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1
Slides Prepared by
JOHN S. LOUCKSSt. Edward’s University
© 2002 South-Western/Thomson Learning™
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) )
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
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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*
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; ;
8.ample! Finger La9es :omes 4"7
#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.ample! Finger La9es :omes 4"7
#ultinomial (istri&ution Goodness of Fit Test
• 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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#ultinomial (istri&ution Goodness of Fit Test
• 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
8.ample! Finger La9es :omes 4"7
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8.ample! Finger La9es :omes 4"7
#ultinomial (istri&ution Goodness of Fit Test
• 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
1* Set up the null and alternati+e h%potheses*
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
1* Set up the null and alternati+e h%potheses*
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
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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>
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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*
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1A 1A
8.ample! Tro% $ar9ing Garage
$oisson (istri&ution Goodness of Fit Test
" 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
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21 21
8.ample! Tro% $ar9ing Garage
$oisson (istri&ution Goodness of Fit Test
• 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
( -
=
−+ +
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22 22
8.ample! Tro% $ar9ing Garage
$oisson (istri&ution Goodness of Fit Test
•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
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2) 2)
8.ample! Tro% $ar9ing Garage
$oisson (istri&ution Goodness of Fit Test
• 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
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2 2
$oisson (istri&ution Goodness of Fit Test
• 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/ ",=
8.ample! Tro% $ar9ing Garage
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2 2
Goodness of Fit Test! 'ormal (istri&ution
1* Set up the null and alternati+e h%potheses*
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
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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>
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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*
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2? 2?
8.ample! itor Computers
'ormal (istri&ution Goodness of Fit Test
" 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
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2A 2A
'ormal (istri&ution Goodness of Fit Test
• :%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?**
8.ample! itor Computers
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)0 )0
'ormal (istri&ution Goodness of Fit Test
• 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*
8.ample! itor Computers
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)1 )1
8.ample! itor Computers
'ormal (istri&ution Goodness of Fit Test
• 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
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)) ))
'ormal (istri&ution Goodness of Fit Test
• 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=
8.ample! itor Computers
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