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CHI-SQUARE & F-TEST

ISI: Kaitan antara Chi-Square dengan ANO A

!F te"t#

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Chi-"quare and F $i"tri%uti n"

Chi'dren ( the N r)a'

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$i"tri%uti n"

There are )an* the reti+a'di"tri%uti n", % th + ntinu u" anddi"+rete

.e u"e / ( the"e a ' t: 0 !unit n r)a'#,t, +hi-"quare, and F

1 and t are +' "e'* re'ated t the

"a)2'ing di"tri%uti n ( )ean"3 +hi-"quare and F are +' "e'* re'ated t the"a)2'ing di"tri%uti n ( 4arian+e"

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Chi-"quare !6#

.hat i( e t 8 6 4a'ue" ( 0 6 at rand ) and added the);

6

666

66

656

5#!

3#!

=

=

y z

y z 6

6656

66

6

656

#6!#!#!

z z y y

+=

+

=

Chi-"quare i" the di"tri%uti n ( a "u) ( "quare"Ea+h "quared de4iati n i" ta8en (r ) the unit n r)a':

N!=,5# The "ha2e ( the +hi-"quare di"tri%uti nde2end" n the nu)%er ( "quared de4iate" that areadded t gether

Sa)e )ini)u) and )a

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Chi-"quare !/#

The e

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$i"tri%uti n ( Sa)2'e

arian+e5

#! 66

=

N

y y s

Sa)2'e e"ti)ate ( 2 2u'ati n 4arian+e!un%ia"ed#

6

6

6 #5!

#5!

s N

N

=

7u'ti2'* 4arian+e e"ti)ate %* N-5 t

get "u) ( "quare" $i4ide %* 2 2u'ati n 4arian+e t n r)a'i0eRe"u't i" a rand ) 4aria%'e di"tri%uteda" +hi-"quare ith !N-5# df

.e +an u"e in( a% ut the "a)2'ing di"tri%uti n ( the4arian+e e"ti)ate t (ind + n(iden+e inter4a'" and+ ndu+t "tati"ti+a' te"t"

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Te"ting E

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E=

6?B:36?B:6

65

6=

==

>B N S either a*

N te: 6 tai'ed ith a'2ha =5

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C n(iden+e Inter4a'" ( r the

arian+e.e u"e t e"ti)ate It +an %e "h n that:6 s 6

D?-#5!#5!

6#DG?3-5!

66

6#=6?3-5!

6

= N N

s N s N p

Su22 "e N 5? and i" 5= Then df 5/ and ( r Q =6?the 4a'ue i" 6B 56 F r Q DG? the 4a'ue i" ? B>

D?-B>-?

#5=#!5/!

56-6B

#5=#!5/! 6 = p

[ ] D?-HG-6/>B-? 6 = p

6 s

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N r)a'it* A""u)2ti n

.e a""u)e n r)a' di"tri%uti n" t (igure"a)2'ing di"tri%uti n" and thu" p 'e4e'"

i 'ati n" ( n r)a'it* ha4e )in r

i)2'i+ati n" ( r te"ting )ean", e"2e+ia''* a" N get" 'arge i 'ati n" ( n r)a'it* are ) re "eri u" ( r

te"ting 4arian+e" 8 at * ur data %e( re+ ndu+ting thi" te"t Can te"t ( r n r)a'it*

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The F $i"tri%uti n !5#

The F di"tri%uti n i" the rati ( t4arian+e e"ti)ate":

A'" the rati ( t +hi-"quare", ea+hdi4ided %* it" degree" ( (reed ):

6

6

65

6

6

65

est

est

s

s F ==

66!

56

#!

J#

J

6

5

v

v F

v

v

=

In ur a22'i+ati n", 4 6 i'' %e 'argerthan 4 5 and 4 6 i'' %e 'arger than 6In "u+h a +a"e, the )ean ( the Fdi"tri%uti n !e

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F $i"tri%uti n !6#

F de2end" n t 2ara)eter": 4 5 and 4 6 !df 5 and df 6# The "ha2e ( F +hange"ith the"e Range i" = t in(init*

Sha2ed a %it 'i8e +hi-"quare F ta%'e" "h +riti+a' 4a'ue" ( r df in

the nu)erat r and df in theden )inat r

F ta%'e" are 5-tai'ed3 +an (igure 6-tai'edi( * u need t !%ut * u u"ua''* d n t#

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Te"ting H*2 the"e" a% ut 6

arian+e" Su22 "e N te 5-tai'ed

.e (ind

Then df 5 df 6 5?, and

66

655

66

65= :3: > H H

G-535B3H-?35B 666655

==== s N s N

/5->G-5

H-?66

65 ===

s

s

F 9 ing t the F ta%'e ith 5?and 5? df , e (ind that ( r a'2ha =? !5-tai'ed#, the +riti+a'4a'ue i" 6 /= There( re there"u't i" "igni(i+ant

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A 8 Ahead

The F di"tri%uti n i" u"ed in )an*"tati"ti+a' te"t" Te"t ( r equa'it* ( 4arian+e"

Te"t" ( r di((eren+e" in )ean" in ANO A Te"t" ( r regre""i n ) de'" !"' 2e"

re'ating ne + ntinu u" 4aria%'e t an ther

'i8e SAT and 9@A#

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Re'ati n" a) ng $i"tri%uti n"

the Chi'dren ( the N r)a' Chi-"quare i" dra n (r ) the n r)a' N!=,5# de4iate" "quared and "u))ed

F i" the rati ( t +hi-"quare", ea+h

di4ided %* it" d( A +hi-"quare di4ided %* it" df i" a 4arian+e e"ti)ate, that i",a "u) ( "quare" di4ided %* degree" ((reed )

F t 6 I( * u "quare t , * u get an F ith 5 d( in the nu)erat r

#,5!6

#! vv F t =