Tavi VIII. statistikuri hipoTezebis Semowmebis Sesavali: z kriteriumi da erTi SerCevis t kriteriumi

statistikuri hipoTezis Semowmebaparametruli da araparametruli kriteriumebistatistikuri hipoTezebikriteriumis statistika: zmniSvnelovnobis doneebi: statistikuri iSviaTobagadawyvetilebis miReba statistikuri hipoTezebis Sesaxeb

kriteriumis statistikisa da mniSvnelovnobis donis gamoyenebiT

statistikuri hipoTezis Semowmebis etapebis SejamebaerTi SerCevis t kriteriumi

t statistikis SerCeviTi ganawileba statistikuri hipoTezis Semowmeba erTi SerCevis t

kriteriumiTerTi SerCevis t kriteriumis Sedegebis prezentaciamniSvnelovani sakiTxebi statistikuri hipoTezis SemowmebaSi

ratom ar viRebT nulovan hipoTezas?I da II gvaris Secdomebi statistikuri kriteriumebis gamoyenebisas

statistikuri da mecnieruli mniSvnelovnobandobis intervalebis ageba populaciis saSualosaTvis t statistikis gamoyenebiT

wina TavebSi ganvixileT statistikis gamoyeneba monacemTa organizebis, aRweris mizniT da populaciis parametrebis Sesafaseblad. am TavSi ki ganvixilavT statistikis gamoyenebas hipoTezis Semowmebis mizniT.

ganvixiloT Semdegi problema. adamianuri faqtori aris bihevioristuli mecnierebis sfero, romelic adamianis SesaZleblobebisadmi manqanisa da teqnologiis Sesatyvisobas swavlobs. adamianuri faqtoris Semswavleli fsiqologebi xSirad atareben adamianTa fizikuri maxasiaTeblebis gazomvebs, raTa Seqmnan skamebis, kontrolis panelisa da monitoris iseTi dizaini, romelic adamians moergeba. magaliTad, warmoidgineT, rom xarT satvirTo manqanebis mwarmoebeli kompaniis TanamSromeli, romelic pasuxismgebelia Sida dizainze. wlebis manZilze skamebi Tqvens satvirTo manqanebSi iseTnairad iyo mowyobili, rom Seesatyviseboda mZRolebis populacias, romelTa saSualo muxlqveSa simaRle iyo 42 santimetri (sm) da _ 3 sm. fexis muxlqveSa simaRle aris vertikaluri manZili mjdomare mZRolis muxlidan iatakamde.

axlaxan Tqvenma kompaniam daiwyo satvirTo manqanebis gayidva mTel msoflioSi da miiRo saCivrebi skamebis arakomfortulobis Taobaze. Tqven ivaraudeT, rom SesaZloa zogierT qveyanaSi mZRolebis muxlqveSa simaRle gansxvavdeba im populaciisagan, romlisTvisac Seqmnili iyo es skamebi. am hipoTezis Sesamowmeblad, Tqven aiyvaneT satvirTo manqanis 25 mZRolisagan Semdgari SerCeva, romelic im qveynis mZRolebis populaciisagan damoukideblad da SemTxveviTad iyo SerCeuli, romelTaganac iyo miRebuli saCivrebi. am SerCevis mZRolebis saSualo muxlqveSa simaRle aris 44 sm. aSkaraa, rom am SerCevis saSualo _ 44 sm ar aris igive, rac im populaciis saSualo, romelic 42 sm-is tolia. magram ras gveubneba es gansxvaveba populaciis Sesaxeb? arsebobs ori SesaZlebloba. pirveli aris is, rom populacias, romlidanac iyo aRebuli realurad SerCeva, saSualo muxlqveSa simaRle 42 sm-is toli aqvs, xolo Tqveni SerCevis saSualo _ 44 sm gansxvavdeba am -gan mxolod SerCevis Secdomis gamo. SemTxveviT Tqven SearCieT iseTi SerCeva, romlis saSualo muxlqveSa simaRle iyo 42 sm-ze meti.

meore SesaZlebloba isaa, rom SerCevis saSualo asaxavs populacias iseTi -iT, romelic gansxvavdeba 42 sm-isagan. mZRolebis populacias am qveyanaSi aqvs saSualo muxlqveSa simaRle, romelic gansxvavdeba 42 sm-sagan. am ori SesaZleblobidan romeli unda airCioT rogorc im situaciis amsaxveli, romelic arsebobs im qveyanaSi, saidanac iyo aRebuli SerCeva?

Cven gvaqvs problema, romelic moiTxovs statistikuri hipoTezis Semowmebas, raTa gavakeToT arCevani, Tu romelia ufro metad mosalodneli im ori SesaZlo situaciidan. statistikuri terminebiT, es problema moiTxovs statistikuri hipoTezis Semowmebas imis gadasawyvetad, mniSvnelovnad gansxvavdeba Tu ara SerCevis saSualo _ 44 sm populaciis saSualosagan, romelic 42 sm-is tolia. suraTi #8.1 am problemis ilustracias warmoadgens.

suraTi #8.1. ori SesaZlo situacia, romelic SeiZleba ganapirobebdes SerCevis realurad miRebul saSualos. SerCevis realuri saSualo = 44 sm

qvemoT mocemuli ori SesaZleblobidan romeli ganapirobebs SerCevis am saSualos?

.

statistikuri hipoTezis Semowmeba parametruli da araparametruli kriteriumebi

parametruli kriteriumebi

parametrulia kriteriumebi Tu hipoTezaSi aris gamoTqmuli varaudi populaciis iseTi parametrebis Sesaxeb rogoricaa, magaliTad, da 2. parametruli kriteriumebi xSirad garkveul pirobebs warmoadgenen, rogorc WeSmarit daSvebas im populaciis monacemebisaTvis, romlidanac aris aRebuli SerCeva. roca gavecnobiT konkretul parametrul kriteriumebs, agreTve, unda ganvsazRvroT am

kriteriumebisaTvis gakeTebuli daSvebebi. am TavSi gavecnobiT statistikuri hipoTezis Semowmebas, romelic iyenebs or gansxvavebul parametrul kriteriums, erTi SerCevis z

kriteriums da erTi SerCevis t kriteriums. sxva parametrul kriteriumebs gavecnobiT IX-XII TavebSi.

araparametruli kriteriumebi

parametruli kriteriumi (parametric

test) _ statistikur kriteriums ewodeba parametruli, Tu hipoTezaSi aris gamoTqmuli varaudi populaciis parametris Sesaxeb.

populaciis saSualo, romlidanac SerCeva iyo aRebuli, aris 42 sm. SerCevis saSualo gansxvavdeba populaciis saSualosagan SerCevis Secdomis gamo.

populaciis saSualo, romlidanac SerCeva iyo aRebuli, ar aris 42sm. SerCevis saSualo gansxvavdeba populaciis saSualosagan ara SerCevis Secdomis gamo, aramed sxva mizeziT.

rogor vupasuxoT SekiTxvas: gamoviyenoT statistikuri kriteriumi, raTa ganvsazRvroT, gansxvavdeba Tu ara

=44 sm mniSvnelovnad =42 sm-sagan.

araparametrulia kriteriumebi, Tu hipoTezaSi ar aris gamoTqmuli varaudi populaciis parametris Sesaxeb. Cveulebriv, araparametruli kriteriumebi ar akeTeben daSvebebs im populaciis

monacemebis ganawilebis Sesaxeb, romlidanac aris aRebuli SerCeva. am mizeziT, isini xSirad miekuTvnebian e.w. ganawilebisagan Tavisufal kriteriumebs. araparametruli kriteriumebi xSirad gamoiyeneba, roca parametruli kriteriumebisaTvis aucilebeli daSvebebi ar Seesabameba monacemebs. parametruli kriteriumebis didi nawilisaTvis, magram ara yvela kriteriumisaTvis, arsebobs alternatiuli araparametruli kriteriumebi igive monacemebis gasaanalizeblad. ramdenime araparametruli kriteriumi warmodgenilia me-13 TavSi. statistikuri hipoTezis Semowmebisas gamoyenebuli ZiriTadi cnebebis umravlesoba erTnairia parametruli da araparametruli kriteriumisaTvis.

statistikuri hipoTezebistatistikuri hipoTezis

Semowmeba, iqneba is parametruli Tu araparametruli, iwyeba statistikuri hipoTezis CamoyalibebiT. parametruli kriteriumisaTvis statistikuri hipoTeza aris debuleba populaciis parametris Sesaxeb. es debuleba SeiZleba WeSmariti iyos da SeiZleba _ ara. is ubralod imisaTvis keTdeba, raTa safuZveli daudos Semowmebas. magaliTad, Cven SegviZlia CamovayaliboT hipoTeza imis

Sesaxeb, rom muxlqveSa simaRlis saSualo im populaciisaTvis, romlidanac ayvanili gvyavda 25 kaciani SerCeva, aris 42 sm. realurad populaciis saSualo SesaZloa iyos 42 sm an ar iyos. Cven ubralod vvaraudobT, rom is 42 sm-ia da am hipoTezas SemdgomSi SevamowmebT. statistikuri kriteriumebi moiTxovs, rom formulirebuli iyos ori hipoTeza, nulovani da alternatiuli.

nulovani hipoTezanulovani hipoTeza aris im pirobis Camoyalibeba, romelic mecniers populaciis

Sesaxeb apriorulad WeSmaritad miaCnia. es aris hipoTeza, romelic mowmdeba statistikuri kriteriumiT. nulovani hipoTeza, Cveulebriv, aRiniSneba simboluri an pirobiTi formiT H0-is

araparametruli kriteriumi (nonparametric test) _ statistikur kriteriums ewodeba araparametruli, Tu hipoTezaSi ar aris gamoTqmuli varaudi populaciis parametris Sesaxeb.

statistikuri hipoTeza (statistical hypothesis) _ debuleba populaciis parametris Sesaxeb(parametruli kriteriumis Sesaxeb).nulovani hipoTeza (null hypothesis) - aris im pirobis Camoyalibeba, romelic mecniers populaciis Sesaxeb apriorulad WeSmaritad miaCnia. es aris hipoTeza, romelic mowmdeba statistikuri kriteriumiT. alternatiuli hipoTeza (statistical alternative hypothesis) _ aris debuleba imis Sesaxeb, Tu ra unda iyos WeSmariti, Tu nulovani hipoTeza mcdaria.

gamoyenebiT imis aRsaniSnavad, rom SemoTavazebulia iseTi hipoTeza, rogoricaa =42 smis. nulovani hipoTezis _ =42 sm gamoxatva statistikur CanawerSi Semdegnairad SegviZlia H0:=42sm, da es gamoiTqmis aseTnairad: `aS nuli: miu tolia 42 sm-is”.

alternatiuli hipoTezaalternatiuli hipoTeza aris debuleba imis Sesaxeb, ra unda iyos WeSmariti, Tu

nulovani hipoTeza mcdaria. magaliTad, Tu H0 : =42 sm ar aris WeSmariti, maSin alternatiuli unda iyos, rom ar aris 42 sm-is toli. es alternatiuli hipoTeza aRiniSneba H1-iT da iwereba Semdegnairad: H1: 42 sm. es gamoiTqmis aseTnairad: `aS erTi: miu ar aris 42 sm-is toli“.

statistikuri hipoTezebis TvisebebiTiToeuli statistikuri kriteriumi moiTxovs nulovani da alternatiuli hipoTezebis

formulirebas. es hipoTezebi iseTnairad unda iyos warmodgenili, rom akmayofilebdnen Semdeg pirobebs:

isini arian urTierTgamomricxavni: urTierTgamomricxavi niSnavs, rom orive hipoTeza ar SeiZleba iyos WeSmariti erTdroulad. magaliTad, SeuZlebelia H0 : = 42 sm da H1 :

42 sm iyos erTdroulad WeSmariti erTi da imave populaciisaTvis. isini moicaven parametris yvela SesaZlo mniSvnelobas, romelic CarTulia hipoTezaSi.

H0 : =42 sm da H1 : 42 sm akmayofilebs am moTxovnas; maT Soris moTavsebulia -s yvela SesaZlo mniSvneloba CvenTvis saintereso populaciisaTvis.

am Tvisebebis gamo mocemuli ori hipoTezidan mxolod erTi unda warmoadgendes populaciisaTvis WeSmarit pirobas. magaliTad, populaciis saSualo an aris 42 sm-is toli da H0 : = 42 sm aris WeSmariti, an ar aris toli da maSin alternatiuli hipoTeza H1 : 42 sm-s aris WeSmariti.amrigad, ganxiluli problemisaTvis statistikuri hipoTezebia:H0 : = 42 sm (nulovani hipoTeza)H1 : 42 sm (alternatiuli hipoTeza)

statistikuri hipoTezebis funqcianulovani hipoTezis funqciaa statistikur kriteriumSi daadginos is pirobebi, romlis drosac

statistikis SerCeviTi ganawileba iyo miRebuli. me-7 TavSi Cven ganvsazRvreT saSualos SerCeviTi ganawileba, rogorc -is mniSvnelobebis ganawileba, roca N moculobis yvela SesaZlo SerCeva populaciidan iyo aRebuli. magaliTad, Tu vivaraudebT, rom H0:=42 sm aris WeSmariti da =3 sm, Cven SegviZlia miviRoT -is SerCeviTi ganawileba am populaciis iseTi SerCevebisaTvis, romelTa moculoba aris 25. suraTze #8.2 mocemuli es

SerCeviTi ganawileba aris normaluri ganawileba populaciis saSualoTi =42.0 sm, standartuli SecdomiT _ -iT, sadac

am ganawilebaSi SerCeviTi saSualo _ 44 sm -isagan daSorebulia saSualos sami standartuli SecdomiT. Sesabamisad, im pirobebSi, roca H0:=42 sm-s, iseTi SerCevis albaToba, romlis saSualoa 44 sm, aris Zalian mcirea, Tuki populaciidan aRebuli SerCevis moculoba iqneba 25. amitomac Cven ufro mivemxrobodiT nulovani hipoTezis _ H0:=42 sm-is uaryofas, Tuki aRebuli SerCevis saSualo iqneboda 44 sm. -is aseTi mniSvneloba aris statistikurad iSviaTi xdomiloba, roca H0 aris WeSmariti. magram sanam gadavwyvetdeT sabolood ganvixiloT Tu ara H0 rogorc WeSmariti hipoTeza, Cven unda:

davadginoT albaToba imisa, rom SeiZleba miviRoT -is iseTi mniSvneloba, romelic iqneba 44 sm-is toli an masze meti, Tu H0:=42 sm iqneba WeSmariti. es nabiji moiTxovs kriteriumis statistikis gamoyenebas;

ganvsazRvroT ras vgulisxmobT statistikurad iSviaT xdomilobaSi. es nabiji moiTxovs mniSvnelovnobis donis gansazRvras.

suraTi #8.2. saSualos Teoriuli SerCeviTi ganawileba agebuli N=25 moculobis toli

SerCevebisaTvis =42.0 sm-

is da _ =0.6-s mqone

populaciidan.

kriteriumis statistika: z

kriteriumis statistika aris ricxvi, romelic nulovani hipoTezis

Semowmebis saSualebas gvaZlevs da gamoiTvleba SerCevis monacemebze dayrdnobiT. magaliTad, Cven gvinda kriteriumi, raTa davadginoT albaToba imisa, rom SerCeviTi saSualo iqneba 44 sm-is toli an masze meti, Tu H0 : =42 sm aris WeSmariti. radgan suraTze #8.2 ilustrirebuli SerCeviTi saSualos ganawileba aris normaluri ganawileba =42 sm-iTa da

=0.6, Cven SegviZlia -is nebismieri mniSvnelobis standartizacia:

z =

sadac:=SerCeviTi saSualo

kriteriumis statistika (test statistic)

aris ricxvi, romelic nulovani hipoTezis Semowmebis saSualebas gvaZlevs da gamoiTvleba SerCevis monacemebze dayrdnobiT.

-3 -2 -1 +1 +2 +3 40.2 40.8 41.4 42.0 42.6 43.2 43.8

muxlqveSa simaRle (sm)

= populaciis hipoTeturi saSualo=saSualos standartuli Secdoma

z-is mniSvneloba gansazRvravs SerCeviTi saSualos mdebareobas standartul normalur ganawilebaSi. aRsaniSnavia, rom es formula identuria im formulisa, romelic mocemuli iyo z-isaTvis me-7 TavSi. Tu =44 sm -s hipoTeturi mniSvneloba _ 42 sm, da =0.6, maSin zrealuris mniSvnelobaa

zrealuri=

z statistikuri sidide gansazRvrulia rogorc zrealuri, radgan is miRebulia SerCevis mniSvnelobebze dakvirvebiT. z statistikis SerCeviTi ganawilebidan, romelic mocemulia cxrilSi #a-1, Cven vpoulobT, rom albaToba imisa, rom z iqneba toli an meti +3.33-ze aris 0.0004 (ix. g sveti cxrilSi #a-1 z=3.33-saTvis). Sesabamisad =44 sm an masze meti mniSvnelobis miRebis albaToba 25-is toli moculobis SerCevisaTvis aris 0.0004, Tu H0:=42sm-is aris WeSmariti. sxva sityvebiT rom vTqvaT, Tu Cven SemTxeviT SevarCevT 25-is toli moculobis 10000 SerCevas populaciidan, romlis =42 sm da =3, maSin mosalodnelia, rom 10000 SerCevidan mxolod oTxs eqneboda SerCeviTi saSualo 44-is toli an masze meti. amrigad, Tuki H0:=42sm aris WeSmariti, iseTi SerCevis arseboba, romlis tolia 44 sm-is, iqneboda statistikurad iSviaTi xdomiloba. Tuki Cven miviRebdiT am , Cven namdvilad uarvyofdiT nulovan hipoTezas, radgan is miRebulia iseTi populaciidan, romlis saSualo muxlqveSa simaRle aris 42 sm da miviRebdiT alternatiul hipoTezas, rom 42 sm.

aseTi gadawyvetilebis miRebis azri isaa, rom Tu populaciis saSualo aris 42 sm, maSin iseTi SerCevis arseboba, romlis saSualo aris 44 sm, SesaZlebeli iqneboda mxolod 4-jer yoveli aRebuli 10 000 SerCevisaTvis. marTalia, =44 sm aris SesaZlebeli xdomiloba, Tuki H0 : =42 sm-a, magram is mainc ar aris maRali albaTobis mqone xdomiloba. misi miReba ufro metad mosalodneli iqneboda, Tuki populaciis saSualo gansxvavebuli iqneboda 42sm-isagan. amrigad, Cven uarvyaviT H0 da miviReT H1. es gadawyvetileba SeiZleba mcdaric iyos. Cven SegveZlo migveRo =44 sm, maSinac ki, roca =42 sm-ia, magram es iqneboda naklebad mosalodneli xdomiloba.

davuSvaT, miviReT SerCeviTi saSualo, romelic 42.5 sm-is tolia, nacvlad 44 sm-isa. aq z toli iqneba:

zrealuri=

cxrili #a-1-is g svetSi vipoviT, rom albaToba imisa, rom z iqneba toli an meti +0.83-ze aris 0.2033. amrigad, Tu H0 aris WeSmariti da Cven aviRebT 10000 SerCevas, romlis moculoba aris 25-is toli, amaTgan 2033-s eqneba -is mniSvneloba 42.5-is toli an masze meti. aris Tu ara 42.5 sm-is toli SerCeviTi saSualo statistikurad iSviaTi xdomiloba, Tu H0 :

=42sm aris WeSmariti? ra gadawyvetilebas miiRebT am SemTxvevaSi nulovani hipoTezis

Sesaxeb? uaryofT H0 : =42sm hipoTezas da miiRebT H1 : 42sm hipoTezas? Tu ver uaryofT H0-s da amitom ar miiRebT H1-s? imisaTvis rom vupasuxoT am SekiTxvas, Cven unda ganvsazRvroT zrealuris romeli mniSvneloba unda miviCnioT statistikurad iSviaT Sedegad, UTuki H0 aris WeSmariti. statistikuri iSviaToba ganisazRvreba im mniSvnelovnobis donis dadgeniT, romelic iqneba miRebuli.

mniSvnelovnobis doneebi: statistikuri iSviaToba

mniSvnelovnobis done – esaa albaTobis is mniSvneloba, romelic iZleva nulovani hipoTezis uaryofis pirobas statistikur kriteriumSi. statistikur kriteriumSi mniSvnelovnobis doneebi mocemulia rogorc alfas anu -s mniSvneloba (berZnuli anbanis pirveli aso). Cveulebriv, bihevioristebi iyeneben alfas doneebs, romlebic tolia 0.05 an 0.01. amrigad, Tu =0.05, maSin zrealuris mniSvnelobebi,

romlebic 100-dan 5-jer an naklebad xorcieldeba, iTvleba iSviaT xdomilobebad, Tu H0 aris WeSmariti, amitom Cven varCevT H0-is uaryofasa da H1-is miRebas.

statistikuri hipoTezebis Sesaxeb gadawyvetilebis miReba kriteriumis statistikisa da mniSvnelovnobis donis gamoyenebiT

mas Semdeg, rac avirCevT kriteriumis statistikas da mniSvnelovnobis dones, Cven ukve mzad varT miviRoT gadawyvetileba statistikuri hipoTezis Sesaxeb. es moicavs sam safexurs:

1. kriteriumis statistikis SerCeviT ganawilebaSi uaryofis aris an areebis mdebareobis gansazRvra.

2. kriteriumis statistikis mniSvnelobis gamoTvla SerCevidan miRebuli monacemebis safuZvelze.

3. statistikuri hipoTezebis Sesaxeb gadawyvetilebis miReba imis safuZvelze, xvdeba Tu ara kriteriumis statistika uaryofis areSi.

uaryofis areebis mdebareobis gansazRvra

uaryofis are warmoadgens kriteriumis statistikis (SerCeviTi ganawilebis) im mniSvnelobebs, romelTa albaToba tolia an naklebia -ze nulovani hipoTezis WeSmaritebis pirobebSi.

mniSvnelovnobis done (significance level) _ esaa albaTobis is mniSvneloba, romelic iZleva nulovani hipoTezis uaryofis pirobas statistikur kriteriumSi.alfa (): mniSvnelovnobis donis sidide (albaToba).

uaryofis are (rejection region) - kriteriumis statistikis (SerCeviTi ganawilebis) is mniSvnelobebi, romelTa albaToba tolia an naklebia -ze nulovani hipoTezis WeSmaritebis pirobebSi. Tuki kriteriumis statistika moxvdeba uaryofis areSi, maSin H0 uaryofili iqneba.ormxrivi kriteriumi (two-tailed test) - statistikuri kriteriumi, roca uaryofis aris asagebad gamoyenebulia kriteriumis statistikis SerCevis ganawilebis orive kudi.calmxrivi kriteriumi (one-tailed test)- statistikuri kriteriumi, roca uaryofis aris asagebad gamoyenebulia kriteriumis statistikis SerCevis

magaliTad, uaryofis are gansazRvravs z-is mniSvnelobebs, romlebic uzrunvelyofen H0-is uaryofas.

z statistikis SerCeviTi ganawileba Tavisi uaryofis areebiT, romlebic ganlagebulni arian ganawilebis TiToeul kudTan , naCvenebia suraTze #8.3. z-is statistikurad iSviaTi mniSvnelobebi SeiZleba Segvxvdes ganawilebis orive kudTan. magaliTad, z-is mniSvnelobebi +3 an -3 Tanabrad SeuZlebelia, roca H0 WeSmaritia. z-is rogorc didi dadebiTi, aseve didi uaryofiTi mniSvneloba ganxiluli problemisaTvis SeiZleba saeWvod xdides H0 : =42 sm-is WeSmaritebas. amrigad, uaryofis areebi z statistikisaTvis mdebareobs SerCeviTi ganawilebis orive mimarTulebiT. kriteriumis statistika uaryofis areebiT kriteriumis statistikis SerCevis ganawilebis orive mimarTulebiT ganisazRvreba rogorc ormxrivi an aramimarTulebiTi kriteriumi. zrealuris mniSvnelobis ganTavseba romelime uaryofis areSi araalbaTuri, an iSviaTia, Tuki H0 WeSmaritia, magram amis albaToba ufro didia H1-is WeSmaritebis pirobebSi. amitom, Tuki zrealuri moxvdeba uaryofis romelime areSi, H0 iqneba uaryofili da H1 miRebuli.

SeiZleba iseTi SemTxvevac moxdes, rom Cven garkveuli mizeziT uaryofis are z-is SerCeviTi ganawilebis mxolod erTi mimarTulebiT ganvaTavsoT. am SemTxvevaSi gamoiyeneba calmxrivi kriteriumi an mimarTulebis mqone kriteriumi.

suraTi #8.3. z statistikis SerCeviTi ganawileba gviCvenebs uaryofis areebs ormxrivi kriteriumisaTvis, roca =0.05. Tu zrealuris mniSvneloba xvdeba erT-erT uaryofil areSi, H0 unda uarvyoT.

calmxrivi kriteriumis upiratesoba isaa, rom zrealuris ufro mcire mniSvneloba SeiZleba moxvdes uaryofis areSi. calmxriv kriteriums ufro detalurad mogvianebiT ganvixilavT am TavSi. am SemTxvevaSi ki ganvixilavT ormxrivi kriteriumis magaliTs.

Kkritikuli mniSvnelobebi. Kkritikuli mniSvnelobebi konkretuli ricxviTi mniSvnelobebia, romlebic adgenen uaryofis aris sazRvrebs. z-is kritikuli mniSvnelobebi mocemulia cxrilSi #a-1 g sveti. pirveli nabiji am kritikuli mniSvnelobis povnaSi

-3 -2 -1 0 +1 +2 +3 z-is mniSvnelobebi

H0 uaryofauaryofis are0.025

H0 uaryofauaryofis are0.025

H0 miRebis zona

kritikuli mniSvnelobebi (critical value) _ konkretuli ricxviTi mniSvnelobebi, romlebic adgenen uaryofis aris sazRvrebs.

z=-1.96 z=+1.96 z-is kritikuli z-is kritikuli mniSvneloba mniSvneloba

aris mniSvnelovnobis donis SerCeva, rogoricaa =0.05. ormxrivi kriteriumisaTvis uaryofis areebi ganlagebulia z-is SerCeviTi ganawilebis orive kudTan. amitomac Cven gvWirdeba are ganawilebis TiToeul kudTan, romelSic moxvedra tolia -s naxevrisa an 0.5/2, rac tolia 0.025. z-is ra mniSvnelobebi akmayofileben moTxovnas, rom z-is SerCeviTi ganawilebis 0.025 wili moxvdes am mniSvnelobebis miRma? cxrilSi #a-1 Cven gavyvebiT g svets, sanam ar vipoviT ares, romelic mdebareobs z-is im mniSvnelobis gaswvriv, romelic tolia 0.025. roca vipoviT am ares, wavikiTxavT z-is Sesatyvis mniSvnelobas a svetSi. z-is es mniSvneloba iqneba 1.96. amrigad, =0.05-saTvis ormxrivi kritikuli mniSvnelobebia z = -1.96 da z=+1.96. z-is es kritikuli mniSvnelobebi (simbolurad aRvniSnoT zkritikuli) gansazRvraven uaryofis areebis mdebareobas z-is SerCeviT ganawilebaSi, rogorc es naCvenebia suraTze #8.3.

gadawyvetilebebi statistikuri hipoTezebis Sesaxeb H0 hipoTezis uaryofisa Tu miRebis Sesaxeb gadawyvetilebis miReba xdeba zrealursa da

zkritikuli mniSvnelobaTa SedarebiT. zrealuris is mniSvneloba, romelic tolia an naklebia zkritikuli =-1.96 an tolia an metia zkritikuli=+1.96 xvdeba uaryofis areSi. zrealuris mniSvnelobebi, romlebic xvdebian uaryofis areSi, arian statistikurad iSviaTni, Tuki H0 aris WeSmariti, magram arian ufro mosalodneli im SemTxvevebSi, roca H1 aris WeSmariti. amrigad, roca zrealuri xvdeba uaryofis areSi, amas mivyavarT H0-is uaryofamde. Tuki uarvyofT H0-s, maSin unda miviRoT H1. Tu zrealuri ar xvdeba uaryofis areSi, maSin H0-s ver uarvyofT da Tuki H0-s ver uarvyofT, maSin ver miviRebT H1-s.

aRsaniSnavia, rom Cven viyenebT zrealuris mniSvnelobas, raTa miviRoT H0-is Sesaxeb gadawyvetileba; uarvyoT H0, TuU CavTvaloT, rom H0-s ver uarvyofT. gadawyvetileba, romelic miiReba H0-is Sesaxeb, SemdgomSi gansazRvravs gadawyvetilebas, romelic miiReba H1-is Sesaxeb. Tu Cven uarvyofT H0-s, maSin unda miviRoT H1. piriqiT, Tu H0-s ver uarvyofT, ver miviRebT H1-s.

muxlqveSa simaRlis magaliTs Tu davubrundebiT, zrealuri= + 3.33 aris ufro meti, vidre zkritikuli=+1.96, amitom is xvdeba maRal uaryofis areSi. Sesabamisad, Cven unda uarvyoT H0

: =42sm da miviRoT H1 : 42sm. Cven uarvyofT nulovan hipoTezas imis Sesaxeb, rom SerCeva iyo aRebuli im populaciidan, romlis saSualo 42.0 sm-is tolia. radgan SerCevis saSualo, =44.0 sm, aris ufro didi, vidre populaciis hipoTeturi saSualo _ 42 sm, Cven gamogvaqvs daskvna, rom im qveyanaSi, romlidanac iyo aRebuli SerCeva, mZRolebis saSualo muxlqveSa simaRle aris ufro meti, vidre =42 sm. saukeTeso wertilovani Sefaseba am populacis saSualosaTvis aris SerCevis saSualo, =44.0. erT-erTi mizezi imisa, rom satvirTo manqanebis skamebi SeiZleba yofiliyo arakomfortuli, isaa, rom zogierT qveyanaSi mZRolebis populacias ufro grZeli muxlqveSa simaRle aqvs, vidre im skamebis simaRle, romelic maTTvis iyo Seqmnili.

SevajamoT, rogor unda miviRoT gadawyvetileba statistikuri hipoTezebis Sesaxeb:

Tu zrealuri xvdeba uaryofis areSi, maSin uarvyofT H0-s da viRebT H1-s. Tu zrealuri ar xvdeba uaryofis areSi, maSin H0-s ver uarvyofT da ar viRebT H1-s.

statistikurad mniSvnelovani gansxvavebastatistikurad mniSvnelovania gansxvaveba, roca kriteriumis statistikis realuri mniSvneloba xvdeba uaryofis areSi da uarvyofT H0-s. z kriteriumisaTvis stastikurad mniSvnelovani gansxvaveba gviCvenebs, rom Cven miviReT gadawyvetileba imis Sesaxeb, rom SerCevis saSualo aRebulia im populaciidan, romlis saSualo gansxvavdeba populaciis hipoTeturi saSualosagan.

statistikurad mniSvnelovani gansxvaveba xSirad warmodgenilia Semdegi formiT: zrealuri =+3.33, p0.05. simbolo niSnavs naklebia an tolia, amrigad, p0.05 ikiTxeba rogorc “p naklebia an toli 0.05”. xSirad naklebobis niSani < gamoiyeneba niSnis nacvlad, ase rom z aris warmodgenli, rogorc zrealuri=+3.33, p<0.05. orive SemTxvevaSi naCvenebia, rom H0 uaryofilia 0.05 mniSvnelovnobis doneze, radganac zrealuris miRebis albaToba (p), roca H0 WeSmaritia, tolia an naklebia 0.05-s. es gadawyvetilebebi da daskvnebi, romelic miRebulia statistikurad mniSvnelovani gansxvavebidan z kriteriumis saSualebiT Sejamebulia cxrili #8.1-is marcxena svetSi.

cxrili #8.1. SerCevis saSualosa da populaciis hipoTeturi saSualos Sefasebisas gadawyvetilebebisa da daskvnebis Sejameba statistikuri hipoTezebis Semowmebis Sesaxeb z kriteriumis gamoyenebiT. gamoyenebulia 0.05 mniSvnelovnobis done. Tu zrealuri xvdeba uaryofis areSi =0.05-saTvis, maSin:

Tu zrealuri ar xvdeba uaryofis areSi =0.05-saTvis, maSin:

zrealuris miRebis albaToba naklebia an toli 0.05, an p<0.05.H0 uaryofilia.H1 miRebulia. SerCevis saSualo aRebulia populaciidan, romlis saSualo gansxvavebulia populaciis hipoTeturi saSualosagan.

gansxvaveba -sa da hipoTetur -s Soris statistikurad mniSvnelovania 0.05 doneze.

miiReba gadawyvetileba imis Sesaxeb, rom -sa da hipoTetur -s Soris gansxvavebis mizezi ara SerCevis Secdomaa, aramed sxva ram.

zrealuris miRebis albaToba metia an toli 0.05, an p>0.05H0 ar aris uaryofiliH1 ar aris miRebuli

gansxvaveba -sa da hipoTetur -s Soris aris statistikurad aramniSvnelovani 0.05 doneze

miiReba gadawyvetileba imis Sesaxeb, rom SerCevis Secdoma yvelaze damajerebelad xsnis gansxvavebas -sa da hipoTetur -s Soris.

aramniSvnelovani gansxvaveba

statistikurad mniSvnelovani gansxvaveba (statistically

significant difference) _ kriteriumis statis-tikis realuri mniSvneloba xvdeba uaryofis areSi da H0 uaryofilia.

aramniSvnelovani gansxvaveba(nonsignificant difference) _ kriteriumis statistikis realuri mniSvneloba ar xvdeba uaryofis areSi da nulovani hipoTeza ar aris uaryofili

gansxvaveba SerCevis saSualosa da populaciis saSualos Soris aris aramniSvnelovani, Tuki kriteriumis statistikis realuri mniSvneloba ar xvdeba uaryofis areSi. nulovani hipoTeza ar aris uaryofili da realurad miRebuli gansxvaveba -sa da hipoTetur -s Soris miewereba mxolod SerCevis Secdomas. aramniSvnelovani gansxvaveba aRiniSneba p>0.05, rac mosdevs z realuris ricxviT mniSvnelobas. gadawyvetilebebi da daskvnebi, romelic miRebulia z kriteriumis saSualebiT statistikurad aramniSvnelovani gansxvavebidan, Sejamebulia cxrilis #8.1 marcxena svetSi.

statistikuri hipoTezis Semowmebis etapebis Sejameba

yvela statistikuri kriteriumi moiTxovs Semdegs: kriteriumis statistikis arCevas, rogoricaa magaliTad, z;

ori statistikuri hipoTezis formulirebas, H0 da H1;

kriteriumis statistikis SerCeviTi ganawilebis miRebas H0-is WeSmaritebis pirobebSi;

mniSvnelovnobis donis SerCevas;

kriteriumis statistikis kritikuli mniSvnelobis an mniSvnelobebis povnas;

kriteriumis statistikis SerCeviT ganawilebaSi uaryofis aris an areebis mdebareobis gansazRvras;

statistikuri hipoTezebis Sesaxeb gadawyvetilebis miRebis wesebis formulirebas;

kriteriumis statistikis mniSvnelobis gamoTvlas SerCevis monacemebiT;

gadawyvetilebis miRebas H0-is uaryofisa Tu misi ver uaryofis Sesaxeb imis safuZvelze, rom kriteriumis statistikis realuri mniSvneloba xvdeba Tu ara uaryofis areSi;

gadawyvetilebis gamotanas H1-is miRebisa Tu ar miRebis Sesaxeb H0-isaTvis miRebuli gadawyvetilebis safuZvelze.

statistikuri hipoTezebis Semowmebis Sesavali SeiZleba rTuli mogeCvenoT, magram is male gaxdeba martivi, radgan TiToeuli statistikiuri kriteriumisaTvis Semdeg erTi da igive etapebia Sesasrulebeli.

SeamowmeT Tqveni codna VIII-11. gansazRvreT: alfa done, , alternatiuli hipoTeza, kritikuli mniSvneloba, H0, H1,

aramniSvnelovani gansxvaveba, nulovani hipoTeza, erTmxrivi kriteriumi, uaryofis are, mniSvnelovnobis done, statistikuri hipoTeza, statistikurad mniSvnelovani gansxvaveba, kriteriumis statistika, ormxrivi kriteriumi, zkritikuli, zrealuri.

2. gansazRvreT statistikuri hipoTezebis ori Tviseba.3. CawereT H0 da H1 pirobiTi niSnebiT Semdegi debulebebisaTvis:

a) veqsleris inteleqtis testis qulebisaTvis aris 100.b) 25-dan 54 wlamde qalebis populaciisaTvis televizoris yurebaze daxarjuli

saSualo dro dReSi 4.4 saaTis tolia.g) martiv stimulze 19 wlis mamakacis reaqciis saSualo dro aris 423.0 miliwami.

4. qvemoT moyvanili TiToeuli monacemisaTvis warmoadgens populaciis saSualos, saSualos standartul Secdomas am populaciisaTvis da _ SerCevis saSualos, romelic aRebulia am populaciidan. qvemoT mocemul mniSvnelobaTa TiToeuli rigisaTvis gansazRvreT H0 da H1 da ipoveT zrealuris mniSvneloba. Semdeg cxrili #a-1-is gamoyenebiT gansazRvreT, xvdeba Tu ara zrealuri ormxriv uaryofis areSi 0.05 mniSvnelovnobis doneze da miuTiTeT Tqveni gadawyvetileba H0-isa da H1-saTvis.

5. gadawyviteT Semdegi amocana 0.05 mniSvnelovnobis donisa da ormxrivi kriteriumis gamoyenebiT. populaciis saSualo inteleqtis skalisaTvis aris 100 da =15. Tqven xarT dawyebiTi skolis xelmZRvaneli soflis skolaSi. Tqveni

skolis 121 moswavlis am skalaze tolia 103.4-is. gansxvavdeba Tu ara Tqveni skolis moswavleTa saSualo mniSvnelovnad populaciis saSualosagan? Tqvens pasuxSi miuTiTeT:a) romel statistikas gamoiyenebT;b) statistikuri hipoTezebi;g) mniSvnelovnobis done;d) zkritikuli; e) uaryofis areebis mdebareoba;v) zrealuri;z) Tqveni gadawyvetilebani H0-is da H1-is Sesaxeb;T) Tqveni gadawyvetilebani Tqveni moswavleebis saSualos Sesaxeb populaciis saSualosTan _ 100 mimarTebaSi.

erTi SerCevis t kriteriumi praqtikaSi xSirad fasdeba s-is _ SerCevaSi qulebis standartuli gadaxris saSualebiT.

amitom -c fasdeba -is saSualebiT. roca Cven -s -is saSualebiT vafasebT, viRebT axal statistikas _ t-s, romelic ganisazRvreba, rogorc

t =

Tu -is naclad CavsvamT s/N ,

t =

a) 50 4.7 40b) 100 25 70g) 143 0.4 143.9d) 87 2.9 92

sadac = SerCevis saSualoa = populaciis hipoTeturi saSualoa = saSualos standartuli Secdomis Sefaseba, romelic miRebuli iyo populaciis standartuli gadaxris Sefasebis _ s-is saSualebiT N= SerCevis moculoba.

t kriteriumi SeiZleba iyos gamoyenebuli im gansxvavebis statistikuri mniSvnelovnobis Sesamowmeblad, romelic arsebobs SerCevis saSualosa da populaciis hipoTetur saSualos Soris, roca Sefasebulia -is saSualebiT. radgan mxolod erTi SerCevis saSualos gulisxmobs, statistikas uwodeben erTi SerCevis t-s, iseTi t kriteriumisagan gansxvavebis mizniT, romelic ori SerCevis saSualos moicavs, rac me-9 TavSi iqneba ganxiluli. t kriteriumi SemoRebuli iyo britaneli maTematikosis

uiliam sili gosetis mier me-20 saukunis dasawyisSi. gosetma gamosca Tavisi naSromi fsevdonim “studentis” gamoyenebiT. amitomac t kriteriums xSirad uwodeben `studentis t kriteriums”.

t-s SerCeviTi ganawileba

t statistikis gamoyenebisaTvis unda vicodeT misi SerCeviTi ganawileba. t-s SerCeviTi ganawileba gansxvavdeba z-is SerCeviTi ganawilebisagan. radgan z statistika iyenebs populaciis parametrs -s, mas aqvs mxolod erTi SerCeviTi ganawileba, romelic warmodgenilia cxrilSi #a-1. t kriteriumi ki, radgan iyenebs populaciis s standartul gadaxris Sefasebas, ganisazRvreba, rogorc

s=

s-is gamoTvla gulisxmobs SerCevis N moculobas. amitom t-s SerCeviTi ganawileba icvleba SerCevis moculobasTan erTad, an ufro zustad rom vTqvaT, SerCevis Tavisuflebis xarisxTan erTad.

Tavisuflebis xarisxiTavisuflebis xarisxi (romlis abreviaturaa df) ganisazRvreba rogorc qulebis raodenoba, romelic Tavisufalia cvalebadobis dros statistikis gamoTvlisas. moviyvanT ramdenime magaliTs imis ukeT gasagebad, Tu ras niSnavs `qulebis raodenoba, romelic Tavisufalia

erTi SerCevis t kriteriumi

(one-sample t test) _t kriteriumi, romelic gamoiyeneba SerCevis saSualosa da populaciis hipoTetur saSualos Soris gansxvavebis statistikuri mniSvnelovnobis Sesamowmeblad, roca Sefasebulia s -is saSualebiT.

Tavisuflebis xarisxi (degrees of freedom): qulaTa raodenoba, romelic Tavisufalia cvalebadobis dros statistikis gamoTvlisas.

cvalebadobis dros” statistikis gamoTvlisas. davuSvaT, rom Tqven gTxoves amoirCioT sami qula testSi. Tqven SeiZleba airCioT nebismieri sami qula, vTqvaT, 75, 62 da 97. am davalebisas ar arsebobda raime SezRudva qulebis arCevanSi; TiToeuli qula SeiZleba Tavisufali iyos Tavisi cvalebadobis dros. Sesabamisad qulebis am simravlisaTvis Tavisuflebis xarisxi df aris N=3. ganvixiloT meore davaleba, roca Tqven gTxoven airCioT sami qula, magram iseTi, romelTa jami unda iyos 254. davuSvaT Tqven airCieT 81 da 78, rogorc pirveli ori qula; maT SeiZleba hqondeT nebismieri mniSvneloba, romelsac Tqven isurvebT. es ori qula SeiZleba Tavisufali iyos Tavisi cvalebadobis dros. SeiZleba mesame qula aseve Tavisuflad Seicvalos? ara. radgan sami qulis jami unda iyos 254, mesame qula unda iyos 95; arc erTi sxva qulis SemTxvevaSi es jami ar miiReba. amitom, Tu Tqven giTxres, rom qulaTa jami aris 254, ori qula SeiZleba Tavisuflad Seicvalos, xolo mesame aris gansazRvruli im ori qulis mniSvnelobiT, romlebic Tavisuflad SeiZleba Seicvalos da am sami qulis jamiT. Sesabamisad, Tu Tqven iciT sami qulis jami, maSin arsebobs N_1=3_1=2 Tavisuflebis xarisxi am sami qulisaTvis.

Tu gamoviyenebT am argumentacias SerCeviTi saSualos gamoTvlisas, -s aqvs N df. yvela qula qulaTa erTobliobaSi aris Tavisufali cvalebadobisas, roca viTvliT -s. magaliTad, Tqven gTxoves gamoiTvaloT SerCevis saSualo 5 qulis erTobliobisaTvis, aqedan 4 aris mocemuli: 7, 10, 3 da 6. mxolod am informaciis mocemulobisas Tqven ver SesZlebTY saSualos povnas. mexuTe qulad SeiZleba CaTvaloT nebismieri mniSvneloba da Tqven ar gecodinebaT mexuTe qulis mniSvneloba mxolod am oTxi mniSvnelobis codnis safuZvelze. amrigad, yvela N qula (sadac N=5 rogorc am magaliTSi) aris Tavisufali cvalebadobisas, roca xdeba SerCevis saSualos gamoTvla.

s-is Tavisuflebis xarisxi aris N_1. amis sailustraciod davuSvaT, rom Tqven gTxoves gamoTvaloT s 5 qulis mqone SerCevis safuZvelze. populaciis standartuli gadaxris Sefaseba ganisazRvreba, rogorc

davuSvaT, rom xuTidan oTx qulas aqvs Semdegi mniSvnelobebi: 13, 15, 13 da 12 da am xuTi qulis saSualo aris 13.4. Cven viciT, agreTve, rom Tuki Cven gamovaklebT am oTxi qulidan TiToeuls, maSin Cven miviRebT

13 _ 13.4= - 0.415 _ 13.4= +1.613 _ 13.4= - 0.412 _ 13.4= -1.4jami = -0.6am oTxi sxvaobis jami tolia -0.6. roca mexuTe qula iqneba CarTuli am erTobliobaSi, es jami toli iqneba nulis. amrigad, mexuTe qulam unda mogvces

=+0.6, +0.6 rom davumatoT -0.6, jami toli iqneba nulis. Tuki X5 _ 13.4= +0.6, maSin X5 toli unda iyos 14. Sesabamisad, Tu cnobilia am xuTi qulis erTobliobisaTvis, maSin am xuTi qulidan mxolod oTxi qulis mniSvnelobaa Tavisufali cvalebadobisas. mexuTe qula ver Seicvleba Tavisuflad. misi mniSvneloba aris mkacrad gansazRvruli im SezRudviT, romelic saSualos mniSvnelobiT iqna Semotanili. radgan unda iyos cnobili, raTa gamoviTvaloT s, yvela qula, garda erTisa, aris Tavisufali cvalebadobis dros s-is gamoTvlisas. amrigad, roca viTvliT s an s2 NN qulaTa erTobliobas aqvs N-1 Tavisuflebis xarisxi df.

Cven SegviZlia Tavisuflebis xarisxis Sesaxeb es codna s-is gamosaTvlel formulaSi gamoviyenoT, raTa is ufro mokled CavweroT. radgan ,

agreTve, kvadratebis jams (SS) uwodeben, da N-1 asaxavs Tavisuflebis xarisxs SS-is gamoTvlisas, populaciis standartuli gadaxris Sefaseba SeiZleba gamovsaxoT rogorc kvadratuli fesvi kvadratebis jamidan gayofili Tavisuflebis xarisxze:

.

momdevno TavebSi Cven vnaxavT, rom s-is es gamosaxuleba sakmaod xSirad gamoiyeneba.

erTi SerCevis t kriteriumis Tavisuflebis xarisxiaxla gamoviyenoT df-is Sesaxeb Cveni codna t kriteriumisaTvis. saSualos standartuli Secdomis Sefaseba, gamoiTvleba rogorc s/N. radgan s-s aqvs N-1 df,

-s, agreTve, aqvs N-1 df. amrigad, erTi SerCevis t statistikis SerCeviTi ganawilebisaTvis Tavisuflebis xarisxi aris, agreTve N-1.

t SerCeviTi ganawileba

TiToeuli df-isaTvis arsebobs t-s sxvadasxva Teoriuli ganawileba. radgan potenciurad qulaTa nebismieri raodenoba SeiZleba iyos miRebuli SerCevaSi, df

SeiZleba icvlebodes 1-dan usasrulobamde. rogorc potenciuri SerCeviTi ganawilebis magaliTi, suraTi #8.4 gviCvenebs t ganawilebebs, roca df aris 2, 4, da 60.

Tumca t ganawilebis forma icvleba Tavisuflebis xarisxis cvalebadobasTan erTad, is normaluri ganawilebis msgavsia ramdenime aspeqtSi. TiToeuli t

ganawileba simetriuli da unimodaluria da aqvs saSualo, romelic nulis tolia. ganvixiloT, ratom Cndeba es Tavisebureba. roca nulovani hipoTeza, iseTi rogoricaa: H0:=populaciis saSualos raime hipoTeturi mniSvnelobisa, WeSmaritia, t-s mosalodneli mniSvneloba iqneba nuli, radgan ( _)-s saSualo im SerCeviT ganawilebaSi, romelic iyo aRebuli aseTi populaciidan iqneba nuli. magram, maSinac roca H0 WeSmaritia, -s mniSvnelobebi Seicvleba nulis toli saSualos garSemo

SerCevis Secdomis gamo. ufro metic, mosalodnelia, rom iqneba -ze meti, TiTqmis iseve xSirad, rogorc, is iqneba -ze naklebi. mosalodnelia, rom, roca nulovani hipoTeza WeSmaritia, mniSvnelobebi iseve xSirad iqneba dadebiTi, rogorc uaryofiTi. amrigad, t-s mosalodneli mniSvnelobebi simetriulad iqneba ganlagebuli nulis toli saSualos garSemo. Tavisuflebis xarisxis usasrulo raodenobisaTvis (Teoriuli da ara realuri SesaZlebloba) t SerCeviTi ganawileba aris normaluri ganawilebis msgavsi. yvela sxva t ganawilebebi arian ufro farTod gaSlilebi, vidre normaluri ganawileba.

ratom aris ufro swori Tavisuflebis xarisxis da ara N-is gamoyeneba? Tu df erTi SerCevis t kriteriumisaTvis aris mocemuli NN-1 saSualebiT, ratom ar SeiZleboda t SerCeviTi ganawilebis warmodgena N-is terminebiT df-is magivrad? erTi SerCevis t statistikisaTvis Cven ver miviRebT verc erT upiratesobas df-is gamoyenebiT N-is magivrad, im df-saTvis, romelic yovelTvis erTiT naklebia N-ze. df-

is gamoyenebis upiratesoba gamoCndeba me-9 TavSi, sadac Cven gagacnobT sxva t statistikas, romelic gamoiyeneba ori SerCevis saSualoTa Sesadareblad. am t-saTvis df N-2 da ara N-1-ia. toli df-saTvis orive t kriteriums aqvs erTi da igive SerCeviTi ganawileba. igive magaliTebs sxva statistikebis ganxilvisas momdevno TavebSi SevxvdebiT.

SeamowmeT Tqveni codna VIII-21. dawereT erTi SerCevis t statistikis formula.

2. ratom uwodeben t statistikas zogjer studentis t-s?3. Tqven mocemuli gaqvT 7 qula: 73, 42, 51, 68, 49, 62 da 57. ramdeni

Tavisuflebis xarisxi aqvs am qulebis simravles, raTa gamoiTvaloT a) b) s da g) s2

4. Tqven gegmavT gamoTvaloT erTi SerCevis t statistika Semdegi moculobebis SerCevebze. TiToeuli SerCevisaTvis, ipoveT df: a) N=10, b) N=16, g) N=7, d) N=54,

e) N= 120, v) N=30

5. Tqven gaqvT t-s ori Teoriuli ganawileba, erTi 13 df da meore 27 df. romeli ganawileba iqneba ufro dabali da gaSlili?6. ra pirobebSi iqneba t statistikis SerCeviTi ganawileba z SerCeviTi ganawilebis identuri?

statistikuri hipoTezebis Semowmeba t-s saSualebiT erTi SerCevis SemTxvevaSi

erTi SerCevis t saSualebas gvaZlevs ganvsazRvroT, gansxvavdeba Tu ara hipoTeturi -sagan ufro metad, vidre es iqneboda mosalodneli mxolod SerCevis Secdomis SemTxvevaSi, roca aris gamoTvlili s-is saSualebiT. t-s saSualebiT statistikuri hipoTezis Semowmebis procedura identuria imisa, rac iyo z

statistikisaTvis. aucilebeli nabijebi moicavs Semdegs: H0-is da H1-is formulireba; t statistikis SerCeviTi ganawilebis miReba H0-is WeSmaritebis pirobebSi; mniSvnelovnobis donis SerCeva; t-s kritikuli mniSvnelobis an mniSvnelobebis povna; uaryofis aris an areebis mdebareobis dadgena t-s SerCeviT ganawilebaSi; gadawyvetilebis miReba statistikuri hipoTezebis Sesaxeb; trealuris mniSvnelobis gamoTvla SerCevis monacemebis safuZvelze; gadawyvetilebis miReba H0-is da H1-is Sesaxeb trealuris mniSvnelobaze

dayrdnobiT.

am nabijebs asaxavs Semdegi amocanis magaliTi. davuSvaT, bavSvTa fsiqologma, romelic ikvlevda sakiTxs, rogor atarebdnen dros mesame klaseli bavSvebi, romlebic CarTulni iyvnen gakveTilebis Semdgom bavSvebis movlis programaSi, aRmoaCina, rom kviris dReebSi bavSvebi uyureben televizors saSualod 83 wuTis ganmavlobaSi, romlis s 26 wuTis tolia. davuSvaT, agreTve, cnobilia, rom dReSi televizoris yurebaze daxarjuli drois saSualo mesame klaselTa populaciisaTvis

Seadgens 95 wuTs. aris Tu ara SerCevis saSualo, romelic tolia 83 wuTis, mniSvnelovnad gansxvavebuli populaciis saSualosagan _ 95 wuTisagan? radgan aris gamoTvlili s-is saSualebiT, am kiTxvaze pasuxis gasacemad unda iyos gamoyenebuli erTi SerCevis t kriteriumi.

statistikuri hipoTezebis formulirebastatistikuri hipoTezebi erTi SerCevis t statistikisaTvis identuria imis, rac iyo gamoyenebuli z statistikis SemTxvevaSi. nulovani hipoTeza H0 gansazRvrulia rogorc populaciis saSualos hipoTeturi mniSvneloba. alternatiuli hipoTeza, H1

ganisazRvreba rogorc debuleba, rom populaciis saSualo ar aris is, rac navaraudevia H0-is SemTxvevaSi. magaliTad, statistikuri hipoTezebis pirobiTi aRniSvna arisH0:=95 wuTiH1:95 wuTi

t statistikis SerCeviTi ganawilebis miRebat-s SerCeviTi ganawileba damokidebulia t-s Tavisuflebis xarisxebze, romelic gamoiTvleba SerCevis qulebis safuZvelze. df erTi SerCevis t-saTvis mocemulia N-1 saxiT. im saSualosaTvis, romelic efuZneba 30 monacems, df t-saTvis aris 30-1=29. mniSvnelovnobis donisa da uaryofis areebis arCevis Semdeg, Tavisuflebis xarisxis am maCvenebels gamoviyenebT t-s SerCeviTi ganawilebis cxrilSi tkritikulis

sapovnelad.

mniSvnelovnobis donis arCevamniSvnelovnobis done aris albaTobis mniSvneloba, romelic uzrunvelyofsH0-is uaryofas. statistikuri hipoTezis testirebisas Cveulebriv gamoiyeneba 0.05 an 0.01 mniSvnelovnobis done. magaliTad, Cven viyenebT =0.05.

uaryofis areebis gansazRvra t kriteriumisaTvis: ormxrivi kriteriumiroca H0 WeSmaritia uaryofis are moicavs t ganawilebis iseT mniSvnelobebs, romlis albaToba aris -s toli an masze naklebi, amrigad, uaryofis are gansazRvravs trealuris

iseT mniSvnelobebs, romlebic uaryofis areSi moxvdebian.ormxriv anu aramimarTulebiT kriteriumSi t-s uaryofis areebi

mdebareoben Teoriuli SerCeviTi ganawilebis orive kudze, rogorc es naCvenebia suraTze #8.4. roca H0 WeSmaritia, -is mniSvnelobaTa umetesoba mxolod mcired iqneba gansxvavebuli -sagan. amrigad, t-s mniSvnelobaTa umravlesoba

dajgufebuli iqneba ganawilebis SuaSi nulis garSemo. SerCevis Secdomis gamo, t-s zogierTi mniSvneloba iqneba ganawilebis romelime boloSi, maSinac ki roca H0

WeSmaritia. es mniSvnelobebi miiReba, roca t-s mricxveli (e.i. sxvaoba) metia mniSvnelze (anu ). magram t-s aseTi didi mniSvnelobebi ufro iSviaTi xdomilobebia nulovani hipoTezis WeSmaritebis pirobebSi.

meore mxriv, Tuki H1 aris WeSmariti da asaxavs SerCevas im populaciidan, romlis saSualo gansxvavdeba -sagan, maSin ufro xSirad mosalodnelia -is da, maSasadame t-s didi mniSvnelobebi. amrigad, uaryofis are mdebareobs t-s Teoriuli SerCeviTi ganawilebis TiToeul kudze da mTeli iyofa or nawilad kudebs Soris. Tuki =0.05, maSin TiToeuli uaryofis are moicavs albaTobas /2, an 0.025. Tuki SerCevis qulebis mixedviT gamoTvlili t-s mniSvneloba, (gansazRvruli, rogorc trealuri ) xvdeba romelime uaryofis areSi, maSin nulovani hipoTeza iqneba uaryofili da miiReba alternatiuli hipoTeza. uaryofis aris mdebareobis gansazRvriT t ganawilebis orive kudze, eqsperimentatori iTvaliswinebs orive SesaZleblobas, romelic SeiZleba iyos nagulisxmevi alternatiul hipoTezebSi: roca metia -ze da roca naklebia -ze. uaryofis am or areSi moxvedris sruli albaToba udris 0.05, roca H0 WeSmaritia.

suraTi #8.4t ganawilebis ormxrivi uaryofis are, roca df aris 29 da =0.05. trealuri mniSvneloba xvdeba uaryofis erT-erT areSi, rasac mivyevarT H0 –is uaryofamde da viRebT H1-s.

t-s SerCeviTi ganawilebis uaryofis ori are, roca df aris 29, naCvenebia suraTze #8.4. Tu df tolia 29-is da H0 WeSmaritia, maSin albaToba imisa, rom t iqneba toli an naklebi, vidre -2.045, aris 0.025 da albaToba imisa, rom t iqneba toli an meti +2.045-ze isev 0.025-ia. trealuris nebismieri mniSvneloba, romelic tolia an naklebia -2.045-ze, an tolia, an metia +2.045-ze xvdeba uaryofis areSi. es Sedegi miiReba xdomilobaTa mxolod 5 procentis SemTxvevaSi, roca nulovani hipoTeza WeSmaritia. mniSvnelobebi -2.045 da +2.045 miRebuli iyo t-s SerCeviTi ganawilebis kritikul mniSvnelobaTa cxrilidan. am cxrilis gamoyeneba SemdgomSi iqneba ganmartebuli.

t kritikul mniSvnelobaTa cxrilis gamoyeneba

t=-2.045 t=+2.045 t-s kritikuli t-s kritikulimniSvneloba mniSvneloba

-3 -2 -1 0 +1 +2 +3 t-s mniSvneloba

H0 uaryofauaryofis are0.025

H0 uaryofauaryofis are0.025

H0 miRebis zona

FmniSvnelovnobis calkeuli donisaTvis saWiroa t-s kritikul mniSvnelobebis gansazRvra (aRiniSneba, rogorc tkritikuli.), romlebic damokidebulni arian trealuris df-ze. t ganawilebis grafikebi naTlad gviCvenebs, rom t ganawilebis forma icvleba Tavisuflebis xarisxebis cvalebadobasTan erTad, magram am grafikebidan Znelia ipovo tkritikulis zusti mniSvneloba. amitom tkritikuli mniSvnelobebi 0.05 da 0.01 mniSvnelovnobis doneebisaTvis gamoTvlilia da Setanilia cxrilebSi.

cxrili #8.2 asaxavs tkritikulis mniSvnelobebs =0.05 da 0.01. ormxrivi kriteriumisaTvis. cxrili #8.2 aris danarTSi mocemuli cxrili #a-2-is nawili. pirvel svetSi mocemulia Tavisuflebis xarisxebi t ganawilebebisaTvis. =0.05 da 0.01 svetebis nebismier striqonSi mocemulia tkritikuli mniSvnelobebi am mniSvnelovnobis donis Sesatyvisi Tavisuflebis xarisxisaTvis. magaliTad, roca =0.05 tkritikuli ormxrivi mniSvneloba im trealurisaTvis, romelsac aqvs 29-is toli Tavisuflebis xarisxi, aris 2.045. miaqcieT yuradReba, rom am mniSvnelobas ara aqvs win + an _ niSani. is warmodgenilia rogorc t-s absoluturi mniSvneloba. Tumca uaryofis areebis mdebareobis gansazRvrisaTvis, es mniSvneloba unda iyos warmodgenili rogorc -2.045 da +2.045. analogiurad =0.01-saTvis ormxrivi tkritikuli mniSvneloba, romlis Tavisuflebis xarisxi 29-is tolia udris 2.756. aRsaniSnavia, rom 0.01 mniSvnelovnobis done ufro mkacri kriteriumia H0-is uaryofisaTvis, vidre =0.05. nebismieri realuri

sxvaoba unda iyos ufro didi, raTa uarvyoT H0, roca 0.05-s nacvlad miRebuli iqneba 0.01 mniSvnelovnobis done.

cxrili #8.2. t ganawilebis kritikuli mniSvnelobebi =0.05 da =0.01-saTvis. mniSvnelobebi mocemulia ormxrivi kriteriumisaTvis.

df =0.05 =0.01

1 12.607 63.6572 4.3030 9.9253 3.182 5.8414 2.776 4.6045 2.571 4.0326 2.447 3.7077 2.365 3.4998 2.306 3.3559 2.262 3.2510 2.228 3.16911 2.201 3.106

12 2.179 3.05513 2.160 3.01214 2.145 2.97715 2.131 2.94716 2.120 2.92117 2.110 2.89818 2.101 2.87819 2.093 2.86120 2.086 2.84521 2.080 2.83122 2.074 2.81923 2.069 2.80724 2.064 2.79725 2.060 2.78726 2.056 2.77927 2.052 2.77128 2.048 2.76329 2.045 2.75630 2.042 2.75040 2.021 2.70460 2.000 2.660120 1.980 2.617 1.960 2.576

30-ze ufro meti Tavisuflebis xarisxisaTvis mxolod calkeuli kritikuli mniSvnelobebi aris mocemuli cxrilSi. roca df trealurisaTvis metia 30-ze, girCevT, rom tkritikulis mniSvneloba gamoTvlili iyos im df-s safuZvelze, romelic cxrilSi yvelaze axlos imyofeba, magram amasTanave naklebia am trealuris df-ze. magaliTad, TuU df=35, cxrili #8.2 gviCvenebs trealuris mniSvnelobebs 30 da 40 df-isaTvis, magram ara maT Soris mdebare mniSvnelobebisaTvis. amitomac tkritikulis gansazRvrisaTvis, roca df=35, Cven viyenebT 30-is tol df (is tolia 2.042 =0.05) nacvlad imisa, rom gamogveyenebina 40 df. am proceduris Sedegad, mniSvnelovnobis done iqneba odnav dabali, vidre 0.05. df=40 rom gamogveyenebina, maSin tkritikulis

mniSvnelovnobis done iqneboda odnav maRali, vidre 0.05. Tu trealuri xvdeba uaryofis areSi df=30-saTvis, maSin is, agreTve, xvdeba uaryofis areSi df=40-saTvisac. Tuki trealuri (magaliTad, trealuri =2.033) moxvdeba cxrilSi im mniSvnelobaTa Soris, romlebic Seesabamebian df-is momdevno dabal (mag., tkritikuli=2.045, roca df aris 30) da momdevno maRal mniSvnelobebs (mag. tkritikuli=2.021, roca df aris 40), maSin tkritikulis

ufro zusti mniSvneloba SeiZleba gamoviTvaloT cxrilis mniSvnelobebidan.

t uaryofis areebis mdebareobis gansazRvra: calmxrivi kriteriumiormxriv SemowmebaSi H0-s uaryofen roca trealuri xvdeba t-s ganawilebis romelime boloSi. magram, roca sakvlevi hipoTeziT nawinaswarmetyvelebia saSualoTa Soris mimarTulebiTi kavSiri, (mag., Tu Cvens magaliTSi mkvlevari varaudobs, rom iqneba ufro naklebi, vidre ) da ar arsebobs interesi navaraudevis sapirisipiro SedegSi (mag., Tu aris ufro meti, vidre ), maSin zogierTi mkvlevari iyenebs calmxriv kriteriums. calmxriv anu mimarTulebis mqone kriteriumSi uaryofis are, romelic SemosazRvrulia -s sruli mniSvnelobiT, ganlagebulia t ganawilebis im kudis mxareze, romelic Seesatyviseba kvlevis hipoTeziT navaraudevi Sedegis mimarTulebas. tkritikuli mniSvnelobebi calmxrivi kriteriumisaTvis 0.01 da 0.05 mniSvnelobaTa doneebiT warmodgenilia, agreTve, cxrilSi #a-2. df=29-saTvis, calmxrivi tkritikuli 0.05 mniSvnelovnobis doneze aris 1.699. calmxrivi uaryofis are, roca tkritikuli=-1.699 naCvenebia suraTze #8.5, t ganawilebis marcxena kudze. es uaryofis are am magaliTisaTvis Sesatyvisi iqneboda, Tu davuSvebdiT Semdeg statistikur hipoTezebs:H0 : 42H1 : <42suraTi aklia (Reject _ H0 uaryofa, Do not reject _ H0 miRebis zona, Rejection region - uaryofis are, value of t_ t-s mniSvneloba, critical value of t _ t-s kritikuli mniSvneloba)suraTi # 8.5 t ganawilebis calmxrivi uaryofis are, roca df aris 29 da =0.05. trealuris mniSvneloba xvdeba uaryofis areSi, rasac mivyavarT H0 –is uaryofamde da viRebT H1.

rodis unda iyos gamoyenebuli calmxrivi kriteriumi?calmxrivi uaryofis aris upiratesoba isaa, rom tkritikuli iqneba ufro mcire, radgan mTeli uaryofis are mdebareobs ganawilebis mxolod erT kudSi. amitomac ufro advilia H0-is uaryofa, radgan -sa da Soris mcire gansxvavebisas trealuri xvdeba uaryofis areSi. TumcaRa am upiratesobas Tan axlavs SezRudvac. gadawyvetileba imis Sesaxeb, Tu romeli uaryofis are iqneba gamoyenebuli, calmxrivi Tu ormxrivi, miiReba im dros,

roca xdeba kvlevis dagegmva da sanam daiwyeba monacemTa Segroveba da analizi. Tuki miRebuli iqneba calmxrivi uaryofis are, maSin eqsperimentatori ver SeZlebs uaryos H0 saSualoTa Soris miRebuli im sxvaobis safuZvelze, romelic am hipoTezis sapirispiro mimarTulebisaa, radgan aseTi SedegisaTvis uaryofis are ar gvaqvs. calmxriv da ormxriv kriteriumebs ufro dawvrilebiT me-9 TavSi ganvixilavT. praqtikaSi ormxrivi kriteriumebi yvelaze farTod gamoyenebadia; amitomac, isini gamoyenebulni arian am Tavis yvela magaliTSi.

trealuris gamoTvlatrealuris gamoTvla moiTxovs -is, da mniSvnelobaTa Casmas formulaSi. magaliTad, 30 mesameklaselisaTvis televizoris yurebis =83 wuTs, s=26wuTs da N=30. -s hipoTeturi mniSvneloba tolia 95 wuTis. -is mniSvneloba miRebulia s/N-dan;

amrigad,

= ,rac tolia 4.75. amis Sedegad,

trealuri =

=

roca df aris N-1=29 .

gadawyvetilebani statistikuri hipoTezebis Sesaxeb

statistikur hipoTezaTa Sesaxeb gadawyvetilebis miRebis wesebi erTi da igivea yvela statistikuri kriteriumisaTvis. Sesabamisad, erTi SerCevis t kriteriumisaTvis:

Tuki trealuri xvdeba uaryofis areSi, maSin H0 iqneba uaryofili da H1 miRebuli; Tuki trealuri ar xvdeba uaryofis areSi, maSin H0-s ver uarvyofT da H1 ar

iqneba miRebuli.

tkritikulis ormxrivi mniSvneloba, roca df=29 da =0.05, aris 2.045. Sesabamisad, rogorc naCvenebia suraTze #8.5, trealuri, romelic tolia an naklebia -2.045-ze an tolia, an metia +2.045-ze, xvdeba uaryofis areSi. trealuri _2.526 aris ufro Sors mdebare (eqstremaluri), vidre tkritikuli =-2.045 da amitomac xvdeba ganawilebis marcxena kudze mdebare uaryofis areSi. Cven uarvyofT H0: ≠ 95 wuTs da viReb H1: ≠ 95. gansxvaveba SerCeviT saSualosa da populaciis saSualos Soris aris statistikurad

mniSvnelovani 0.05 doneze. Cven vaskvniT, rom SerCevis saSualo, romelic tolia 83 wuTis aris im populaciidan, romlis naklebia 95 wuTze. im mesameklaselebis mier televizoris yurebaze daxarjuli dro, romlebic CarTulni iyvnen gakveTilebis Semdgom bavSvebis movlis programaSi, aris mniSvnelovnad naklebi, vidre populaciis saSualo, romelic tolia 95 wuTisa dReSi.

Tuki trealuri xvdeba or kritikul mniSvnelobas _ (-2.045) da +2.045 Soris (magaliTad, aris 90 wuTi da amrigad, trealuri =-1.056), maSin H0 -s ver uarvyofT da ver miviRebT H1-s. realuri gansxvaveba SerCeviT saSualosa da populaciis hipoTetur saSualos Soris iqneboda aramniSvnelovani. am SemTxvevaSi ar gveqneboda sakmarisi mtkicebuleba imisaTvis, raTa CagveTvala, rom H0 ar aris WeSmariti.

gadawyvetilebani statistikuri hipoTezebisa da im daskvnebis Sesaxeb, romlebic miRebuli iqneboda romelime statistikurad mniSvnelovani an aramniSvnelovani erTi SerCevis t kriteriumisaTvis aris Sejamebuli cxrilSi #8.3.

daSvebebi erTi SerCevis t kriteriumisaTviserTi SerCevis t kriteriumi ayalibebs Semdeg daSvebebs SerCevis monacemebisaTvis: monacemebi SerCevaSi SemTxveviTad arian miRebuli populaciis

monacemebidan da TiToeuli monacemi damoukidebelia nebismieri sxva monacemisagan;

monacemebi populaciis SerCevaSi normaluradaa ganawilebuli.praqtikaSi, SerCevidan aRebulma monacemebma SeiZleba srulyofilad ar daakmayofilos es daSvebebi, Tumca mkvlevrebi ganagrZoben am monacemebisaTvis t kriteriumis gamoyenebas. am sakiTxs ufro dawvrilebiTY me-9 TavSi ganvixilavT.

cxrili #8.3 erTi SerCevis t kriteriumis gamoyenebiT statistikuri hipoTezebis Semowmebis Sesaxeb gadawyvetilebebisa da daskvnebis Sejameba. gamoyenebuli iyo 0.05 mniSvnelovnobis done.

magaliTi #8-1erTi SerCevis t kriteriumis gamoyeneba

amocanafsiqologebma wamoayenes hipoTeza, rom zogierTi individi Sejibrebis dros gaurbis did warmatebas, raTa Tavi aaridos misi Tanatolebis kritikulobasa da jgufidan gariyvas (gandevnas). TviTSefasebis kiTxvari, romelic zomavs warmatebisagan garidebas, SemuSavebuli iyo Ggudisa da gudis mier (Good and

Good, 1973). davuSvaT cnobilia, rom kolejis damamTavrebeli klasis studenti gogonebisaTvis populaciis saSualo am kiTxvaris mixedviT iyo 8.6 qula. qulebis diapazoni, romelic SeiZleboda yofiliyo miRebuli am kiTxvaris mixedviT icvleboda 0 qulidan, rac niSnavda warmatebisaTvis Tavis aridebis dabal motivacias, 28 qulamde, rac niSnavda warmatebisaTvis Tavis aridebis maRal motivacias. konsultantma moawyo konkurentuli situacia. man misca warmatebisaTvis Tavis aridebis motivaciis sazomi skala kolejis damamTavrebeli klasis student 33 gogonas, vinc gaira es seminari da miiRo saSualo 8.1 da s=1.31. gansxvavdeba Tu ara es mniSvnelovnad =8.6-sagan 0.05 mniSvnelovnobis doneze? gamoiyeneT ormxrivi kriteriumi.

Tu trealuri xvdeba uaryofis areSi =0.05-saTvis, maSin:

Tu trealuri ar xvdeba uaryofis areSi =0.05-saTvis, maSin: trealuris miRebis albaToba naklebia an toli

0.05, an p0.05H0 aris uaryofiliH1 aris miRebuli. SerCevis saSualo aRebulia populaciidan, romlis saSualo gansxvavebulia populaciis hipoTeturi saSualosagan.gansxvaveba -sa da hipoTetur -s Soris aris statistikurad mniSvnelovani 0.05 doneze.miiReba gadawyvetileba imis Sesaxeb, rom -sa da hipoTetur -s Soris gansxvavebis mizezi aris ara SerCevis Secdoma, aramed sxva mizezi.

trealuris miRebis albaToba metia an toli 0.05, an p>0.05H0 ar aris uaryofiliH1 ar aris miRebuli

gansxvaveba -sa da hipoTetur -s Soris aris statistikurad aramniSvnelovani 0.05 donezemiiReba gadawyvetileba imis Sesaxeb, rom SerCevis Secdoma yvelaze damajerebelad xsnis gansxvavebas -sa da hipoTetur -s Soris.

saWiro statistika: radganac gamoiTvleba s-is saSualebiT, am SemTxvevaSi unda gamoviyenoT erTi SerCevis t kriteriumi. daSvebebi:1. individebi, romlebmac gamokiTxvaSi miiRes monawileoba, SemTxveviTad

iyvnen SerCeulebi kolejis damamTavrebeli klasis studenti gogonebis populaciisagan;

2. warmatebis Tavidan aridebis motivaciis qulebi arian normalurad ganawilebulni populaciis SerCevaSi.

statistikuri hipoTezebi: H0: =8.6; H1: ≠8.6.mniSvnelovnobis done: =0.05df t-saTvis: N-1=33-1=32t-s kritikuli mniSvneloba: tkritikuli =2.042. t-s kritikuli mniSvneloba, roca df=32 ar aris warmodgenili cxrilSi #a-2. Cven gamoviyeneT tkritikuli df-is ufro dabali yvelaze axlo mdgomi mniSvneloba, df=30.uaryofis areebi: trealuris mniSvnelobebi, romlebic arian toli an naklebi -2.042, an arian toli, an meti +2.042.gamoTvla:

amitom,

gadawyvetileba: trealuri ufro eqstremaluria, vidre t kritikuli = -2.042 da xvdeba uaryofis areSi. amrigad, Cven uarvyofT H0-s da viRebT H1-s.

daskvna: warmatebisagan garidebis motivaciis saSualo kolejis damamTavrebeli klasis studenti gogonebis im SerCevisaTvis, vinc daeswrnen seminarebs (kerZod, =8.1) mniSvnelovnad naklebia, vidre kolejis damamTavrebeli klasis studenti gogonebis populaciis saSualo _ 8.6.

erTi SerCevis t kriteriumis Sedegebis prezentacia

amerikis fsiqologiuri asociaciis publikaciebis saxelmZRvanelos (1994) stili moiTxovs daskvniTi statistikis kriteriumebis Sedegebis warmodgenas, raTa specialuri forma mieces statistikuri kriteriumis, kriteriumis df-is, statistikis realuri mniSvnelobis, gamoyenebuli alfa donis da statistikis realuri mniSvnelobis albaTobisa da eqstremaluri mniSvnelobis Caweras nulovani hipoTezis WeSmaritebis pirobebSi. aseve saWiroa, Sesatyvisi aRweriTi statistikis _ SerCeviTi saSualosa da standartuli gadaxris warmodgena. Cven giCvenebT Sedegebis prezentaciis am stils erT-erTi sanimuSo magaliTis saxiT, romelic exeba kvlevas mesameklaseli moswavleebis mier televizoris yurebaze daxarjuli drois Sesaxeb.

saSualo dro, romelsac 30 mesameklaseli xarjavda yoveldRiurad televizoris yurebaze iyo 83 wuTi (SD=26 wuTis). alfa tolia 0.05-is da ormxrivi kriteriumisaTvis erTi SerCevis t kriteriumi gviCvenebs, rom SerCeviTi saSualo mniSvnelovnad gansxvavdeba populaciis hipoTeturi saSualosagan, romelic tolia 95 wuTis; t(29)= -2.53, p<0.05.

am Sedegebis prezentaciisas:

=0.05 gviCvenebs am kriteriumis mniSvnelovnobis dones;t(29) gansazRvravs testis statistikas rogorc t-s. es t gansazRvravs trealurs,

Tumca indeqsi `realuri” SeiZleba ar axldes. radgan t ganisazRvreba, rogorc erTi SerCevis t, Cven viciT, rom kriteriumi gulisxmobs SerCevis saSualos Sedarebas populaciis hipoTetur saSualosTan. Tavisuflebis xarisxi trealurisaTvis naCvenebia frCxilebSi. amrigad, Cven viciT, rom t-s gamoTvlisas gamoyenebuli iyo 30 monacemi (radgan df=N-1, amitom 29=30-1);

=-2.53 trealuris mniSvneloba (ara tkritikulisa, romelsac vpoulobT cxrilSi #a-2). t-s mniSvneloba unda iyos gamosaxuli ricxviT, romelsac eqneba mZimis Semdeg sul cota ori cifri;

p<0.05 gviCvenebs Semdegs: a) p da naklebobis niSani (p<) gviCvenebs, rom trealuris miRebis albaToba H0-is WeSmaritebis pirobebSi aris naklebi an toli 0.05-s. es mniSvneloba aris trealuris albaToba an ufro Sors mdebare trealuris

albaToba, roca H0 aris WeSmariti;

b) H0: =95.0 wuTi aris uaryofili, H0: 95.0 wuTi aris miRebuli;g) gansxvaveba SerCevis saSualosa da hipoTetur -s Soris aris statistikurad mniSvnelovani;d)SerCevis saSualo ganixileba rogorc aRebuli im populaciidan, romlis saSualo naklebia hipoTetur =95 wuTze;e) Tu p>0.05 iqneba SedegebSi warmodgenili, maSin metobis niSani aRniSnavs imas, rom H0 ar aris uaryofili da ar gansxvavdeba mniSvnelovnad hipoTeturi -sagan mniSvnelovnobis 0.05 doneze.

SeamowmeT Tqveni codna VIII-31. dawereT H0 da H1 erTi SerCevis t kriteriumisaTvis qvemoT mocemuli

TiToeuli debulebisaTvis.a) Ramis Zilis saSualo xangrZlivoba populacisaTvis aris 493 wuTi.b) depresiisaken midrekilebis Sefasebis kiTxvarSi populaciis saSualo aris 36.1.g) populaciis saSualo standartizebuli inteleqtis testisaTvis aris 100.

2. cxrilis #a-2 gamoyenebiT, ipoveT tkritikulis mniSvnelobebi miTiTebuli df-sa da gaTvaliswinebiT. gamoiyeneT ormxrivi kriteriumi.

df a. 6 0.05b. 17 0.01g. 23 0.05d. 32 0.05e. 39 0.05v. 75 0.01z. 103 0.05T. 256 0.05

3. miuTiTeT, xvdeba Tu ara trealuris Semdegi mniSvnelobebi uaryofis areSi. Semdeg miuTiTeT Tqveni gadawyvetileba H0 da H1-saTvis. gaxsovdeT, rom Tqven unda airCioT Semdegi: an ver uaryofT H0-s, an uaryofT H0-s, da ar miiRebT H1-s, an

miiRebT H1-s. gamoiyeneT =0.05 da ormxrivi kriteriumi. df TiToeuli trealurisaTvis

mocemulia frCxilebSi.a. trealuri(19)= +3.014b. trealuri(19)=-1.989g. trealuri(16)=-2.179d. trealuri(16)=+2.04e. trealuri(30)=-3.961b. trealuri(30)=+2.743z. trealuri(6)=-2.364T. trealuri(6)=+2.447i. trealuri(9)=-2..262k. trealuri(37)=+2.938l. trealuri(42)=-2.414. gansazRvreT daSvebebi, romlebic safuZvlad udevs erTi SerCevis t kriteriums.5. amoxseniT Semdegi amocana da upasuxeT qvemoT mocemul kiTxvebs. gamoiyeneT ormxrivi kriteriumi da 0.05 mniSvnelovnobis done. Tqven gainteresebT Tu ramdenad kargad SeuZlia adamians Seafasos dro saaTis gareSe. Tqven sTxoveT 11 cdis pirs SeefasebinaT drois 7 wuTiani intervali da eniSnebinaT TqvenTvis misi gasvlis dro saaTis gareSe. am 11 Sefasebis saSualo iyo 5.8 wuTi, s=0.8 wuTi. aris Tqveni SerCevis saSualo mniSvnelovnad gansxvavebuli populaciis saSualosagan, romelic tolia 7 wuTis? Tqvens pasuxSi miuTiTeT Semdegi: a. kriteriumi, romelsac gamoiyenebT b misi gamoyenebisas gakeTebuli daSvebebi g. statistikuri hipoTezebi d. mniSvnelovnobis done e. kriteriumis statistikis kritikuli mniSvneloba v. uaryofis areebi z. trealuris mniSvneloba T. Tqveni gadawyvetilebani statistikuri hipoTezebis Sesaxeb i. Tqveni daskvnebi SerCevis saSualos Sesaxeb6. es savarjiSo warmoadgens t kriteriumis gamoyenebis Sedegebs, romelic adaptirebulia gamoqveynebuli kvlevis Sedegebidan. TiToeuli warmodgenili SedegisaTvis upasuxeT Semdeg SekiTxvebs:a. ras udris SerCevis saSualos mniSvneloba?b. ras udris populaciis standartuli gadaxris Sefasebis mniSvneloba?

g. romeli t kriteriumi gamoiyeneba am kvlevaSi?d. ra aris H0 da H1 am t kriteriumisaTvis?e. ramdeni monacemi iyo miRebuli am kvlevaSi?v. risi tolia trealuris mniSvneloba am kvlevaSi?z. risi tolia df am trealurisaTvis?T. risi tolia tkritikuli =0.05-sa da ormxrivi uaryofis aresaTvis?i. uaryofT H0-s, Tu ver uaryofT H0-s?k. miiRebT H1-s, UTu ar miiRebT H1-s?l. daaskvniT Tu ara, rom SerCevis saSualo mniSvnelovnad gansxvavdeba populaciis hipoTeturi saSualosagan?

angariSi I: morbenali mamakacis saSualo wona iyo 74 kg (SD=7.6 kg), =0.05 da ormxrivi t kriteriumisaTvis erTi SerCevis t kriteriumma gviCvena, rom SerCevis saSualo ar gansxvavdeba mniSvnelovnad populaciis hipoTeturi saSualosagan, romelic tolia 75 kg-is 25 wlis mamakacebisaTvis; t(5)= -1.48, p<0.05.

angariSi II: araletaluri daavadebebis mqone 65 wlis da zemoT pacienti mamakacebis saavadmyofoSi gatarebuli dReebis saSualo Seadgenda 10.4 dRes (SD=2.3). =0.05 da ormxrivi kriteriumisaTvis erTi SerCevis t kriteriumma gviCvena, rom SerCevis saSualo mniSvnelovnad gansxvavdeba populaciis hipoTeturi saSualosagan, romelic tolia 8.7 dRisa; t(24) = +3.7, p<0.01.

mniSvnelovani sakiTxebi statistikuri hipoTezis SemowmebaSiratom ar viRebT nulovan hipoTezas?

Tu kriteriumis statistika ar xvdeba uaryofis areSi, maSin Cven ar uarvyofT nulovan hipoTezas. am gadawyvetilebas xSirad ase gamovTqvamT:`Cven ver uarvyofT nulovani hipoTezas”. Tqven SeiZleba gainteresebT, Tu ratom vxmarobT am rTul (Znelad gasageb) frazas; rogorc wesi, Tu Cven ar uarvyofT raimes, maSin Cven viRebT mas. ratom ar SegviZlia ubralod miviRoT nulovani hipoTeza, Tuki kriteriumis statistika ar xvdeba uaryofis areSi? am kiTxvaze pasuxis gasacemad Cven unda davubrundeT magaliTs, romelic satvirTo manqanebSi skamis simaRles exeboda. skami morgebuli iyo im mZRolebis populaciisaTvis, visi muxlqveSa simaRlec iyo 42 sm da =3.0sm. davuSvaT, rom Tqven gyavT satvirTo manqanis mZRolebis 25 kaciani SerCeva da aRmoCnda, rom am mZRolebis saSualo

muxlqveSa simaRle aris 42,5 sm. zrealuris mniSvneloba am SerCevisaTvis iqneba +0.83.

zrealuri =

es zrealuri ar xvdeba uaryofis areSi, romelic gansazRvrulia zkritikuls plus an minus 1.96 =0.05-is mniSvnelovnobis doneze, amitom is aris aramniSvnelovani. Cven ver uarvyofT H0:=42sm-is. magram niSnavs Tu ara es gadawyvetileba imas, rom im populaciis saSualo, romlidanac aris aRebuli es SerCeva unda iyos zustad 42sm? ra Tqma unda, ara; SerCevis saSualo 42.5 sm ar niSnavs imas, rom populaciis saSualo, romlidanac SerCeva aris aRebuli, unda iyos 42 sm-is toli. meore mxriv, =42.5 sm, agreTve, ar gvaZlevs imis mtkicebis saSualebas, rom ar SeiZleba iyos 42 sm toli, anu =42.5 sm ar gvaZlevs imis mtkicebis saSualebas, rom Cven unda uarvyoT H0: =42 sm, magram arc imis mtkicebis saSualebas gvaZlevs, rom zustad 42 sm-is tolia. is gvaCvenebs mxolod imas, rom Cven ar SegviZlia uarvyoT H0: =42 sm.

davuSvaT, rom toli iyo 44 sm, ase rom zrealuri =+3.33. es zrealuri aris statistikurad mniSvnelovani 0.05 doneze da Cven uarvyofT H0: =42 sm-s da miviRebT H1: 42sm-s. am SemTxvevaSi sworia miviRoT H1, radgan H1 amtkicebs, rom aris garkveuli mniSvneloba, romelic gansxvadeba 42sm-sagan da =44sm gviCvenebs, rom marTlac aris garkveuli mniSvneloba, romelic gansxvavdeba 42sm-sagan. am SemTxvevaSi, SerCevis saSualo 44sm uzrunvelyofs -s saukeTeso wertilovan Sefasebas mZRolebis im populaciisaTvis, saidanac iyo aRebuli SerCeva.

swori daskvnebi statistikuri hipoTezebis Sesaxeb SeiZleba Semdegnairad CamovayaliboT: TuU kriteriumis statistika xvdeba uaryofis areSi, maSin H0 uaryofili iqneba da

miRebuli iqneba H1;

Tu kriteriumis statistika ar xvdeba uaryofis areSi, maSin H0-s ver uarvyofT da ver miviRebT H1-s.

pirveli da meore gvaris Secdomebi statistikuri kriteriumebis gamoyenebisasstatistikuri kriteriumi mkvlevars aZlevs saSualebas uaryos H0 (1) an ver uaryos H0. (2) gadawyvetilebis siswore damokidebulia imaze, aris Tu ara nulovani hipoTeza

pirveli gvaris Secdoma (Type I error) Secdoma statistikuri gadawyvetilebis miRebis dros, romelic gamoixateba nulovani hipoTezis uaryofaSi, maSin roca is realurad WeSmaritia. meore gvaris Secdoma (Type II error) - Secdoma statistikuri gadawyvetilebis miRebis dros, romelic gamoixateba imiT, rom ar xdeba nulovani hipoTezis uaryofa, maSin roca is realurad mcdaria da alternatiuli H1 hipoTeza ki WeSmariti.beta(): meore gvaris Secdomis albaToba.

faqtiurad WeSmariti populaciisaTvis. pirveli gvaris Secdoma - Secdoma statistikuri gadawyvetilebis miRebis dros, romelic gamoixateba nulovani hipoTezis uaryofaSi, maSin roca is realurad WeSmaritia. meore gvaris Secdoma - Secdoma statistikuri gadawyvetilebis miRebis dros, romelic gamoixateba imiT, rom ar xdeba nulovani hipoTezis uaryofa, maSin roca is realurad mcdaria da alternatiuli H1

hipoTeza ki WeSmariti. cxrili #8.4 gviCvenebs am Secdomebs da SesaZlo swor gadawyvetilebebs, romelic SeiZleba miviRoT statistikuri kriteriumis gamoyenebisas.

pirveli da meore gvaris Secdomebi xdeba imis gamo, rom statistikuri gadawyvetilebis miReba albaTuri xasiaTiaa. TuU H0 aris WeSmariti, pirveli gvaris Secdomis albaToba udris -s. amitom, swori daskvnis albaToba da imis albaToba, rom H0 ar iqneba uaryofili, maSin roca is WeSmaritia, aris mocemuli 1--s saSualebiT. meore gvaris Secdomis albaToba udris -s. aqedan gamomdinare, H1-is sworad miRebis albaToba, roca H1 WeSmaritia aris mocemuli 1--s saSualebiT. mniSvneloba 1- aris statistikuri kriteriumis simZlavre. amitomac statistikuri kriteriumis simZlavre an ubralod simZlavre

aris H0-is uaryofis albaToba, roca H0 ar aris samarTliani da H1 aris WeSmariti. mniSvneloba damokidebulia mTel rig faqtorebze da Cven am faqtorebs me-9 TavSi ganvixilavT. TiToeuli es albaToba gansazRvrulia cxrilSi

#8.4.I gvaris Secdoma xdeba mxolod maSin, roca H0 aris WeSmariti. Aamitom, I

gvaris Secdomis daSvebis albaToba pirdapir damokidebulia arCeuli mniSvnelovnobis doneze. gaixseneT, rom mniSvnelovnobis done gviCvenebs kriteriumis statistikis mniSvnelobis miRebis albaTobas (mag., an z an t), romelsac mivyavarT H0-is uaryofamde, roca H0 aris realurad WeSmariti. Sesabamisad, roca statistikuri kriteriumisaTvis aris arCeuli , iqmneba I gvaris Secdomis daSvebis riski. roca =0.05 WeSmariti H0-is SecdomiT uaryofis albaToba tolia 0.05. analogiurad, roca =0.01, maSin arsebobs mxolod 1 Sansi 100-dan, rom daSvebul iqnes I gvaris Secdoma H0-is uaryofiT, roca is realurad WeSmaritia. mecnierebi, Cveulebriv, irCeven ufro frTxil gzas, raTa garantirebuli iyos I gvaris Secdomis daSvebis dabali albaToba maT kvlevebSi. Sesabamisad, -s mniSvnelobad irCeven xolme mis dabal mniSvnelobebs, rogoricaa, =0.05 an =0.01.

cxrili #8.4

simZlavre (Power) – H0-is uaryofis albaToba, roca H0 ar aris samarTliani da H1 ki aris WeSmariti. statistikuri kriteriumis simZlavre mocemulia 1--s saSualebiT.

Secdomebi da swori gadawyvetilebani statistikur gadawyvetilebebSi da TiToeul SedegTan dakavSirebuli

albaToba

Eeqsperimentatoris gadawyvetilebaH0-s ver uarvyofT H0 uaryofilia, H1

miRebuliaH0 WeSmaritia swori gadawyvetileba:

p=1-

I gvaris Secdoma:p=

WeSmariti situacia populaciaSiH1 WeSmaritia II gvaris Secdoma:

p=

swori gadawyvetileba:p=1-

statistikuri da mecnieruli mniSvnelovnobastatistikur kriteriumSi mniSvnelovnoba gviCvenebs mxolod imas, rom H0 aris uaryofili. Tu H0 uaryofilia, maSin statistikuri kriteriumi arc miRebuli Sedegis mizezs xsnis, arc imas gviCvenebs, Tu ramdenad mniSvnelovania, mecnieruli TvalsazrisiT, es gansxvaveba. mTeli pasuxismgebloba rCeba mkvlevars, romelmac yuradRebiT unda Seamowmos kvlevis pirobebi da miRebuli monacemebis xasiaTi.

kriteriumis statistikis realuri mniSvnelobebi, romlebic ar xvdeba uaryofis areSi iTvleba aramniSvnelovnad da ara umniSvnelod. umniSvnelo gulisxmobs rom raimes ara aqvs mniSvneloba. maSin, roca aramniSvnelovani gansxvaveba SeiZleba mecnierulad iyos Zalian mniSvnelovani. magaliTad, davuSvaT, rom gakveTilebis Semdeg bavSvebis movlis centris mier SemoTavazebuli iyo programa, romelic miznad isaxavda Seemcirebina is dro, romelsac bavSvebi atarebdnen televizoris yurebisas. Tuki da Sedarebisas aRmoCnda aramniSvnelovani gansxvaveba, es niSnavs, rom programa araefeqturia da ver cvlis im dros, romelsac isini atareben televizoris yurebisas da es SeiZleba mkvlevrisaTvis iyos mniSvnelovani Sedegi.

populaciis saSualosaTvis ndobis intervalebis ageba t statistikis gamoyenebiT

ndobis intervalebi populaciis saSualosaTvis gansazRvruli iyo me-7 TavSi rogorc intervalebi, romlebSic SeiZleba iyos moTavsebuli populaciis saSualo Cveni

darwmunebulobis garkveuli xarisxiT. im TavSi Cven avageT es intervalebi -is, z-is da -is gamoyenebiT. magaliTad, Tu cnobilia, 95- procentiani ndobis intervali populaciis saSualosaTvis mocemulia intervaliT

-1.96 -dan +1.96 -mde.xSirad ucnobia da misi mniSvneloba gamoiTvleba -iT. roca gamoiyeneba -is gamosaTvlelad, ndobis intervali agebulia t statistikis da ara z-is gamoyenebiT. t-sa da -is gamoyenebiT 95-procentiani ndobis intervali populaciis saSualosaTvis mocemulia intervaliT

_(t0.05)( )-dan + (t 0.05)( )-mde,sadac =realuri SerCevis saSualos = im saSualos gamoTvlili standartuli Secdoma, romelic miRebulia s/N-dan t 0.05= tkritikulis ormxrivi mniSvnelobisa 0.05 mniSvnelovnobis doneze N-1 Tavisuflebis xarisxiT.analogiurad, 99%-iani ndobis intervali populaciis saSualosaTvis mocemulia

- (t0.01)( )-dan + (t0.01)( )-mde.amis sailustraciod, Cven davadgineT 95-procentiani ndobis intervali 30 mesameklaselis mier televizoris yurebisas gatarebuli drois monacemebisaTvis =83 wuTisa da s=26 wuTisaTvis. radgan N=30, am SerCevisaTvis Tavisuflebis xarisxi iqneba 29 da ormxrivi tkritikulis(29) mniSvneloba 0.05 mniSvnelovnobis doneze aris 2.045. saSualos standartuli Secdoma, , tolia 26/30, anu 4.75 wuTis. amrigad, 95-procentiani ndobis intervali -saTvis iqneba

83-(2.045)(4.75)-dan 83+(2.045)(4.75),an Tu CavatarebT maTematikur gamoTvlebs, miviRebT ndobis intervals

83-9.7-dan 83+9.7 wuTamde,rac 73.3-dan 92.7 wuTamde intervalis tolia. albaToba imisa, rom moTavsebulia am intervalSi aris 0.95.

Tumca ndobis intervalebi ar aris farTod ganxiluli kvlevebSi, amJamad mkvlevrebs urCeven xolme CarTon isini TavianT kvlevebSi. (Streiner, 1996; Wilkinson da sxva, 1999). magaliTad, gvartnei-gibma (Gwartney-Gibb, 1986) Seiswavla wyvilebis qorwinebamde Tanacxovreba saarqivo masalebis gamoyenebiT, romelic aRebuli iyo qorwinebaze Tanxmobis ganacxadebidan, raTa mieRo demografiuli informacia im wyvilebze, romlebic qorwinebamde erTad cxovrebdnen da romlebic ar cxovrobdnen. misi kvlevis erT nawilSi, man warmoadgina qorwinebis asakis saSualo maCvenebeli im mamakacebisaTvis da

qalebisaTvis, romlebic qorwinebamde Tanacxovrobdnen da imaTTvisac, vinc qorwinebamde ar Tanacxovrobdnen. asakis saSualos Tan axlda 95-procentiani ndobis intervali populaciis saSualosaTvis. rogorc magaliTi, gvartnei-gibma warmoadgina Semdegi Sedegebi im qalebisaTvis, romlebic qorwinebamde Tanacxovrobdnen da ar Tanacxovrobdnen da amasTanave ar iyvnen adre namyofi qorwinebaSi.

Qqorwinebamde Tanacxovrobdnen

Qqorwinebamde ar Tanacxovrobdnen

Qqorwinebis saSualo asaki 806 877

22.8 21.7s 4.6 4.495-procentiani ndobis intervali -saTvis

22.48-dan 23.12-mde

21.41-dan 21.29-mde

Cven SegviZlia viyoT 95 procentiT darwmunebulebi, rom qorwinebis asakis im qalebisaTvis, romlebic qorwinebamde Tanacxovrobdnen, xvdeba 22.48-dan 23.12 wlebis SualedSi im populaciisaTvis, romelic SerCeuli iyo gvartnei-gibis mier. xolo im qalebisaTvis, romlebic qorwinebamde ar Tanacxovrobdnen, -s 95- procentiani ndobis intervali aris 21.41-dan 21.29 wlamde. ndobis es ori intervali ar faravs erTmaneTs, amitom gvartnei-gibsma daaskvna, rom is qalebi, romlebic qorwinebamde Tanacxovroben, Cveulebriv qorwindebian erTi wliT gvian, vidre is qalebi, romlebic qorwinebamde ar Tanacxovroben im populaciaSi, romelic mis mier iyo Seswavlili.

SeamowmeT Tqveni codna VIII-41. axseniT, Tu ratom ar iRebs mkvlevari H0-s, Tuki trealuri ar xvdeba uaryofis areSi.2. upasuxeT Semdeg kiTxvebs:

a. ra aris I gvaris Secdoma?b. risi tolia I gvaris Secdomis daSvebis albaToba, roca H0 aris WeSmariti?g. risi tolia I gvaris Secdomis daSvebis albaToba, roca H0 aris mcdari?d. ra aris II gvaris Secdoma?

e. SesaZlebelia Tu ara, rom eqsperimentatorma dauSvas orive I da II gvaris Secdomebi erTi da imave statistikur kriteriumSi? axseniT Tqveni pasuxi.

3. ra gansxvavebaa statistikur mniSvnelovnobasa da mecnierul mniSvnelovnobas Soris? 4. eqsperimentatori swavlobda, Tu ramden dros xarjavdnen mozardebi video TamaSebze video salonebSi. aRmoCnda, rom TamaSis saSualo dro dReSi Seadgenda 97 wuTs 23 mozardi biWisaTvis da standartuli gadaxra iyo 39 wuTi. aageT 95- procentiani ndobis intervali mozardi biWebis mier TamaSze daxarjuli drois populaciis saSualosaTvis.

Sejameba

statistikuri kriteriumebi SeiZleba iyo parametruli da araparametruli. statistikuri hipoTezis Semowmeba moicavs Semdeg nabijebs:

kriteriumis statistikis arCeva (mag. z an t) H0-is da H1-is formulirebas. H0 gamoiyeneba kriteriumis statistikis

SerCeviTi ganawilebis dasadgenad. mniSvnelovnobis donis dadgena. mniSvnelovnobis done, gansazRvruli

mniSvnelobiT, aris albaTobis mniSvneloba, romelic adgens nulovani hipoTezis uaryofis kriteriums.

kriteriumis statistikis kritikuli mniSvnelobis an mniSvnelobebis gansazRvra cxrilis gamoyenebiT. kriteriumis statistikis kritikuli mniSvnelobebi gansazRvraven uaryofis aris sazRvars an sazRvrebs.

uaryofis aris an areebis mdebareobis gansazRvra. es uaryofis are warmoadgens SerCevis ganawilebaSi kriteriumis statistikis mniSvnelobebs, romelTa miRebis albaToba -s tolia an masze naklebia, Tu H0 WeSmaritia.

statistikuri hipoTezebis Sesaxeb gadawyvetilebis miRebis wesebis formulireba. es wesebia:

_ Tu kriteriumis statistika xvdeba uaryofis areSi, maSin unda uarvyoT H0

da miviRoT H1. gansxvaveba SerCevis saSualosa da populaciis hipoTetur saSualos Soris aris statistikurad mniSvnelovani.

_ Tu kriteriumis statistika ar xvdeba uaryofis areSi, maSin ver uarvyofT H0-s da ver miviRebT H1-s. gansxvaveba SerCevis saSualosa da populaciis hipoTetur saSualos Soris aramniSvnelovania.

kriteriumis statistikis realuri mniSvnelobis gamoTvla SerCevis monacemebidan.

H0-isa da H1-is Sesaxeb gadawyvetilebis miReba damokidebulia imaze, kriteriumis statistikis realuri mniSvneloba moxvdeba Tu ara uaryofis areSi.

z statistika, mocemuli z=( -)/ formuliT, gamoiyeneba saSualosa da populaciis hipoTetur saSualos Sesadareblad, roca cnobilia.

t statistika, mocemuli t=( -)/ formuliT, gamoiyeneba saSualosa da populaciis hipoTeturi saSualos Sesadareblad, roca gamoiTvleba saSualebiT.

t statistikis SerCeviTi ganawileba dmokidebula t-s df-ze, sadac df=N-1. statistikuri daskvnebis gakeTeba imTaviTve albaTuria. I gvaris Secdoma

xdeba WeSmariti H0-is uaryofisas. II gvaris Secdoma xdeba maSin, roca ar aris uaryofili mcdari H0 .

I gvaris Secdomis albaToba aris , xolo II gvaris Secdomis albaToba aris .

statistikuri kriteriumis simZlavre aris H0-is .uaryofis albaToba, roca H0 .aris mcdari da H1 aris WeSmariti.

t ganawileba da SeiZleba gamoyenebuli iyos populaciis saSualosaTvis ndobis intervalebis dasadgenad.

mniSvnelovani terminebi da simboloebi

alfa ()alternatiuli hipoTeza (H1)

beta ()kritikuli mniSvnelobaTavisuflebis xarisxiaraparametruli kriteriumebiaramniSvnelovani gansxvavebanulovani hipoTeza

erTi SerCevis t kriteriumicalmxrivi kriteriumiparamatruli kriteriumebisimZlavreuaryofis aret-s SerCeviTi ganawilebamniSvnelovnobis donestatistikuri hipoTezastatistikurad mniSvnelovani gansxvavebatkritikuli

trealuri

kriteriumis statistikaormxrivi kriteriumiI gvaris SecdomaII gvaris Secdomaz

zrealuri

zkritikuli

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kiTxvebi1. ra moTxovnebi unda daakmayofilos nebismierma statistikurma hipoTezam?2. ra daniSnulebas emsaxureba nulovani hipoTeza statistikur kriteriumSi?3. ra aris mniSvnelovnobis done?4. ra aris kriteriumis statistikis kritikuli mniSvneloba?5. ra gadawyvetileba miiReba H0-sa da H1-is mimarTebaSi, Tuki:a. kriteriumis statistikis realuri mniSvneloba xvdeba uaryofis areSi?b. kriteriumis statistikis realuri mniSvneloba ar xvdeba uaryofis areSi?6. Tqven gTxoves SeamowmoT, gansxvavdeba Tu ara SerCevis saSualo _ 73

mniSvnelovnad populaciis hipoTeturi saSualosagan, romelic tolia 75-s. ra gansazRvravs Tqvens arCevans _ gamoiyenebT z kriteriums, Tu erTi SerCevis t kriteriums?

7. ra gadawyvetilebas miiRebT SerCevis saSualosa da populaciis hipoTeturi saSualos mimarTebis Sesaxeb, Tuki:

a. zrealuri xvdeba uaryofis areSi?b. zrealuri ar xvdeba uaryofis areSi?8. zrealuris TiToeuli mniSvnelobisaTvis gansazRvreT, moxvdeba Tu ara zrealuri

ormxriv uaryofis areSi mniSvnelovnobis 0.01 doneze. Semdeg miuTiTeT Tqveni gadawyvetilebis Sesaxeb H0-isa da H1-saTvis. gaxsovdeT, rom Tqven an (1) H0-s ver uaryofT an unda uaryoT H0 da, an (2) ar miiRoT, an miiRoT H1. (a) zrealuri = +3.71, (b) zrealuri = -2.6, (g) zrealuri = +1.74, (d) zrealuri=-3.05, (e) zrealuri = -1.96,

(v) zrealuri= +2.58.9. populaciis saSualo inteleqtis skalaze jgufisaTvis aris 100 da =16. skolis konsultantma am skalaze Seafasa 17 aTi wlis bavSvi da miiRo =94.1. SeasruleT statistikuri Semowmeba am monacemebis gamoyenebiT, raTa gansazRvroT, aris Tu ara gansxvaveba populaciis saSualosa da SerCevis saSualos Soris statistikurad mniSvnelovani. gamoiyeneT 0.05 mniSvnelovnobis done da ormxrivi kriteriumi.10. z kriteriumis Sedegebze dayrdnobiT daadgines, rom arsebobs mniSvnelovani gansxvaveba SerCevis saSualosa da populaciis saSualos Soris. axseniT, ras niSnavs statistikuri mniSvnelovnoba am SemTxvevaSi.11. ra gadawyvetilebas miiRebT SerCevis saSualosa da populaciis hipoTeturi saSualos mimarTebis Sesaxeb, Tuki:

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a. trealuri xvdeba uaryofis areSi? b. trealuri ar xvdeba uaryofis areSi?12. trealuris TiToeuli mniSvnelobisaTvis gansazRvreT, moxvdeba Tu ara is uaryofis areSi. CaTvaleT, rom =0.01 da gamoiyeneT ormxrivi kriteriumi. df trealuris

TiToeuli mniSvnelobisaTvis mocemulia frCxilebSi.a. trealuri(18) = +3.713b. trealuri(7) = -3.015g. trealuri(35) = +2.794d. trealuri(44) = -3.0.58e. trealuri(74) = -1.96v. trealuri(15) = -2.368

13. dedebi, romelTac patara bavSvebi hyavT xSirad Civian, rom maT akliaT Zili. davuSvaT, rom Tqven gazomeT im cxra dedis Zilis xangrZlivobis maCveneblebi, romelTac hyavT wlamde bavSvebi da miiReT Semdegi monacemebi (saaTebSi): 6.4, 7.5, 6.9, 7.3, 7.6, 7.1, 6.5, 7.7, 7.8. a. gansxvavdeba Tu ara mniSvnelovnad am dedebis Zilis xangrZlivobis saSualo populaciis hipoTeturi saSualosagan, romelic tolia 7.7 saaTisa RameSi? gamoiyeneT ormxrivi kriteriumi da 0.05 mniSvnelovnobis done.

b. aRwereT Tqveni Sedegebi im nimuSis Sesabamisad, romelic naCvenebi iyo am Tavis erT-erT qveTavSi _ erTi SerCevis t kriteriumis Sedegebis prezentacia.g. SeiZleba Tu ara statistikuri hipoTezis Sesaxeb Tqvens mier miRebul gadawyvetilebaSi iyos I gvaris Secdoma?d. SeiZleba Tu ara statistikuri hipoTezis Sesaxeb Tqvens mier miRebul gadawyvetilebaSi iyos II gvaris Secdoma?e. gamoiTvaleT 95 procentiani ndobis intervali populaciis saSualosaTvis am monacemebis mixedviT. xvdeba Tu ara populaciis hipoTeturi saSualo _7.7 saaTi RameSi am intervalSi?14. mkvlevari samedicino sferoSi swavlobda sakiTxs, axlavs Tu ara avadmyofobas sxeulis temperaturis mateba. man gazoma im 12 adamianis sxeulis temperatura, romelsac dausves daavadebis diagnozi da miiRo Semdegi monacemebi (farengeitebSi): 99.3, 99.1, 100.4, 98.4, 98.2, 98.9, 99.7, 100.1, 100.7, 99, 98.8, 99.2.

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a. gansxvavdeba Tu ara am individebis sxeulis temperatura populaciis hipoTeturi saSualo temperaturisagan _ 98.6F? gamoiyeneT ormxrivi kriteriumi da 0.05 mniSvnelovnobis done.

b. SeiZleba Tu ara statistikuri hipoTezis Sesaxeb Tqvens mier miRebul gadawyvetilebaSi iyos II gvaris Secdoma? ganmarteT Tqveni pasuxi.15. TiToeuli qvemoT mocemuli debulebisaTvis miuTiTeT, aris gadawyvetileba swori, Tu Secdomas Seicavs? Tuki is Seicavs Secdomas, miuTiTeT, is I gvaris, Tu II gvaris Secdomaa?.

a. mocemuli populaciisaTvis H0 aris WeSmariti, magram eqsperimentatorma zrealuris safuZvelze uaryo H0.

b. mocemuli populaciisaTvis H1 aris WeSmariti, magram eqsperimentatorma zrealuris safuZvelze CaTvala, rom H0-s ver uaryofs.

g. mocemuli populaciisaTvis H0 aris WeSmariti da eqsperimentatorma zrealuris safuZvelze CaTvala, rom H0-s ver uaryofs. d. mocemuli populaciisaTvis aris WeSmariti da eqsperimentatorma zrealuris

safuZvelze uaryo H0.

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