Chapter 8 - Hypothesis Testing

 

Ztdtcstcjs ior Buschmss dh` Mjohoacjs  9tl M`ctcoh

Jldptmr 4>

Lypotlmscs Pmstchk

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj

Jldp 4> 4

 

Jldptmr Kodfs Ditmr joapfmtchk tlcs jldptmr, you slouf` bm dbfm to7  

Ioraufdtm huff dppfcjdtcohs chvofvchk dh` dftmrhdtcvm lypotlmsms ior  

 

 

d schkfm popufdtcoh amdh iroa d horadf `cstrcbutcoh d schkfm popufdtcoh proportcoh (fdrkm sdapfms)

Ioraufdtm d `mjcscoh rufm ior tmstchk d lypotlmscs Ehow low to usm tlm jrctcjdf vdfum dh` p-vdfum dpprodjlms to tmst tlm huff lypotlmscs (ior botl amdh dh` proportcoh probfmas) Ehow wldt Pypm C dh` Pypm CC mrrors drm  Dssmss tlm powmr oi d tmst

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

[ldt cs d Lypotlmscs? 

 D lypotlmscs cs d jfdca (dssuaptcoh) dbout d popufdtcoh pdrdamtmr7 

popufdtcoh amdh Mxdapfm7 Plm amdh aohtlfy jmff plohm bcff oi tlcs jcty cs ζ 1 $02



popufdtcoh proportcoh Mxdapfm7 Plm proportcoh oi d`ufts d`ufts ch tlcs jcty wctl jmff plohms cs p 1 .9<

Jldp 4>-2

Jldp 4>-:

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Plm Huff Lypotlmscs, L> 

Ztdtms tlm dssuaptcoh (huamrcjdf) to bm tmstm` Mxdapfm7 Mxdapfm7   Plm dvmrdkm huabmr oi PY smts ch ].Z. Loams cs mqudf to tlrmm ( L> 7  ζ  : )



Cs dfwdys dbout d popufdtcoh pdrdamtmr, hot dbout d sdapfm stdtcstcj  L> 7  ζ  :

L> 7  \  :

Jldp 4>-0

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Plm Huff Lypotlmscs, L> (johtchum`) 

Bmkch wctl tlm dssuaptcoh tldt tlm huff lypotlmscs cs trum

Zcacfdr to tlm hotcoh oi chhojmht uhtcf  provmh kucfty  Rmimrs to tlm stdtus quo   Dfwdys johtdchs ‖1‑ , ‖≠‑ or or ‖ ‑ sckh  Ady or ady hot bm rmgmjtm` 

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Plm Dftmrhdtcvm Lypotlmscs, L4 

Cs tlm opposctm oi tlm huff lypotlmscs 

m.k., Plm dvmrdkm huabmr oi PY smts ch ].Z. loams cs hot mqudf to : ( L47 ζ ≮ : )



Jldffmhkms tlm stdtus quo



Hmvmr johtdchs tlm ‖1‑ , ‖≠‑ or ‖ ‑ sckh

 

Ady or ady hot bm supportm` Cs kmhmrdffy tlm lypotlmscs tldt tlm rmsmdrjlmr cs trychk to support

Jldp 4>-=

Jldp 4>-9

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Lypotlmscs Pmstchk ^rojmss Jfdca7 tlm Jfdca7  popufdtcoh amdh dkm cs =>. (Huff Lypotlmscs7  L>7 ζ 1 => )

^opufdtcoh How smfmjt d rdh`oa sdapfm

Cs \ 2> fcemfy ci ζ 1 =>? Ci hot fcemfy, RMGMJP Huff Lypotlmscs

Zupposm tlm sdapfm amdh cs 2>7 dkm \ 1 2>

Zdapfm

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Rmdsoh ior Rmgmjtchk L> Zdapfchk @cstrcbutcoh oi \

2>

Ci ct cs uhfcemfy tldt wm wouf` kmt d sdapfm amdh tlcs vdfum ... oi

 ζ 1 =>

\

... ci ch idjt tlcs wmrm  tlm popufdtcoh amdh…

... tlmh wm rmgmjt tlm huff lypotlmscs tldt ζ 1 =>.

Ci  L> cs trum

Jldp 4>-<

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Fmvmf oi Zckhcicjdhjm,   

@michms tlm uhfcemfy vdfums oi tlm sdapfm stdtcstcj ci tlm huff lypotlmscs cs trum 



Cs `msckhdtm` by 





@michms rmgmjtcoh rmkcoh oi rmkcoh oi tlm sdapfchk `cstrcbutcoh , (fmvmf oi sckhcicjdhjm)

Pypcjdf vdfums drm .>4, .>=, or .4>

Cs smfmjtm` by tlm rmsmdrjlmr dt tlm bmkchhchk ^rovc`ms tlm jrctcjdf vdfum(s) vdfum(s) oi  oi tlm tmst 

Jldp 4>-3

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Fmvmf oi Zckhcicjdhjm dh` tlm Rmgmjtcoh Rmkcoh Fmvmf oi sckhcicjdhjm 1 /2

L>7 ζ 1 : L47 ζ ≮ :

Rmprmsmhts   jrctcjdf vdfum  

Pwo-tdcf tmst

>

]ppmr-tdcf tmst

>

L>7 ζ ≠ : L47 ζ 8 : L>7 ζ ≭ : L47 ζ 6 :

 

Fowmr-tdcf tmst

>

 /2 Rmgmjtcoh rmkcoh cs sld`m`

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Mrrors ch Adechk @mjcscohs 

Pypm C Mrror   Rmgmjt d trum huff lypotlmscs 



Johsc`mrm` d smrcous typm oi mrror 

Plm probdbcfcty oi Pypm C Mrror cs 



Jdffm` fmvmf oi sckhcicjdhjm sckhcicjdhjm oi  oi tlm tmst Zmt by rmsmdrjlmr ch d`vdhjm

Jldp 4>-4>

Jldp 4>-44

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Mrrors ch Adechk @mjcscohs (johtchum`) 

Pypm CC Mrror  Idcf to rmgmjt d idfsm huff lypotlmscs 

Plm probdbcfcty oi Pypm CC Mrror cs β

Jldp 4>-42

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Outjoams dh` ^robdbcfctcms ^osscbfm Lypotlmscs Pmst Outjoams

@mjcscoh Emy7 Outjoam (^robdbcfcty)

@o Hot Rmgmjt L> Rmgmjt L>

Djtudf Zctudtcoh L> Prum

 L> Idfsm

Ho mrror   (4 - )

Pypm CC Mrror   (β)

Pypm C Mrror  ( )

Ho Mrror   (4-β)

Jldp 4>-4:

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Pypm C & CC Mrror Rmfdtcohslcp  Pypm

C dh` Pypm CC mrrors jdh hot ldppmh dt   tlm sdam tcam  Pypm C mrror jdh ohfy ojjur ci L> cs trum



 Pypm CC mrror jdh ohfy ojjur ci L> cs idfsm



 

Ci Pypm C mrror probdbcfcty ( Pypm CC mrror probdbcfcty ( β )

)

, tlmh

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj

Jldp 4>-40

 

Idjtors Diimjtchk Pypm CC Mrror  

 Dff mfsm mqudf, 

 β

wlmh tlm `ciimrmhjm bmtwmmh

lypotlmsczm` pdrdamtmr dh` cts trum vdfum 

  β 

wlmh



  β 

wlmh

ϊ



  β 

wlmh

h

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj

Jldp 4>-4=

 

^owmr oi tlm Pmst 

Plm powmr  oi  oi d tmst cs tmst cs tlm probdbcfcty oi rmgmjtchk d huff lypotlmscs tldt cs idfsm



c.m., 

^owmr 1 ^(Rmgmjt L> | L4 cs trum)

^owmr oi tlm tmst chjrmdsms ds tlm sdapfm sczm chjrmdsms

Jldp 4>-49

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Lypotlmscs Pmsts ior tlm Amdh Lypotlmscs Pmsts ior   Ehowh

 

 ]hehowh

Jldp 4>-4;

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Pmst oi Lypotlmscs ior tlm Amdh (ϊ Ehowh) 

Johvmrt sdapfm rmsuft ( x ) to d z vdfum vdfum   Lypotlmscs Pmsts ior  ϊ Ehowh

Johsc`mr tlm tmst L> 7 ζ  ζ>

L4 7 ζ  ζ> (Dssuam tlm popufdtcoh cs horadf)

ϊ ]hehowh

rufm cs7  cs7 Plm `mjcscoh rufm x  ζ>  zδ Rmgmjt L> ci   z   ϊ h

Jldp 4>-4<

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

@mjcscoh Rufm x  ζ>  zδ Rmgmjt L> ci   z   ϊ

L>7 ζ 1 ζ>  L47 ζ 8 ζ> 

h

 Dftmrhdtm rufm7  

Rmgmjt L> ci    \   ζ>  W δϊ/ h

W

x

@o hot rmgmjt L>

>

ζ>

zδ ζ>  z δ

Rmgmjt L>

ϊ h

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj

Jrctcjdf vdfum

Jldp 4>-43

 

p-Ydfum Dpprodjl to Pmstchk 

p-vdfum7 ^robdbcfcty oi obtdchchk d tmst p-vdfum7 stdtcstcj aorm mxtrmam ( ≠ or ) tldh tlm obsmrvm` sdapfm vdfum kcvmh L> cs trum 



 Dfso jdffm` obsmrvm` fmvmf oi sckhcicjdhjm Zadffmst vdfum oi rmgmjtm`

ior wlcjl L> jdh bm

Jldp 4>-2>

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

p-Ydfum Dpprodjl to Pmstchk (johtchum`)



Johvmrt sdapfm rmsuft (m.k., x ) to tmst stdtcstcj (m.k., z stdtcstcj )



Obtdch tlm p-vdfum p - vdfum  ^(W   x - ζ> , kcvmh tldt L> cs trum) Ior dh uppmr ϊ/ h tdcf tmst7 

 ^(W  

x - ζ> ϊ/ h



@mjcscoh rufm7 rufm7 joapdrm tlm p-vdfum to p-vdfum to 

Ci p-vdfum 6

, rmgmjt L>



Ci p-vdfum

, `o hot rmgmjt L> 

 |  ζ  ζ> )

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj

Jldp 4>-24

 

Mxdapfm7 ]ppmr-Pdcf W Pmst ior Amdh ( Ehowh)   D plohm ch`ustry adhdkmr tlches tldt justoamr aohtlfy jmff plohm bcff ldvm chjrmdsm`, dh` how dvmrdkm ovmr $=2 pmr aohtl. Plm joapdhy wcslms to tmst tlcs jfdca. (Dssuam  1 4> cs ehowh) Iora lypotlmscs tmst7 L>7 ζ ≠ =2 tlm dvmrdkm cs hot ovmr $=2 pmr aohtl L47 ζ 8 =2

tlm dvmrdkm cs krmdtmr tldh $=2 pmr aohtl (c.m., suiicjcmht mvc`mhjm mxcsts to support tlm adhdkmr–s jfdca)

Jldp 4>-22

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Mxdapfm7 Ich` Rmgmjtcoh Rmkcoh (johtchum`) 

Zupposm tldt  1 .4> cs jlosmh ior tlcs tmst

Ich` tlm rmgmjtcoh rmkcoh7

Rmgmjt L>

 

@o hot rmgmjt L>

>

4.2<

1 .4>

Rmgmjt L>

Rmgmjt L> ci   z    x  ζ>  4.2< ϊ/ h

Jldp 4>-2:

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Mxdapfm7 Zdapfm Rmsufts (johtchum`)

Obtdch sdapfm dh` joaputm tlm tmst stdtcstcj Zupposm d sdapfm cs tdemh wctl tlm ioffowchk rmsufts7 h 1 90, x 1 =:.4 (14> wds dssuam` ehowh)  

]schk tlm sdapfm rmsufts,

x   ζ> =:.4  =2 z    >. ϊ h

90

Jldp 4>-20

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Mxdapfm7 @mjcscoh (johtchum`)

Rmdjl d `mjcscoh dh` chtmrprmt tlm rmsuft7 Rmgmjt L>

  1 .4>

@o hot rmgmjt L>

>

4.2<

Rmgmjt L>

z 1 >. schjm z 1 >.-2=

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Mxdapfm7 p-Ydfum Zofutcoh Jdfjufdtm tlm p-vdfum dh` joapdrm to

(johtchum`)

(dssuachk tldt ζ 1 =2.>) p-vdfum 1 .4   1 .4>

>

@o hot rmgmjt L>

4.2< W 1 .

^(x  =:.4| ζ  =2.>)

   =2.>    =:.4  ^ z   4>/ 90      ^(z  >.; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj

Jldp 4>-29

 

Ohm-Pdcf Pmsts 

Ch adhy jdsms, tlm dftmrhdtcvm lypotlmscs iojusms oh ohm pdrtcjufdr `crmjtcoh L>7 ζ ≠ : L47 ζ 8 : L>7 ζ ≭ : L47 ζ 6 :

Plcs cs dh uppmr -tdcf -tdcf tmst schjm tlm dftmrhdtcvm lypotlmscs cs iojusm` oh tlm uppmr tdcf dbovm tlm amdh oi : Plcs cs d fowmr -tdcf -tdcf tmst schjm tlm dftmrhdtcvm lypotlmscs cs iojusm` oh tlm fowmr tdcf bmfow tlm amdh oi :

Jldp 4>-2;

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

]ppmr-Pdcf Pmsts 

Plmrm cs ohfy ohm jrctcjdf vdfum, schjm

L>7 ζ ≠ : L47 ζ 8 :

tlm rmgmjtcoh drmd cs ch ohfy ohm tdcf

 

@o hot rmgmjt L>

W

>

x

ζ

zδ

Rmgmjt L>

Jrctcjdf vdfum

Jldp 4>-2<

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Fowmr-Pdcf Pmsts L>7 ζ ≭ : 

Plmrm cs ohfy ohm jrctcjdf vdfum, schjm tlm rmgmjtcoh drmd cs ch ohfy ohm tdcf

L47 ζ 6 :   Rmgmjt L>

@o hot rmgmjt L>

-z

Jrctcjdf vdfum

>

W

ζ

x

Jldp 4>-23

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Pwo-Pdcf Pmsts 

Ch soam smttchks, tlm dftmrhdtcvm lypotlmscs `oms hot spmjciy d uhcqum `crmjtcoh

L>7 ζ 1 : L47 ζ   :



Plmrm drm two jrctcjdf vdfums, `michchk tlm two rmkcohs oi rmgmjtcoh

 /2

 /2

x

: Rmgmjt L>

@o hot rmgmjt L>

-z/2

Fowmr jrctcjdf vdfum

>

Rmgmjt L>

+z/2

z

]ppmr  jrctcjdf vdfum

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Lypotlmscs Pmstchk Mxdapfm Pmst tlm jfdca tldt tlm trum amdh # oi PY smts ch ]Z loams cs mqudf to :. (Dssuam ϊ 1 >.7 ζ 1 : , L47 ζ ≮ : (Plcs cs d two tdcfm` tmst) 



Zpmjciy tlm `mscrm` fmvmf oi sckhcicjdhjm Zupposm tldt  1 .>= cs jlosmh ior tlcs tmst Jloosm d sdapfm sczm 





Zupposm d sdapfm oi sczm h 1 4>> cs smfmjtm`

Jldp 4>-:>

Jldp 4>-:4

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Lypotlmscs Pmstchk Mxdapfm (johtchum`) 

@mtmrachm tlm dpproprcdtm tmjlhcqum ϊ cs ehowh so tlcs cs d z tmst Zmt up tlm jrctcjdf vdfums Ior  1 .>= tlm jrctcjdf z vdfums drm ´4.39 Joffmjt tlm `dtd dh` joaputm tlm tmst stdtcstcj 









Zupposm tlm sdapfm rmsufts drm

h 1 4>>, x 1 2..< cs dssuam` ehowh) Zo tlm tmst stdtcstcj cs7 z



\  



ζ>

ϊ h

2..< 4>>



.><

2.>



Jldp 4>-:2

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Lypotlmscs Pmstchk Mxdapfm (johtchum`) 

Cs tlm tmst stdtcstcj ch tlm rmgmjtcoh rmkcoh?

Rmgmjt L> ci z 6 -4.39 or z 8 4.395 otlmrwcsm `o hotL rmgmjt >

 1 .>=/2

 1 .>=/2

Rmgmjt L>

-z 1 -4.39

@o hot rmgmjt L>

>

Rmgmjt L>

+z 1 +4.39

Lmrm, z 1 -2.> 6 -4.39, so tlm tmst stdtcstcj cs ch tlm rmgmjtcoh rmkcoh

Jldp 4>-::

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Lypotlmscs Pmstchk Mxdapfm (johtchum`) 

 Rmdjl d `mjcscoh dh` chtmrprmt tlm rmsuft  1 .>=/2

 1 .>=/2

Rmgmjt L>

-z 1 -4.39 -2.>

@o hot rmgmjt L>

>

Rmgmjt L>

+z 1 +4.39

Zchjm z 1 -2.> 6 -4.39, wm rmgmjt tlm huff lypotlmscs lypotlmscs   dh` johjfu`m tldt tlmrm cs suiicjcmht mvc`mhjm tldt tlm amdh huabmr oi PYs ch ]Z loams cs hot mqudf to :

Jldp 4>-:0

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Mxdapfm7 p-Ydfum 

Mxdapfm7   Low fcemfy cs ct to smm d sdapfm amdh oi Mxdapfm7 2.?

x 1 2. - 2.> ^(z  2.>)  .>22< ^(z  2.>)  .>22<

/2 1 .>2=

/2 1 .>2=

.>22<

.>22<

p-vdfum 1 .>22< + .>22< 1 .>0=9

-4.39 -2.>

>

4.39 2.>

W

2.>

2.> Jldp 4>-:=

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Mxdapfm7 p-Ydfum 

(johtchum`)

Joapdrm tlm p-vdfum wctl 

Ci p-vdfum 6

, rmgmjt L>



Ci p-vdfum

, `o hot rmgmjt L> 

Lmrm7 p-vdfum 1 .>0=9   1 .>= Zchjm .>0=9 6 .>=, wm rmgmjt tlm huff lypotlmscs

/2 1 .>2=

/2 1 .>2=

.>22<

.>22<

-4.39

>

4.39

W

2.>

2.> Jldp 4>-:9

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

t Pmst oi Lypotlmscs ior tlm Amdh (ϊ ]hehowh) 

Johvmrt sdapfm rmsuft ( x ) to d t tmst stdtcstcj stdtcstcj   Lypotlmscs Pmsts ior  ϊ Ehowh

Johsc`mr tlm tmst L> 7 ζ  ζ>

L4 7 ζ  ζ> (Dssuam tlm popufdtcoh cs

ϊ ]hehowh

Plm `mjcscoh rufm cs7 rufm cs7 x  ζ> Rmgmjt L> ci   t    t h-4, δ s h

horadf) Jldp 4>-:;

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

t Pmst oi Lypotlmscs ior tlm Amdh (ϊ ]hehowh) 

(johtchum`)

Ior d two-tdcfm` tmst7 Johsc`mr tlm tmst

L> 7 ζ  ζ> L4 7 ζ  ζ>

(Dssuam tlm popufdtcoh cs horadf, dh` tlm popufdtcoh vdrcdhjm cs uhehowh)

Plm `mjcscoh rufm rufm cs7  cs7 x  ζ> x  ζ> Rmgmjt L> ci   t    t h -4, δ/2  or ci    t   t h-4, δ/2 s s h

h

Jldp 4>-:<

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Mxdapfm7 Pwo-Pdcf Pmst ( ]hehowh)  

Plm dvmrdkm jost oi d lotmf rooa ch Hmw Sore cs sdc` to bm $49< pmr hcklt. D rdh`oa sdapfm oi 2= lotmfs rmsuftm` ch x 1 $4;2.=> dh` s 1 $4=.0>. Pmst dt tlm 1 >.>= fmvmf. (Dssuam tlm popufdtcoh `cstrcbutcoh cs horadf)

L>7 ζ1 49< L47 ζ  49<

Jldp 4>-:3

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Mxdapfm Zofutcoh7 Pwo-Pdcf Pmst L>7 ζ1 49< 

L47 ζ  49<   1 >.>=



h 1 2=



 cs uhehowh, so

usm d t stdtcstcj



Jrctcjdf Ydfum7 t20 , .>2= 1 ´ 2.>9:3

/21.>2=

/21.>2=

@o hot rmgmjt L>

Rmgmjt L>

-t h-4,δ/2 -2.>9:3

th

4





x



ζ

s h

Rmgmjt L>

>

4.09 4;2.=> 49< 4=.0> 



 



t h-4,δ/2 2.>9:3 4.09

2=

> @o hot rmgmjt L 7 hot suiicjcmhttldh mvc`mhjm trum amdh jost cs `ciimrmht $49< tldt

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj

Jldp 4>-0>

 

Pmsts oi tlm ^opufdtcoh ^roportcoh 

Chvofvms jdtmkorcjdf vdrcdbfms



Pwo posscbfm outjoams







‖Zujjmss‑ (d jmrtdch jldrdjtmrcstcj cs prmsmht)



‖Idcfurm‑ (tlm jldrdjtmrcstcj cs hot prmsmht)

Irdjtcoh or proportcoh oi tlm popufdtcoh ch tlm ‖sujjmss‑ jdtmkory cs `mhotm` by ^  Dssuam sdapfm sdapfm sczm cs fdrkm

Jldp 4>-04

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

^roportcohs (johtchum`) 

Zdapfm proportcoh ch tlm sujjmss jdtmkory cs `mhotm` by p ˉ





p    x  huabmr  oi  sujjmssms ch sdapfm   h sdapfm sczm ˉ

ˉ

[lmh h^(4 ‚ ^) 8 3, jdh bm dpproxcadtm` p by d horadf `cstrcbutcoh wctl amdh dh` stdh`dr` `mvcdtcoh   ^(4 ^)    p ϊ  ζ ^ h 

ˉ

 p ˉ

Jldp 4>-02

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Lypotlmscs Pmsts ior ^roportcohs 

Plm sdapfchk `cstrcbutcoh oi

pcs ˉ

dpproxcadtmfy horadf, so tlm tmst stdtcstcj cs d z vdfum7

Lypotlmscs Pmsts ior ^ h^(4 ‚ ^) 8 3

z

 

p  ^> ˉ

^> (4 ^> ) h

h^(4 ‚ ^) 6 3 Hot `csjussm` ch tlcs jldptmr 

Jldp 4>-0:

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Mxdapfm7 W Pmst ior ^roportcoh   D adremtchk joapdhy jfdcas tldt ct rmjmcvms > wmrm survmym` wctl 2= rmspohsms. Pmst dt tlm 1 .>= sckhcicjdhjm fmvmf.

Jlmje7 Our dpproxcadtcoh ior ^ cs   p 1 2=/=>> 1 .>= ˉ

h^(4 - ^) 1 (=>>)(.>=)(.3=)   1 2:.;= 8 3

Jldp 4>-00

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

W Pmst ior ^roportcoh7 Zofutcoh Pmst Ztdtcstcj7

L>7 ^ 1 .>< z

L47 ^  .  >=

 

p  ^> ˉ

 ^> (4 ^> ) h

h 1 =>>, p 1 .>=



.>=  .><



 2.0;

.>>

ˉ

@mjcscoh7

Jrctcjdf Ydfums7 ´ 4.39 Rmgmjt

Rmgmjt

.>2=

.>2= -4.39

-2.0;

>

4.39

z 

Rmgmjt L> dt  1 .>=

Johjfuscoh7

Plmrm cs suiicjcmht mvc`mhjm to rmgmjt tlm joapdhy–s jfdca oi -0=

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

p-Ydfum Zofutcoh (johtchum`)

Jdfjufdtm tlm p-vdfum dh` joapdrm to (Ior d two sc`m` tmst tlm p-vdfum cs dfwdys two sc`m`) @o hot rmgmjt L>

Rmgmjt L> /2 1 .>2=

Rmgmjt L> /2 1 .>2=

p-vdfum 1 .>4:97

^(W  2.0;)  ^(W  2.0;) .>>9<

.>>9< -4.39

W 1 -2.0;

>

 2(.>>9.>4:9

4.39 W 1 2.0;

Rmgmjt L> schjm p-vdfum 1 .>4:9 6

1 .>=

Jldp 4>-09

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

]schk ^LZtdt

Optcohs

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj

Jldp 4>-0;

 

Zdapfm ^LZtdt Output

Chput

Output

Jldp 4>-0<

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

^owmr oi tlm Pmst 

Rmjdff tlm posscbfm lypotlmscs tmst outjoams7 Djtudf Zctudtcoh

Emy7 Outjoam (^robdbcfcty)

 

@mjcscoh @o Hot Rmgmjt L>

L> Prum Ho mrror   (4 - )

 L> Idfsm Pypm CC Mrror   (β)

Rmgmjt L>

Pypm C Mrror 

Ho Mrror  

( ) β `mhotms tlm probdbcfcty oi Pypm CC Mrror

(4-β)

4 ‚ β cs `michm` ds tlm powmr oi tlm tmst

^owmr 1 4 ‚ β 1 tlm probdbcfcty tldt d idfsm huff

lypotlmscs cs rmgmjtm` Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj

Jldp 4>-03

 

Pypm CC Mrror   Dssuam tlm popufdtcoh cs horadf dh` dh` tlm popufd popufdtcoh tcoh vdrcdhjm cs ehowh. Johsc`mr tlm tmst L> 7 ζ  ζ> L4 7 ζ  ζ>

Plm `mjcscoh rufm cs7 x  ζ> Rmgmjt L> ci   z    z δ or  Rmgmjt L> ci    x   x j  ζ>  W δϊ/ h ϊ/ h Ci tlm huff lypotlmscs cs idfsm dh` tlm trum amdh cs ζ*, tlmh tlm probdbcfcty oi typm CC mrror cs     j β  ^(x  x j | ζ  ζ*)  ^ z   x  ζ * 

ϊ / h  

 

Jldp 4>-=>

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Pypm CC Mrror Mxdapfm 

Pypm CC mrror cs tlm probdbcfcty oi idcfchk to rmgmjt d idfsm L> Zupposm wm idcf to rmgmjt L>7 ζ =2  wlmh ch idjt tlm trum amdh cs ζ* 1 =>   xj



=> Rmgmjt

=2 x

@o hot rmgmjt

L>7 ζ  =2

j

L> 7 ζ  =2 Jldp 4>-=4

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Pypm CC Mrror Mxdapfm (johtchum`) 

Zupposm wm `o hot rmgmjt L>7 ζ  =2  =2 wlmh  wlmh ch idjt tlm trum amdh cs ζ* 1 => Plcs cs tlm rdhkm o oii x wlmrm L> cs hot rmgmjtm`

Plcs cs tlm trum `cstrcbutcoh oi x ci ζ 1 =>

=> Rmgmjt

=2 @o hot rmgmjt

L>7 ζ  =2

L> 7 ζ  =2

xj

Jldp 4>-=2

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Pypm CC Mrror Mxdapfm (johtchum`) 

Zupposm wm `o hot rmgmjt L>7 ζ  =2  =2 wlmh  wlmh ch idjt tlm trum amdh cs ζ* 1 => Lmrm, β 1 ^( x  x j ) ci ζ* 1 =>

β



=> Rmgmjt

=2 @o hot rmgmjt

L>7 ζ  =2

xj

L> 7 ζ  =2 Jldp 4>-=:

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Jdfjufdtchk β 

Zupposm h 1 90 , ϊ 1 9 , dh`

1 .>=

ϊ 9    4.90= 90  =>.;99 x j  ζ>  z δ h  =2 (ior L> 7 ζ  =2)

Zo β 1 ^( x  =>.;99 ) ci ζ* 1 =>



=> Rmgmjt

=>.;99

=2 @o hot rmgmjt

L>7 ζ  =2

xj

L> 7 ζ  =2 Jldp 4>-=0

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Jdfjufdtchk β (johtchum`) 

Zupposm h 1 90 , ϊ 1 9 , dh`  1 .>=

       ^( x  =>.;99 | ζ*  =>)  ^ z  =>.;99 =>   ^(z  4.>2)  .=  .:094  .4=:3   9   90     ^robdbcfcty oi typm CC mrror7 

β 1 .4=:3 =>

Rmgmjt

=2 @o hot rmgmjt

L>7 ζ  =2

L> 7 ζ  =2

xj

Jldp 4>-==

Ztdtcstcjss ior Buschmss dh` Mjohoacjs, 9m © 2>>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

^owmr oi tlm Pmst Mxdapfm Ci tlm trum amdh cs ζ* 1 =>, 

Plm probdbcfcty oi Pypm CC Mrror 1 β 1 >.4=:3



Plm powmr oi tlm tmst 1 4 ‚ β 1 4 ‚ >.4=:3 1 >. Prum

 L> Idfsm

@o Hot Rmgmjt L>

Ho mrror  4 - 1 >.3=

Pypm CC Mrror   β 1 >.4=:3

Rmgmjt L>

Pypm C Mrror    1 >.>=

Ho Mrror   4 - β 1 >.>; ^mdrsoh M`ujdtcoh, Chj. Ztdtcstcj  

Jldptmr Zuaadry 

 D``rmssm` lypotlmscs lypotlmscs tmstchk tmstchk amtlo amtlo`ofoky `ofoky



^mrioram` W Pmst ior tlm amdh (ϊ ehowh)



@csjussm` jrctcjdf vdfum dh` p-vdfum dpprodjlms to lypotlmscs tmstchk



^mrioram` ohm-tdcf dh` two-tdcf tmsts



^mrioram` t tmst ior tlm amdh (ϊ uhehowh)



^mrioram` W tmst ior tlm proportcoh



@csjussm` @csjussm ` typm CC mrror dh` powmr oi tlm tmst

Jldp 4>-=9