cursul 4 - statistica inferentiala
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]tatlstlma lj`frfjtlaha
Mursuh jr* 5
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Elstrlkutla fgplrlma
Elstrlkutla fgplrlmaelstrlkutla eatfhor gasuratffxpfrlgfjtah
Akatfrfa stajeare fstf ogsur a varlaklhltll saurspjelrll `ata ef mfjtru aeatfhor `ata ef gfelf*
[fiuha fgplrlma2 eatfhfsf aproxlgfaza mu
elstrlkutla jorgaha eama2
03& elj eatf sujt ljtrf 1 sl 1 s*
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Vj stuefjt sf ljtfrfsfaza ef elstrlkula pujmtabfhor ha fxagfj eup mf aoktljut 39 elj 1>>* A a`hat ma elstrlkula pujmtabfhor sf aproxlgfazamu `orga ef mhopot( mu gfela 96 l akatfrfa stajeare ef 1>* ]fmojstata ma stuefjtuh a huat uj pujmtab gal garf efmat gfela( ear jusf stlf mat ef kuj fstf pujmtabuh*
]f aphlma rfiuha fgplrlm2
03& elj stuefjtl au pujmtabuh ljtrf 06 sl 36*
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Uforfga hlglt mfjtrah
]f mojslefr uj fajtloj ahfatorlu `orgat elj j varlaklhfhfahfatoarf2 S1( S;( ***( Sj4
Slau amffal elstrlkulf ef prokaklhltl l sujt rfmlpromljefpfjefjtf*
Gfela fajtlojuhul fstf o varlaklh ahfatoarf*Mf `ujmlf ef elstrlkulf a prokaklhltlhor l morfspujef gfelflfajtlojuhul?
^fjtru a rspujef ha amfast jtrfkarf( sf f`fmtufaz urgtoruhfxpfrlgfjt2
]f arujm ef jorl zarurlhf l sf eftfrglj gfela arujmarlhor4 sfmrfaz @E^ pfjtru el`frltf vahorl ahf hul j*
x
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Uforfga hlglt mfjtrah
j 7 =>
j 7 1j 7 ;
j 7 1>
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Uforfga hlglt mfjtrah
Mojmhuzll fxpfrlgfjt2Mu mrfstfrfa hul g2
Gfela rgjf amffal* Akatfrfa stajeare sf rfeumf*
@E^ a xsf aproplf ef elstrlkula jorgah*
Uforfga hlglt mfjtrahEam S1( S;( ***( Sjaparlj ujul fajtloj ahfatorlu prfhfvat eljtr'
o popuhalf mu gfela l akatfrfa stajeare l j fstf
su`umufjt ef garf atujml2
x jx
=* Elstrlkula gfelfl x fstf aproxlgatlv jorgah
1* ;*
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Uforfga hlglt mfjtrah
Eam sf mogklj mfhf trfl oksfrvall jtr'o sljiur `orguh sfokljf2 rfprfzljt elstrlkula jorgah stajeare*
j
x
R /
R rfprfzljt elstrlkula jorgah stajeare*
Amfst rfzuhtat pfrgltf s sf utlhlzfzf gfela fajtlojuhul mafstlgator ah gfelfl popuhalfl *
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Fstlgatorl pujmtuahl
popuhatlf
fsajtloj
]tuelu lj`frfjtlah
1* Elj popuhatlf sf prfhfvfaza uj fsajtloj;* ]f ajahlzfaza fsajtlojuh=* ]f `amf o prfelmtlf pfjtru popuhatlf pf kaza fsajtlojuhul*5* ]f gasoara bustftfa prfelmtlfl*
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Fstlgatorl pujmtuahl
x
p
fajtloj popuhalf
statlstlm paragftru
gfela akatfrfa stajeare s proportla p
Fstlgarfa pujmtuah ' mfa gal kuj fstlgarf a ujul paragftruah popuhalfl( ear mfh gal prokaklh ju fstf fiah mu paragftruh*
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Ljtfrvahuh ef fstlgarf
Ljtfrvahuh ef fstlgarf ' ljtfrvah ef vahorl marf ljmhuef vahoarfaparagftruhul popuhalfl*
Eam ljtfrvahuh ef fstlgarf fstf slgftrlm( elstajta ef ha
fstlgatoruh pujmtuah ha gariljfa ljtfrvahuh ef fstlgarf sfjugfstf garb ef froarf*
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Ljtfrvahuh ef fstlgarf
Garla vrfa s'l vje masa l arf jfvolf ef vahoarfa ef pla amasfl*^opuhala ' masfhf slglharf elj martlfr*
Aifjtuh lgoklhlar ahfif ma fajtloj( masfhf slglharf elj martlfr
marf s'au vjeut j uhtlgfhf asf hujl sl apol hf abustfazprfurlhf ef vjzarf pfjtru a ljf sfaga ef el`frfjfhf eljtrfmasa Garlfl l masfhf vjeutf
5;>*>>> 55>*>>> 59>*>>>
5=>*>>> 56>*>>> 59>*>>>
5=>*>>> 50>*>>> 53>*>>>
Fsajtloj prfturl abustatf ef masf
Aifjt lgoklhlarmahmuhfaz gfela
fajtlojuhul l o rfmogaje mavahoarfa ef pla a masfl ef 56>*>>>eoharl*
Aifjtuh lgoklhlara `muto lj`frfja vahorll gfell a popuhalfl*
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Ljtfrvahuh ef fstlgarf
Gfela popuhalflfstf( prokaklh( ljtrf 5;6*>>>% l 596*>>>%(Garba ef froarffstf ef ;6*>>>% `a ef fstlgarfa ljllah ef56>*>>>%* Fstlgarfa fstf2 56>*>>>% ;6*>>>%
Garba ef froarf;6*>>>%
Fstlgator pujmtuah56>*>>>%
Ljtfrvahuh ef fstlgarf5;6*>>>% ' 596*>>>%
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Ljtfrvahf ef jmrfefrf
Ljtfrvah ef jmrfefrf ' ljtfrvahuh a marul prokaklhltatf ef amojljf vahoarfa aefvrat a paragftruhul popuhalfl fstfmujosmut
Ljtfrvahuh ef jmrfefrf pfjtrugfela popuhalfl eam2 akatfrfa stajeare a popuhalfl fstf mujosmut( j fstf su`lmlfjt ef garf pfjtru a utlhlza ujmla R ha mahmuhuh
vahorlhor mrltlmf ahf ljtfrvahuhul ef ljmrfefrf*
jRx
jlvfh efjmrfefrf
R vahoarfmrltlm
& 1(056
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Arla 7 jlvfhuhef ljmrfefrf
Ljtfrvahf ef jmrfefrf
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Fxfgphu2Efmajuh vrfa s fstlgfzf jugruh gfelu ef orf pfsptgja humratf ef stuefjl* Vj fajtloj ef 5< efstuefjl humrfaza lj gfelf ;5 orf( mu o akatfrf stajeare ef5 orf* Fstlgatoruh pujmtuah2 gfela fajtlojuhul 7 ;5 c*
Marf fstf ljtfrvahuh ef jmrfefrf pfjtru uj jlvfh ef jmrfefrfef
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Eama jlvfhuh ef jmrfefrf mrftf( mrftf l garba ef froarf*Ef fxfgphu( eam mrftf jlvfhuh ef jmrfefrf mrfstf ha &( fstf
lgposlklh ef rfahlzat `r a `amf rfmfjsgjtuh jtrfill popuhall*
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Ljtfrvahf ef jmrfefrf
Gfela popuhalflfstfma o osifat marfjtotefauja atljifmfjtruh takfhfl ef Earts
Ljtfrvahuh ef jmrfefrffstf spatluh efsmrls ef ljta j glmarfpfjtru a prljef sifata lj mfjtru*
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Ljtfrvah ef jmrfefrf pfjtru Ljtfrvah ef jmrfefrf pfjtru maje sfmujoastf akatfrfa
stajeare a fsajtlojuhul sf mahmuhfaza mu elstrlkutla t ]tuefjt2
j
stxm
mu j '1 iraef ef hlkfrtatf
Fxfgphu]ahhy aparljfa ujfl oriajlzall ef sajatatlf #O],( marf a avutratljiuh ef 0;* ]ahhy a trfmut ha o jou O]* ^fjtru a fvahua
ratljiuh gfelu ah joll O]( ;> ef gfgkrl au `ost mcfstlojal l arfzuhtat uj ratlji gfelu ef 06( mu o akatfrf stajeare ef 1>*Marf fstf ljtfrvahuh ef jmrfefrf mu uj jlvfh ef jmrfefrf ef
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003(506;>
1>>
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Ljtfrvah ef jmrfefrf pfjtru p
Lj elstrlkula kljoglaha( p fstf proporla ef summfsf apopuhalfl*
j pramtlm( sf utlhlzfaza statlstlmlhf fajtlojuhul pfjtru a fstlgaproporla popuhalfl p*
^roporla fajtlojuhul ef elgfjslujf j fstf fstlgatoruhpujmtuah pfjtru p*Elj tforfga hlgltfl mfjtrahf( eam jp : 6l j#1'p, : 6( elstrlkula
proporlfl fajtlojuhul arf o elstrlkulf aproxlgatlv jorgah*
j
pp
pp
1
p
Ljtfrvah ef jmrfefrf pfjtru proporla popuhalfl( p*
j
ppRp
j
ppRp
11
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Fxfgphu;>> ef o`frl elj Mahl`orjla au `ost ljmhul jtr'uj fajtloj ahfatorlu l s'
a efsmopfrlt m ;6 eljtrf fl au vorklt ha tfhf`ojuh goklh( `r cajes'`rff* ] sf mahmuhfzf fstlgatoruh pujmtuah ah proporlfl efmojeumtorl auto marf utlhlzfaz tfhf`oajfhf goklhf lhfiah l s sf
mojstrulasm ljtfrvah ef jmrfefrf pfjtru 7 >(1;6 sau 1;(6&
^fjtru uj jlvfh ef jmrfefrf ef (>1;6(>;>>
1;6(>11;6(>690(;1;6(>
Garba ef froarf pfjtru sojeab fstf ef 0&( l sf a`lrga mu o jmrfefrfef
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Fstlgatoruh pujmtuah pfjtru ^fjtru a fstlga varlala sau vohatlhltatfa ujul sft ef eatf(
sf utlhlzfaz akatfrfa stajeare(s ma fstlgatorpujmtuah ah akatfrll stajeare a popuhalfl *
FxfgphuLjvfstlllhf A l K au rata rfjtaklhltll ef 0& pf aj* Lj
uhtlgfhf ;5 ef hujl( ljvfstlla A a avut akatfrfa stajearea fajtlojuhul ef =& pf huj( j tlgp mf pfjtru ljvfstltlaK( akatfrfa stajeare a fajtlojuhul a `ost ef 6& pfhuj* ]f poatf a`lrga m ljvfstlla K fstf gal vohatlh lgal rlsmajt efmt ljvfstlla A elj mauza vahorll gal garla akatfrll stajeare fstlgatf*
^fjtru a mrfa uj ljtfrvah ef jmrfefrf pfjtru fstlgarfaakatfrll stajeare( sf utlhlzfaza elstrlkula ;*
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Ljtfrvahuh ef jmrfefrf ;Ljtfrvahuh ef jmrfefrf ju fstf slgftrlm efoarfmf elstrlkula ;
fstf aslgftrlm*
;
;
;
; 1(
1
H[
sjsj
Arla 7jlvfhuh ef
ljmrfefrf
Arla 7 jlvfhuhef ljmrfefrf
Ljtfrvahuh ef ljmrfefrf2
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Fxfgphuj gsurarfa pfr`orgajfl ljvfstllhor( akatfrfa stajeare fstf o
gsur a vohatlhltll sau a rlsmuhul* Eouzfml ef rajeagfjtfhujarf ahf ujul `oje gutuah ljelm uj rajeagfjt gfelu hujar
ef 1& l o akatfrf stajeare a fajtlojuhul ef 6&* ] sfmahmuhfzf ljtfrvahuh ef jmrfefrf morfspujztor jlvfhuhul efjmrfefrf ef '1 7 1< iraef ef hlkfrtatf
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]f poatf spujf mu jmrfefrfa ef
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Ufstarfa lpotfzfhor ujfl popuhall
]tatlstlma lj`frfjlah sf utlhlzfaz2
pfjtru a fstlga uj paragftru ah popuhalfl l a gsura
`laklhltatfa fstlgrll pf uj ljtfrvah ef jmrfefrf*
ha tfstarfa ujfl lpotfzf mu prlvlrf ha uj paragftru ahpopuhalfl( aphlmalf marf mojstltulf fhfgfjtuh mfjtrah ahgftoefl tlljl`lmf ef mfrmftarf*
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@orguharfa prokhfgfhorifjfrahf ahf mfrmftarll
@orguharfa lpotfzfhorsl prolfmtarfa
fxpfrlgfjtuhul
@orguharfamojmhuzllhor lj hlgkabjfstatlstlm
Mohfmtarfa sl ajahlzaeatfhor fxpfrlgfjtahf
Ufstarfa lpotfzfhor ujfl popuhall
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@orguharfa jtrfkrlhor ifjfrahf efmfrmftarf
Mfhf gal guhtf jtrfkrl porjfsm ef ha o surs ef ljsplralf sauo leff efsprf suklfmtuh sau `fjogfjuh ef ljtfrfs*
Mtfva fxfgphf ef jtrfkrl ifjfrahf2
]jtatf2 ^oatf uj slstfg pukhlm ef jirlblrf ah sajatatf s`lf gal f`lmlfjt efmt slstfguh prlvat ef asliurrl?
Gujm2Marf fstf f`fmtuh lgliralfl lhfiahf l fxtfrjahlzarfahomurlhor ef gujm asupra ratfl ogabuhul*?
Fmojoglf2Fstf pamcftuh ef stlguharf fmojoglm ahiuvfrjuhul f`lmlfjt j elgljuarfa rfmfslujll? Feumalf2 Fstf prfa smugp ljvatagajtuh ujlvfrsltar pfjtru
stuefjll?
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Lpotfzfhf sujt efmharatll efsprf paragftrl popuhalfl* Mtfvafxfgphf2
Mfh pulj ;>& elj lj`ramtorll gljorl sujt prljl l mojeagjal
p : >(; Pfjltuh gfelu hujar ah aksohvfjlhor ef `amuhtatf fstf ef mfhputlj 6>> furo : 6>>
^ujmtabuh gfelu ah irupfl 1 fstf amfhasl mu ah irupfl ;1 7 ; ^romfjtuh kohjavlhor ef majmfr puhgojar elj @rajta fstf galglm efmt efmat mfh elj Ahkajla p1 8 p; Akatfrfa stajeare a kursfl ef vahorl elj Jfw \ord fstf mu ef
1> & gal garf lj amft aj : 1>
Ufstarfa lpotfzfhor ujfl popuhall
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Ufstarfa lpotfzfhorfstf o promfeur( kazat pf fvlefjf ahffajtlojuhul l pf tforla prokaklhltll( pfjtru a eftfrglja eamlpotfza 2
fstf o efmharalf rfzojaklh l ju trfkulf rfspljs(
fstf jfrfzojaklh l trfkulf rfspljs*Lpotfza tfstat sf jugftf lpotfza juh( C>* Eam lpotfza juh fstf
jfrfzojaklha sf tfstfaza lpotfza ahtfrjatlva( Ca
Lpotfza juh( C>2 efmharalf mu prlvlrf ha vahoarfa ujul paragftru
ah popuhalfl( marf sf prfsupujf aefvratpfjtru smopuh tfstrll* Lpotfza ahtfrjatlv( Ca2 efmharalf mu prlvlrf ha vahoarfa ujulparagftru ah popuhalfl marf sf prfsupujf aefvrat( eamlpotfza juh fstf rfspljs prlj tfstarf*
Ufstarfa lpotfzfhor ujfl popuhall
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Oksfrvatll2
1* Lpotfza ahtfrjatlv mojtrazlmf j goe jfmfsar lpotfzajuh4 agkfhf lpotfzf ju pot `l aefvratf j amfhal tlgp*
;* Lpotfzfhf trfkulf s `lf efmharall efsprf paragftrlpopuhalfl( l efsprf statlstlmlhf fajtlojuhul*
=* Lpotfza juh tfstfaza fiahltatfa #7( ( ,( lar lpotfza
ahtfrjatlv ljfiahltatf #8* :( ,*
5* ^rlj tfstarf sf jmfarm ma lpotfza juh s sfrfspljif l s sf suslj lpotfza ahtfrjatlv*
Ufstarfa lpotfzfhor ujfl popuhall
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Fxfgphf ef lpotfzf juhf sl ahtfrjatlvf2
C>2 p >(;> Ca2 p : >(;> C>2 6>>> Ca2 : 6>>> C>2 1 7 ; Ca2 1 ;
C>2 p1 p; Ca2 p1 8 p; C>2 1> Ca2 : 1>
Ufstarfa lpotfzfhor ujfl popuhall
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^fjtru a tfsta o lpotfza fstf jfmfsar uj goefh statlstlm marf efsmrlfmogportagfjtuh eatfhor l paragftruh popuhalfl marf fstf tfstat*
Goefhf statlstlmf sujt elj `aglhla jorgah( mfhf gal lgportajtfelstrlkull sujt2 ; R( U sl @*
@lfmarf goefh sf `orguhfaz pf kaza ujor lpotfzf a mror oportujltatfsf vfrl`lma*
Ef fxfgphu2 guhtf goefhf mojslefr m gfela fajtlojuhul arf oelstrlkulf aproxlgatlv jorgah( lpotfz marf ju sf aphlma eamsfturlhf ef eatf sujt glml sau elstrlkultf aslgftrlm*
]f ahfif uj tfst statlstlm marf fstf utlhlzat pfjtru a efmlef eam lpotfzajuh sf rfspljif sau ju sf rfspljif*
Ufstarfa lpotfzfhor ujfl popuhall
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@orguharfa lpotfzfhor
Lpotfza juh Lpotfza ahtfrjatlv
Eftfrgljarfa goefhuhul aefmvat
Ufst statlstlm Ufstarfa ljtfrvahuhul mu ujasau eoua hlgltf
]taklhlrfa mrltfrlhor ef efmlzll
Jlvfhuh ef sfgjl`lmalf l putfrfa tfstuhul
Ufstarfa lpotfzfhor ujfl popuhall
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Frorl j huarfa efmlzllhor
Huarfa ujfl efmlzll lgphlma o prokaklhltatf ef a `amf o ahfifrf irflt*
Ufstarfa lpotfzfhor mfrf2 s sf rfsplji lpotfza juhl s sf suslj lpotfza ahtfrjatlv s ju sf rfsplji lpotfza juh*
]f pot `amf eou tlpurl ef frorl2 Froarf ef tlp L sf rfspljif lpotfza juha mje amfasta fstf
aefvrat* Froarf ef tlp LL ju sf rfspljif lpotfza juha mje amfsta fstf
`ahs*
Ha tfstarfa lpotfzfhor( sf la j mojslefrarf prokaklhltatfa ef a `amf ujaelj amfstf frorl
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C>fstf ammfptat C>fstf rfspljsC>fstf aefvarat Efmlzlf morfmta Froarf ef tlp L
C>fstf `ahs Froarf ef tlp LL Efmlzlf morfmta
Frorl j huarfa efmlzllhor
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Fxfgphu ef froarf ef tlp L
O mogpajlf ef gfelmagljtf mogfrmlahlzfaz uj slrop( marf s'a eovfelta `l ljf`lmlfjt j tratagfjtuh tusfl* jaljtf ef mogfrmlahlzarf(mogpajla a mfrmftat l staklhlt m gfelmagfjtuh fstf f`lmlfjt jtratagfjtuh tusfl( mffa mf jsfagj s'a rfspljs lpotfza juh mslropuh ju arf jlml uj f`fmt asupra kohll*
O mogpajlf marf tfsta utlhlzarfa Ljtfr`frojuhul ha tratagfjtuh ujfl kohlpuhgojarf( a akajeojat mfrmftarfa mje tfstarfa a artat mLjtfr`frojuh fstf ljf`lmlfjt* Mogpajla ju a rfult s rfspljilpotfza juh m gfelmagfjtuh fstf ljf`lmlfjt* Mf sf jtgpheam gfelmagfjtuh ar `l `ost mu aefvrat f`lmlfjt? Eam mogpajla`amf froarf ef tlp LL? ^oslklh( ear elj gogfjt mf gfelmagfjtuh ju a
`ost mogfrmlahlzat( ju sf mujoatf aefvruh*
^rokaklhltatfa gaxlg ef a `amf o froarf ef tlp Lsf jugftf jlvfhef sfgjl`lmalf ah tfstuhull sf jotfaz mu *
^rokaklhltatfa ef a `amf o froarf ef tlp LL sf jotfaz mu *
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Pahoarfa mrltlm l zoja ef rfspljifrf
Eama jlvfhuh ef sfgjl`lmatlf ah tfstuhul fstf ahfas( sfeftfrglja elj `ujmla ef elstrlkulf ef prokaklhltatf atfstuhul statlstlm o zoj j marf lpotfza juh fstfrfspljs ' zoj ef rfspljif
Pahoarfa mrltlm ' hlglta eljtrf zoja ef rfspljifrf l efjfrfspljifrf*
^fjtru o sltualf eat( fxlst o vahoarf mrltlm l o zoj efrfspljifrf*
]upra`aa totah a zojfl ef rfspljifrf fstf fiah mujlvfhuhul ef sfgjl`lmalf *
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Pahoarf mrltlma
]f rfspljif C>Ju sf rfspljif C>
Arla zojfl efrfspljifrf 7
Arla zojfl lj marfju sf rfspljif 7 1'
>
Ufstarfa ujlhatfraha a lpotfzfl
Vj tfst fstf ujlhatfrah
eam lpotfza ahtfrjatlv(Ca( a`lrg o sltuatlf( mugar `l2C>2 Pfjltuh gfelu ah`fgflhor fstf mu mfh ah
krkalhor*Ca2 Pfjltuh gfelu ah`fgflhor : efmtkrkalhor*
Efoarfmf fiahltatfa fstfspfml`lm lpotfzfl juhf(lpotfza ahtfrjatlv vaeftfrglja marf gariljfa ljtfrvahuhul sf va tfsta*
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>
Arla zojfl efrfspljifrf 7 /;Arla zojfl effspljifrf 7 /;
Pahoarf mrltlmaPahoarf mrltlma
]f rfspljif C>]f rfspljif C>
Ju sf rfspljif C>
Arla zojfl lj marfju sf rfspljif 7 1'
Ufstarfa klhatfraha a lpotfzfl
Vj tfst fstf klhatfraheama lpotfzfl ahtfrjatlvfCa ju a`lrga o sltuatlf(mug ar `l2
C>
2 Pfjltuh gfelu ah`fgflhor fstf fiah mumfh ah krkalhor*Ca2 Pfjltuh gfelu ah`fgflhor ju fstf fiahmu vfjltuh gfelu ah
krkalhor*
j tfstuh klhatfrah( sfgpartf j eou prl(efoarfmf fxlst eou
zojf ef rfspljifrf*
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Fxfgphf ef tfstarf a gfelf `a ef o vahoarfprfsupus >2
Ca2 : >tfstuh gfelfl pfjtru hlglta supfrloarl fstf efjuglt tfstuh hlgltfl elj erfpta*
Ca2 8 >tfstuh gfelfl pfjtru hlglta lj`frloar l
fstf efjuglt tfstuh hlgltfl elj stjia*
Ca2 >tfstuh gfelfl pfjtru agkfhf hlgltf*
Ufstarfa lpotfzfhor ujfl popuhall
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Mohfmtarfa l ajahlza eatfhorfxpfrlgfjtahf
Mohfmtarfa sl vfrl`lmarfa eatfhor
F`fmtuarfa fxpfrlgfjtuhul Pfrl`lmarfa vahorlhor akfrajtf
]taklhlrfa tfstuhul statlstlm l /sau vahoarfa hul p
Mogpararf mu vahorlhf mrltlmf Mogpararf mu
]f la o efmlzlf pfjtru C>
[fspljifrfa C>sl susljfrfa Ca Ju sf rfspljifC>
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Uratarfa vahorlhor akfrajtf
O vahoarf akfrajt fstf eata marf el`fr `oartf guht efmfhfhahtf eatf elj sft*
Pahorlhf akfrajtf pot `l eftfrgljarf ef2
irffhl ha jrfilstrarfa eatfhor( eatf marf ju am partf elj popuhalf( fvfjlgfjtf rarf rfahf*
FxfgphuPjzrlhf trlgfstrlahf ef masf a 1> aifjl sujt2 ;( ;( =( 5( 6(
6( 0( 0( 9( 6>
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Mu vahorl akfrajtf @r vahorl akfrajtf
Gfela > 5(55
Gfelaja 6(>> 6(>>
Akatfrfa stajeare 15(61 1(31
Ha mahmuharfa statlstlmlhor( sf oksfrv m gfela l akatfrfa stajearesujt a`fmtatf ef vahoarfa akfrajt( j tlgp mf gfelajaju sf smclgk*
O sohulf pfjtru popuhala mu vahorlhf fxtrfgf fstf utlhlzarfa statlstlmlllj`frfjlahf*
Elairaga Kox sf utlhlzfaz pfjtru a eftfrgljarfa vahorlhor akfrajtf*
Uratarfa vahorlhor akfrajtf
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Hoilma tfstarll lpotfzfhor
Ha tfstarfa lpotfzfhor sf la efmlzla eff rfspljif sau ef jfrfspljifrf alpotfzfl juhf* ^romfsuh efmlzlojah slglhar mu mfh elj gatfgatlm ef +aeovfel prlj mojtraelmlf fstf2
]f prfsupujf m C>fstf aefvrat l Cafstf mogphfgfjtar hul C>* ]f ajahlzfaz eam eatfhf morfspuje mu C>
Eam eatfhf ju morfspuje( sf eovfeftf m C>fstf `ahs2 sfrfspljif C>l sf propujf Ca ' efmharalf putfrjlm*
Eam ajahlza eatfhor ju fstf mojmhuefjt pfjtru rfspljifrfa C>sf a`lrg m eatfhf sujt jfmojmhuefjtf ' efmharala shaka*
Oksfrvatlf2
1* Efmharatlf shaka ' ju s'a efgojstrat C>( eoar a `ost lj`lrgata*;* Lj statlstlm a efgojstra o lpotfza jsfagj m lpotfza fstf
susljut mu mfrtltueljfa ef #1',1>>&*
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Fxfgphu
Vj mfrmfttor eorftf s sutlja a`lrgala m( j gfelf( stuefjtll trlglt galguht ef 1>>> ef ]G]'url pf huj*
Lpotfzfhf ef mfrmftarf sujt2
C>2 7 1>>> vs Ca2 : 1>>>*
Efmlzla prlj marf C>fstf rfspljs2 jugruh gfelu ef ]G] trlglsf efstuefjl efpftf 1>>>*
Efmlzla prlj marf C>ju fstf rfspljs2 ju fxlst eovfzl su`lmlfjtfpfjtru a susljf a`lrgala m jugruh gfelu ef ]G]'url trlglsf ef
stuefjl efpftf 1>>>*
Oksfrvatlf2Efmlzla sf hlgltfaz ha popuhala elj marf s'a prfhfvatfajtlojuh
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Mogparfa vahorll p mu Ufst statlstlm ' vahoarf mahmuhat elj eatf j moj`orgltatf mu
goefhuh statlstlm ah eatfhor*
]f mahmuhfaza vahoarfa mrltlm p pfjtru jlvfhuh efsfgjl`lmalf ahfs( *
Ufstuh statlstlm sf mogpar mu vahoarfa mrltlm a tfstuhullpotfzfl*
Eam vahoarfa p 8 ( C>fstf rfspljs Eam vahoarfa p : ( C>ju fstf rfspljs*
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Pahoarf mrltlma
]f rfspljif C>Ju sf rfspljif C>
Ufststatlstlm
Efmlzla ef rfspljifrfa a C>
C>2 7 1>Ca2 : 1>
]f ahfif sl sf mahmuhfaza
vahoarfa mrltlm p 7 tfststatlstlm*
Ufstuh statlstlm sf jmaerfazj zoja ef rfspljifrf2 p 8
Efmlzlf ' sf rfspljif C>*
Efmharalf putfrjlm2 Eatfhfsuslj lpotfza ahtfrjatlv*
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Pahoarf mrltlma
]f rfspljif C>Ju sf rfspljif C>
Efmlzlf ' ju sf rfspljif C>
C>2 7 1>Ca2 : 1>
]f ahfif sl sf mahmuhfazavahoarfa mrltlm p 7 tfststatlstlm**
Ufstuh statlstlm ju sfjmaerfaz j zoja efrfspljifrf2 p :
Efmlzlf ju sf rfspljif C>*
Efmharalf shak2 Eatfhf sujtjfmojmhuefjtf l ju sprlblj
lpotfza ahtfrjatlv*
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Ufst ah vahorll gfell vs vahoarfa lpotftlmFxfgphu2 O mogpajlf ef proeusf ahlgfjtarf aphlm pohltlma j marf
mojljutuh efmharat ah ujul proeus fstf lefjtlm mu ah proeusuhul`akrlmat*
Ljtrfkarf2 Irfutatfa efmharat a proeusuhul morfspujef mu irfutatfarfah?
Vj statlstlmlaj j mojtrohuh mahltll efmlef s tfstfzf stlmha mu sos ef solaef 10 ujmll* Lj amfst sfjs prolfmtfaz fxpfrlgfjtuh*
]tatlstlmlajuh fstf autorlzat sa prfhfvfzf =0 ef stlmhf tlgp ef o ora( ef hao sljiura gaslja ef lgkutfhlat4 elj eatf ajtfrloarf 7 >(6 oz*
]f tfstfaza gfela popuhalfl vs vahoarfa lpotftlm ef 10 ujmll*
]f ahfif uj tfst mu eou hlgltrl( efoarfmf mogpajla fstf prfomupatatt supraugphfrfa mt l ef sukugphfrfa stlmhfhor*
Lpotfzf ef mfrmftarf2C>2 7 10 ' gaslja ef ugphfrf `ujmlojfaz morfmt
Ca2 10 ' gaslja ef ugphfrf ju `ujmlojfaz morfmt*
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Efoarfmf fstf mujosmut tfstuh statlstlm fstf2
Ufst ah vahorll gfell vs vahoarfa lpotftlm
Goefhuh sf aphlma eama elgfjslujfa fajtlojuhul asliur elstrlkulajorgah a gfelfl fajtlojuhul*Froarfa ef tlp L ' sf rfspljif lpotfza juh l sf a`lrga m gaslja eflgkutfhlat ju `ujmlojfaz morfmt atujml mje fa `ujmlojfaz morfmt*jtrumt somlftatfa ju vrfa s sf oprfasm ljutlh proeumla l s
rfmahlkrfzf gaslja( statlstlmlajuh ahfif s hlgltfzf prokaklhltatfa frorllef tlp L prlj staklhlrfa ha 6&*
]tatlstlmlajuh a prfhfvat =0 ef stlmhf ef sos ef sola elj marf o sljiurvahoarf fstf jfokljult ef 19(60 ujmll* Pahoarfa fstf ammfptat j sftuhef eatf*
j
xR
/
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]f mahmuhfaz vahoarfa gfelf a fajtlojuhul l a tfstuhul statlstlm*
55(1=0>(6/
10'10(1;R1;(10 ujmllx
Gfeoea 12 Lpotfza juh sf rfspljif eama R fstf j zoja ef rfspljifrf('1*
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Pahoarf mrltlmaPahoarfmrltlma ]f rfspljif C>]f rfspljif C> Ju sf rfspljif C>
Gftoea R
Ufst ah vahorll gfell vs vahoarfa lpotftlm
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p
Ufst ah vahorll gfell vs vahoarfa lpotftlm
Gftoea p
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Mojmhuzllhf l rfmogajearlhf statlstlmlajuhul pfjtrumojeumfrfa mogpajlfl2
+Ju fxlst eovfzl su`lmlfjtf pfjtru a mojmhuzloja mgalja marf ugphf stlmhf ef sos sola ef 10 ujmll`ujmlojfaz jfmorfspujztor* Amfast mojmhuzlf sfkazfaz pf gsurtorl f`fmtuatf pf =0 ef stlmhf pf eurataujfl sljiurf orf ef proeumlf*
P rfmogaje gojltorlzarfa mojtlju a galjll pfel`frltf smclgkurl( efoarfmf fstf poslklh o froarf ugaj
j aparlla varlallhor ef irfutatf*+
Ufst ah vahorll gfell vs vahoarfa lpotftlm
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Froarf ef tlp LL l putfrfa statlstlm
j fxfgphuh ajtfrlor( statlstlmlajuh ju a rfult s rfsplji lpotfza juh(efoarfmf prokaklhltatfa ef a `amf frorl ef tlp L #ef rfspljifrf a ujfllpotfza juh aefvrat, a efplt jlvfhuh ef sfgjl`lmalf ef 6&*
]tatlstlmlajuh ar `l putut `amf o froarf ef tlp LL( eama galja `ujmlojajfmorfspujztor*
Vja elj sarmljlh fstf ef a ajahlza prokaklhltatfa ef a `amf froarfa eftlp LL ' *
' prokaklhltatfa ma s ju `lf rfspljs lpotfza juh eam fstf `ahs
]tatlstlmlfjl utlhlzfaz putfrfa statlstlm( marf fstf mogphfgfjtuh hul *
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Lpotfz
Lporfza juh Lpotfza ahtfrjatlv
Eftfrgljarfa goefhuhul morfspujztor
Ufst statlstlm Hlgltarf ujl sau klhatfraha
Mrltfrll pfjtru huarfa efmlzlhor
Jlvfhuh ef sfgjl`lmalf l ajahlza putfrll
Froarf ef tlp LL l putfrfa statlstlm
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Ju sf rfspljif C> ]f rfspljif C
>
C>fstf aefvrat 1' Froarf ef tlp L
C>fstf `ahs Froarf ef tlp LL
1'putfrf
Froarf ef tlp LL l putfrfa statlstlm
Eam tfstuh lpotfzfl arf putfrf glm( fstf el`lmlh s sf rfsplji C>( mclaream C>fstf `ahs4 mfrmftarfa ar `l o plfrefrf ef tlgp sl kajl*
j fxfgphuh ajtfrlor( lpotfza ahtfrjatlva `ost m 10 ujmll( aelmgalja ugphf stlmhfhf a 7 10*>>>1 ujmll( lar Cafstf pramtlm aefvrat*Eam sf ajahlzfaz putfrf l froarfa ef tlp LL sf ahfif o vahoarf pfjtrugfela popuhalfl pfjtru lpotfza ahtfrjatlv #a,( marf fstf +pramtlm el`frlt+ef gfela lpotfzfl juhf #>,*
Amfast el`frfj pramtlm sf jugftf elgfjslujfa f`fmtuhul*
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El`frfjaeljtrf gfellhf lpotfzfhor sf jugftfelgfjslujfa f`fmtuhul
>2 vahoarfa gfelfl a popuhalfl pfjtru lpotfza juh a2 vahoarfa gfelfl a popuhalfl pfjtru lpotfza
ahtfrjatlv Elgfjslujfa f`fmtuhul2 +el`frfj pramtlm+ jtrf >l a
7 |> ' a|*
Froarf ef tlp LL l putfrfa statlstlm
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] prfsupujfg m sf f`fmtufaz uj tfst ujlhatfrah ah gfelflpopuhalfl2
C>2 7 > Ca2 : >
]f lau j mojslefrarf mfhf eou ira`lmf2
Ira`lmuh ef sus fstf elstrlkula gfelfl fajtlojuhul pfjtrulpotfza juh* Roja elj erfapta a vahorll mrltlmf fstf zojaef rfspljifrf*
Ira`lmuh ef bos fstf elstrlkula gfelfl fajtlojuhul pfjtrulpotfza ahtfrjatlv pfjtru vahoarfa spfml`lm a a* ^famfst ira`lm sf poatf gasura slputfrfa tfstuhul
Froarf ef tlp LL l putfrfa statlstlm
Pahoarf mrltlma
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Pahoarf mrltlma
]f rfspljif C>Ju sf rfspljif C>
fstf asomlata muprokaklhltatfa frorll ef tlp L(sf rfspljif C> maje fstfaefvarata
fstf asomlata muprokaklhltatfa frorllef tlp LL( ju sfrfspljif C> maje fstf`ahsa
^utfrfa fstfasomlata mu
prokaklhltatfa ef arfspljif lpotfza C>`ahsa
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Gftoef ef mrftfrf a putfrll statlstlmf
Goeahltl ef mrftfrf a putfrll [fahlzarf
Mrftfrfa elgfjslujll fajtlojuhul Mrftf mostuh l sf rfeumfvarlaklhltatfa eatfhor
Mrftfrfa jlvfhuhul efsfgjl`lmalf( Mrftf prokaklhltatfa ef rfspljifrfa lpotfzfl C>aefvratf froarf eftlp L
Ahfifrfa vahorll a el`frlt ef > [fzuhtatfhf pot `l gal puljsfgjl`lmatlvf
[fef`ljlrfa popuhalfl pfjtru aglmora akatfrfa stajeare
[fzuhtatfhf pot `l prfa hlgltatf
]f rfmogaje f`fmtuarfa ujultfst ujlhatfrah l ju klhatfrah
^ot proeumf rfzuhtatf prtljltoarf
F h t t t h l
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Fxfgphu putfrfa tfstuhul
^hmufhf ef `rj ahf autokuzfhor sujt prolfmtatf s eurfzf jgfelf( mfh pulj 0>*>>> ef dg l o mogpajlf vrfa s tfstfzfamfast a`lrgalf*
Mogpajla ef autokuzf vrfa s eftfrgljf vahoarf +pramtlm+ musmopuh ef grlrll sliuraja autokuzuh( efoarfmf uhtlgfhf
tagpojrl s'au proeus ha mfh pulj 63(>>> dg* Eam akatfrfastajeare fstf ef 6*>>>( lar elgfjslujfa fajtlojuhul ef 6>(marf fstf putfrfa tfstuhul eam gfela pramtlma fstf ef 63(>>>dg* #sf prfsupujf 7 >*>6,
]f eftfrglja vahoarfa mrltlm a tfstuhul* C>sf rfspljif eam R 8 '1*056 ]f mahmuhfaza vahoarfa marf morfspujef vahorll mrltlmf2
6333=96>/6>>>056(10>>>>>
j
Rx
x
F h h l
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C>sf rfspljif eam 8 633=9]f mahmuhfaza prokaklhltatfa ef rfspljifrf a C>( eam Ca fstf
aefvrat2
x
331(>13(16>/6>>>63>>>633=9
/
633=9
633=9>
R^R^jR^x^
^utfrfa amfstul tfst fstf ef 33& l fstf ef 1;&*
Pahorl ljllahf
> 7 0>*>>> dga 7 63*>>> dg 7 >(>6j 7 6>
7 6>>> dg
Pahorl mahmuhatfF`fmtuh grlgll 7 ;*>>> dgPahoarfa mrltlm 7 63*3=9 dg 7 >(11 sau 1;&^utfrfa 7 >*331> sau 33&
Fxfgphu2 putfrfa tfstuhul
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^utfrfa 7 33&
Pahoarf mrltlma
]f rfspljif C> Ju sf rfspljif C>
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