" #ultinomial $opulation

1* Set up the null and alternati+e h%potheses*

2* Selet a random sample and reord theo&ser+ed

fre,uen% f i for eah of the k ategories*

)* "ssuming H0 is true ompute the e.peted

fre,uen% ei in eah ategor% &% multipl%ing

the ategor% pro&a&ilit% &% the sample sie*

ontinued





8.ample! Finger La9es :omes 4"7

#ultinomial (istri&ution Goodness of Fit Test

Finger La9es :omes manufatures four modelsof

prefa&riated homes a t5o-stor% olonial aranh a

split-le+el and an "-frame* To help in prodution

planning management 5ould li9e to determine if

pre+ious ustomer purhases indiate that thereis a

preferene in the st%le seleted*



; ;


#ultinomial (istri&ution Goodness of Fit

Test

The num&er of homes sold of eah modelfor 100

sales o+er the past t5o %ears is sho5n

&elo5*

#odel Colonial anh Split-Le+el "-Frame

< Sold )0 20 )1




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• 8.peted Fre,uenies

e1 > *241007 > 2 e2 > *241007 >

2

e) > *241007 > 2 e

> *241007 >

2

• Test Statisti

> 1 @ 1 @ @

> 10

χ 2

2 2 2 2!" 2#

2#

2" 2#

2#

!# 2#

2#

1# 2#

2#=

−+

−+

−+

−( ( ( ( χ

22 2 2 2!" 2#

2#

2" 2#

2#

!# 2#

2#

1# 2#

2#=

−+

−+

−+

−( ( ( (


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• e3etion ule

With α > *0 and

k - 1 > - 1 > )

degrees of freedom

χ2χ2

=*?1 =*?1

(o 'ot e3et H0(o 'ot e3et H0 e3et H0

e3et H0



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• Conlusion

χ 2 > 10 B =*?1 so 5e re3et the assumptionthere is

no home st%le preferene at the *0 le+elof signi6ane*


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Test of Independene! Contingen% Ta&les


2* Selet a random sample and reord theo&ser+ed

fre,uen% f ij for eah ell of the ontingen%

ta&le*

)* Compute the e.peted fre,uen% eij for eah

ell*e

i jij =

($%w &%tal(C%l'n &%tal

Saple Si)e


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Test of Independene! Contingen% Ta&les

* Compute the test statisti*

* e3et H0

if 45here α is the

signi6ane le+el and 5ith n ro5s and m olumns there are

4n - 174m - 17 degrees of freedom7*

χ 2

2

=−

∑∑( f e

e

ij ij

ij ji

χ χ α

2 2>


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8.ample! Finger La9es :omes 47

Contingen% Ta&le 4Independene7 Test

8ah home sold an &e lassi6ed aordingto prie and to st%le* Finger La9es :omesDmanager 5ould li9e to determine if the prie ofthe home and the st%le of the home areindependent +aria&les*

The num&er of homes sold for eah modeland prie for the past t5o %ears is sho5n&elo5* For on+eniene the prie of the homeis listed as either $65,000 or less or more than

$65,000*

Price Colonial Ranch Split-LevelA-Frame

E ;000 1? ; 1A

12




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Contingen% Ta&le 4Independene7 Test

• Test Statisti

> *1); @ 2*2=2= @ * * * @ 2*0?)) >A*1?;

• e3etion ule

With α > *0 and 42 - 174 - 17 > ) d*f*

e3et H0 if χ 2

B =*?1• Conlusion

We re3et H0 the assumption that the prie

of the home is independent of the st%le of

the home that is purhased*

χ 2

2 2 21* 1+ #

1+ #

+ 11

11

! + ,#

+ ,#=

−+

−+ +

−( .

.

( . .

( .

. .

χ . ."#2 , *1=

8.ample! Finger La9es :omes 47


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Goodness of Fit Test! $oisson (istri&ution


2* Selet a random sample and

a* eord the o&ser+ed fre,uen% f i for eah

of the

k +alues of the $oisson random +aria&le*

&* Compute the mean num&er of ourrenes µ *

)* Compute the e.peted fre,uen% ofourrenes ei for eah +alue of the $oisson

random +aria&le*

ontinued



1=

Goodness of Fit Test! $oisson (istri&ution

* Compute the +alue of the test statisti*

* e3et H0

if

45here α is the signi6ane le+el and thereare k - 2 degrees of freedom7*

χ 2

2

1

= −

∑=

( f e

ei i

ii

k

χ 2

2

1

= −

∑=

( f e

ei i

ii

k

χ χ α

2 2> χ χ α

2 2>



1? 1?

8.ample! Tro% $ar9ing Garage

$oisson (istri&ution Goodness of Fit Test

In stud%ing the need for an additionalentrane to a it% par9ing garage a onsultanthas reommended an approah that isapplia&le onl% in situations 5here the num&er

of ars entering during a spei6ed time periodfollo5s a $oisson distri&ution*



1A 1A



" random sample of 100 one-minute timeinter+als resulted in the ustomer arri+alslisted &elo5* " statistial test must &eonduted to see if the assumption of a

$oisson distri&ution is reasona&le*

< "rri+als 0 1 2 ) ; = ? A10 11 12

Fre,uen% 0 1 10 1 20 12 12 A ?; ) 1





21 21



• 8stimate of $oisson $ro&a&ilit% Funtion

Τotal "rri+als > 0407 @ 1417 @ 247 @ * * * @12417 > ;00

Total Time $eriods > 100

8stimate of µ > ;00/100 > ;

:ene f x

e

x

x

( -

=

−+ +

f x e

x

x

( -

=

−+ +



22 22



•8.peted Fre,uenies

x f 4 x 7 xf 4 x 7 x f 4 x 7 xf 4 x 7

0 *002 *2 = *1)?A1)*?A

1 *01A 1*A ? *10110*1

2 *0; *; A *0;A;*A

) *0?A2 ?*A210 *01=*1=

*1))A 1)*)A 11 *022= 2*2=

*1;20 1;*20 12 *01 1*

; *1;0; 1;*0; Total 1*0000 100*00



2) 2)



• H&ser+ed and 8.peted Fre,uenies

i f i ei f i

- ei

0 or 1 or 2 ;*20 -1*20) 10 ?*A2 1*0?

1 1)*)A*;1

20 1;*20)*?0

; 12 1;*0;-*0;= 12 1)*?A-1*?A

? A 10*1 -1*1

A ? ;*A 1*0;

10 or more10 =*AA 2*01



2 2


• Test Statisti

• e3etion ule

With α > *0 and k - p - 1 > A - 1 - 1 > = d*f*45here k > num&er of ategories and p >num&er of population parametersestimated7

e3et H0 if χ 2 B 1*0=

• Conlusion

We annot re3et H0* ThereDs no reason to

dou&tthe assumption of a $oisson distri&ution*

χ 2

2 2 21 2"

+ 2"

1 "*

* 2

2 "1

, ! /2=

−+ + + =

( .

.

( .

.. . .

( .

.. χ

22 2 21 2"

+ 2"

1 "*

* 2

2 "1

, ! /2=

−+ + + =

( .

.

( .

.. . .

( .

..

χ . ."#2 1/ ",= χ . ."#2 1/ ",=




2 2

Goodness of Fit Test! 'ormal (istri&ution


2* Selet a random sample and

a* Compute the mean and standard de+iation*

&* (e6ne inter+als of +alues so that thee.peted

fre,uen% is at least for eah inter+al*

* For eah inter+al reord the o&ser+edfre,uenies

)* Compute the e.peted fre,uen% ei for eah

inter+al*

ontinued



2; 2;

Goodness of Fit Test! 'ormal (istri&ution

* Compute the +alue of the test statisti*

* e3et H0 if

45here α is the signi6ane le+el

and there are k - ) degrees of freedom7*

χ 2

2

1

= −

∑=

( f e

ei i

ii

k

χ 2

2

1

= −

∑=

( f e

ei i

ii

k

χ χ α

2 2> χ χ α

2 2>



2= 2=

8.ample! itor Computers

'ormal (istri&ution Goodness of Fit Test

itor Computers manufatures and sells a

general purpose miroomputer* "s part of astud% to e+aluate sales personnelmanagement 5ants to determine if the annual

sales +olume 4num&er of units sold &% asalesperson7 follo5s a normal pro&a&ilit%distri&ution*



2? 2?



" simple random sample of )0 of thesalespeople 5as ta9en and their num&ers ofunits sold are &elo5*

)) ) 2 2 ; ? ;);; ; ;; ;? =0 =2 =) =) =

=?) ? ? ?; A1 A2 A A? 102

10 4mean > =1 standard de+iation > 1?*7



2A 2A


• :%potheses

H0! The population of num&er of units

sold has a normal distri&ution 5ithmean =1 and standard de+iation

1?**

Ha! The population of num&er of units

sold does not ha+e a normaldistri&ution 5ith mean =1 and standard

de+iation 1?**




)0 )0


• Inter+al (e6nition

To satisf% the re,uirement of an e.peted

fre,uen% of at least in eah inter+al 5e

5ill di+ide the normal distri&ution into )0/> ;

e,ual pro&a&ilit% inter+als*




)1 )1



• Inter+al (e6nition

"reas > 1*00/; > *1;;=

"reas > 1*00/; > *1;;=

=1=1)*02)*02

;)*0);)*0) =?*A==?*A=

??*A? > =1 @ *A=41?*7??*A? > =1 @ *A=41?*7





)) ))


• Test Statisti

• e3etion ule

With α > *0 and k - p - 1 > ; - 2 - 1 > )d*f*

e3et H0 if χ 2 B =*?1

•

Conlusion We annot re3et H0* There is little e+idene

to

support re3eting the assumption the

population is normall% distri&uted 5ith µ > =1and σ > 1?**

χ 2

2 2 2 2 2 21

#

2

#

1

#

"

#

1

#

1

#1 +"= +

−+ + +

−+ =

( ( ( ( ( ( . χ

22 2 2 2 2 21

#

2

#

1

#

"

#

1

#

1

#1 +"= +

−+ + +

−+ =

( ( ( ( ( ( .

χ . ."#2 , *1= χ . ."#2 , *1=




))

8nd of Chapter 12

Statistics for Business and Economics: bab 12

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