Tavi 1 - TSU Learn · Web viewmocemul formulaSi ricxviTi mniSvnelobebis Setanis Semdeg X-is mniSvnelobisgan miRebuli z-is mniSvnelobas z realurs vuwodebT da zrealuri-iT aRvniSnavT

Tavi VI. normaluri ganawileba, albaToba da standartuli qulebi

normaluri ganawilebanormaluri ganawileba da bihevioristuli mecnierebebinormaluri ganawilebis Tvisebebinormaluri ganawilebis qveS moTavsebuli farTobi

standartuli normaluri ganawilebastandartuli normaluri ganawilebis cxrilis gamoyenebaz qulebis gamoyenebis magaliTi

albaTobadiskretuli albaTobis ganawilebis TvisebebiTeoriuli da empiriuli albaTobastandartuli normaluri ganawileba, rogorc albaTuri ganawileba

standartuli qulebi

თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები

me-3-5 TavebSi SerCevidan miRebuli qulebis mowesrigebisa da aRweris sakiTxebze visaubreT. axla ganvixiloT populaciebi da maTi maxasiaTeblebi; kerZod, populaciis erT-erTi ganawileba _ normaluri ganawileba.Y pirvel TavSi populacia ganvmarteT, rogorc saerTo maxasiaTeblis mqone adamianebis, cxovelebis, sagnebisa Tu movlenebis sruli erToblioba. SerCeva aris populaciidan arCeuli qvesimravle anu qvejgufi. me-3, me-4 da me-5 TavebSi ganvixileT SerCevidan miRebuli qulebis mowesrigeba da aRwera sixSireTa ganawilebisa da iseTi aRweriTi statistikebis gamoyenebiT rogoricaa mediana, saSualo, kvartilTSorisi gabnevis diapazonis naxevari da standartuli gadaxra; Tumca, rogorc wina TavebSi iTqva, populaciis qulebis aRwerac SesaZlebelia saSualoTi da standartuli gadaxriT. vinaidan axla vambobT, rom vzomavT mTel populacias, saSualos aRvniSnavT simboloTi μ (miu, berZnuli anbanis meoce aso), xolo standartul gadaxras _ simboloTi σ (sigma, berZnuli anbanis meTvramete aso). garda amisa, miRebul sidideebs axla ukve vuwodebT ara statistikebs,

aramed parametrebs. parametri aris ricxvi, magaliTad, μ an σ,

romelic aRwers populacias. terminologiis aseTi Secvlis mizani isaa, rom gasagebi iyos,

raze vsaubrobT: _ SerCevaze Tu populaciaze. mniSvnelovania am gansxvavebis gageba, radgan normaluri ganawileba, romelzec qvemoT visaubrebT, xasiaTdeba μ-Ti da σ-Ti da ara _ da s-iT. aRniSnuli terminologia mocemulia cxrilSi N6-1.

normaluri ganawileba

bihevioristul mecnierebebSi populaciis erT-erT mniSvnelovan ganawilebas warmoadgens normaluri ganawileba. normaluri ganawileba aris Teoriuli maTematikuri ganawileba, romelic aRwers populaciis qulebis fardobiT sixSireebs. normalurad ganawilebul qulaTa erTobliobis sixSireTa ganawileba SeiZleba srulad aRiweros, Tu cnobilia ganawilebis saSualo (μ) da standartuli gadaxra (σ). normaluri ganawileba, romlis μ = 70 da σ = 10,

mocemulia suraTze N6-1. mocemuli ganawileba warmoadgens maTematikur

ganawilebas, romelic ar asaxavs romelime realurad gazomili populaciis qulebs. realuri populaciis qulebi arasodes aris ise zustad ganawilebuli, rom ganawileba zustad normaluri iyos; magram zogjer gamosadegia imis daSveba, rom zogierTi populaciis

cxrili N6.1. SerCevisa da populaciis Sesatyvisi aRniSvnebi.SerCeva populacia

aRweriTi maxasiaTeblebi

statistikebi parametrebi

saSualostandartuli gadaxradispersia

s

s2

μ

σ

σ2

normaluri ganawileba (Normal distribution): _ Teoriuli maTematikuri ganawileba, romelic aRwers populaciis qulebis fardobiT sixSireebs.

2


qulebi daaxloebiT normalurad aris ganawilebuli; Tumca, winamdebare TavSi normalur ganawilebas ganvixilavT, rogorc Teoriul maTematikur ganawilebas.

normaluri ganawileba da bihevioristuli mecnierebebi

normaluri ganawileba iakob bernulis (Jakob Bernoulli, 1654-1705), abraam de moiavris (Abraham de Moivre, 1667-1754), pier remond de monmortis (Pierre Remond de Montmort, 1678-

1719), karl fridrix gausisa (Karl Friedrich Gauss, 1777-1855) da sxvebis SromebSi ganviTarda. maTi interesis sagani iyo sxvadasxva azartul TamaSebSi mosalodneli Sedegebis albaTobebis gamoTvla da astronomiul da fizikur cdebSi gazomvebis Secdomebis ganawilebebis ageba. normaluri ganawileba maleve gadaiqca statistikosebisTvis mniSvnelovan ganawilebad, radgan statistikaSi arsebuli mravali problemis gadaWra mxolod normaluri ganawilebis daSvebis Semdeg gaxda SesaZlebeli.

normaluri ganawileba bihevioristuli mecnierebebis warmomadgenlebisTvisac mniSvnelovania, radgan gamarTlebulia daSveba imisa, rom mravali damoukidebeli cvladiT ganpirobebuli nebismieri qcevis gazomvis Sedegebi popoulaciaSi daaxloebiT normalurad iqneba ganawilebuli. adamianis mravali unari, iqneba es maTematikuri Tu verbaluri unarebi, ganpirobebulia damoukidebeli faqtorebis mTeli rigiT: _ genetikuri TandayolilobiT, mSoblebis mxridan miRebuli waxalisebiT, ganaTlebiT, konkretuli maswavlebliT, socialuri mdgomareobiT, kulturuli Zalebis zemoqmedebiT da a.S. Sesabamisad, aseTi unarebis mocemulobis xarisxi mosalodnelia, rom populaciaSi daaxloebiT normalurad iyos ganawilebuli. Cven imitom viyenebT sityvas „daaxloebiT“, rom realurad miRebuli qulebi arasodes araa zustad normalurad ganawilebuli.

normaluri ganawilebis Tvisebebi

3

40-

3σ

70μ

50-

2σ

60-

1σ

100+3σ

90+2σ

80+1σ

.10

.20

.30

.40

qulebi

fardobiTi sixSire

suraTi N6.1. normaluri ganawileba, romlis μ =70 da σ =10.


normaluri ganawilebis Tvisebebia: _ simetriuloba, asimptoturoba da uwyvetoba.

simetriulobaTeoriuli normaluri ganawileba simetriulia modis, medianisa da saSualos mimarT;

Sesabamisad, normalur ganawilebaSi moda, mediana da saSualo erTmaneTis tolia.

asimptoturobayuradReba miaqcieT, rom suraTze N6.1 gamosaxuli normaluri ganawilebis boloebi

(kudebi) rac ufro daSorebulia μ-sgan, miT metad uaxlovdeba X RerZs anu abscisas, Tumca, ganawileba asimptoturia, anu ramdenadac ar unda daSordes μ-s, kudebi mainc ar Seexeba X RerZs.

uwyvetobanormaluri ganawileba uwyvetia plius da minus usasrulobebs Soris moTavsebuli

yvela qulisTvis. es niSnavs, rom nebismieri ori qulisTvis moiZebneba mesame qula, romelic maT Soris mdebareobs.

normaluri ganawilebis qveS moTavsebuli farTobi

normaluri ganawileba fardobiT sixSireTa Teoriuli ganawilebaa. gavixsenoT me-3 Tavidan, rom qulis fardobiTi sixSire aris ganawilebaSi qulis Sexvedris sixSire gayofili ganawilebaSi Semaval qulaTa saerTo raodenobaze. amrigad, fardobiTi sixSire gamoisaxeba SefardebiT da ganawilebis saerTo dagrovili fardobiTi sixSire aris 1.0 (ix. cxrili N3.4, (i) sveti). vinaidan normaluri ganawileba fardobiTi sixSireebis ganawilebaa, ganawilebis qveS moTavsebuli sruli farTobi 1.0-s udris; amitom normalur ganawilebaSi qulaTa fardobiTi sixSireebi dagrovilia 1.0-mde.

yvela normaluri ganawilebaSi qulaTa garkveuli nawili moTavsebulia saSualos garSemo garkveul intervalSi. nebismier normalurad ganawilebul populaciaSi qulaTa .3413 wili moTavsebulia intervalSi μ-sa da μ-s plus erT standartul gadaxras Soris, anu qulaTa 34.13% moTavsebulia intervalSi (μ, μ+1σ). vinaidan ganawileba simetriulia, qulaTa igive procenti moTavsebulia intervalSi (μ_1σ, μ). saSualodan erT da or standartul gadaxras Soris anu intervalSi (μ_2σ, μ_1σ) moTavsebulia qulaTa 13.59%. intervalSi (μ_2σ, μ+2σ) moTavsebulia qulaTa (13.59+34.13+34.13+13.59)%=95.44%. normaluri ganawilebis intervalebSi Semaval qulaTa fardobiTi sixSirebi naCvenebia suraTze N6-2. es damokidebuleba ZalaSia nebismieri normaluri ganawilebis SemTxvevaSi, garda im SemTxvevebisa, rodesac μ da σ zogierT konkretul mniSvnelobas iRebs.

garda amisa, normaluri ganawilebis qveS moTavsebuli farTobebi SeiZleba Seikribos. ase magaliTad, saSualos minus erTi standartuli gadaxra da saSualos plius erTi standartuli gadaxra anu intervali (μ_1σ, μ+1σ) moicavs qulaTa (.3413+.3413)-s anu .6826 wils. analogiurad, saSualos minus ori standartuli gadaxra da saSualos plius ori standartuli gadaxra anu (μ_2σ, μ+2σ) intervalSi Sedis

4


qulaTa .1359+.3413+.3413+.1359 anu .9544 wili. qulaTa ufro didi intervali, romelic Seadgens sam standartul gadaxras saSualos orive mxares anu intervali (μ_3σ, μ+3σ), moicavs qulaTa (.0215+.1359+.3413+.3413+.1359+.0215)-s anu .9974 wils. es damokidebulebac naCvenebia surTze N6.2. es fardoba ucvleli rCeba μ-sa da σ-s mniSvnelobebisgan damoukideblad.

aRniSnuli mimarTebebis saCveneblad, davuSvaT, rom qulaTa populacia normalurad aris ganawilebuli. ganawilebis μ=100 da σ =15. ganawilebaSi qula 85 aris erTi standartuli gadaxra saSualos marcxniv (anu μ_1σ = 100-15=85), 115 qula ki _ erTi standartuli gadaxra saSualos marjvniv (anu μ+1σ = 100+15=115). Sesabamisad, ganawilebaSi qulaTa .6826 wili Sedis intervalSi (85, 115). analogiurad, qulaTa .9544 wili moTavsebuli iqneba 70-sa da 130 qulas Soris, intervalSi (μ_2σ, μ+2σ), xolo qulaTa .9974 wili _ 55-sa da 145 qulas Soris, intervalSi (μ_3σ, μ+3σ).

kidev erTi magaliTi. davuSvaT, gvaqvs qulaTa normalurad ganawilebuli populacia, romlis μ=50 da σ =5. am ganawilebaSi qulaTa .6826 wili moTavsebulia intervalSi 45-sa da 55-s Soris anu intervalSi (μ_1σ, μ+1σ), qulaTa .9544 wili _ intervalSi 40-sa da 60-s Soris anu intervali (μ_2σ, μ+2σ) da qulaTa .9974 wili _ intervalSi 35-sa da 65-s Soris anu intervali (μ_3σ, μ+3σ).

amrigad, normalur ganawilebaSi μ-sa da σ-s Semdegi damokidebulebaa: _ saSualos orive mxares mdebare konkretuli intervalebi ganawilebis qulaTa konkretul wils da konkretul procents moicaven.

-3σ -2σ -1σ μ +1σ +2σ +3σ

suraTi N6.2. normaluri ganawilebis konkretul intervalebSi Semaval qulaTa fardobiTi sixSireebi.

.0013 .0013.0215

.1359

.1334 .1334

.1359.0215

.6826

.9544

.9974

5


intervaliintervalSi Semaval

qulaTa wili

intervalSi Semaval qulaTa

procenti

(μ_1σ)-dan (μ+1σ)-mde(μ_2σ)-dan (μ+2σ)-mde(μ_3σ)-dan (μ+3σ)-mde

.6826

.9544

.9974

68.2695.4499.74

Semdeg qveTavSi viswavliT cxrilis gamoyenebas imis gansasazRvrad, qulaTa ra nawili xvdeba normaluri ganawilebis saSualos irgvliv mdebare konkretul intervalebSi. ase rom, ar aris saWiro aq warmodgenili mniSvnelobebis damaxsovreba; Tumca, sasurveli da gamosadegi iqneba miaxloebiTi mniSvnelobebis damaxsovreba, radgan amocanebis amoxsnisas SeZloT Tqvens mierve warmoebuli gamoTvlebis sisworis Semowmeba. amrigad, amocanebis amoxsnisas gamogadgebaT, Tu daimaxsovrebT, rom:

intervali (μ_1σ)-dan (μ+1σ)-mde daaxloebiT moicavs qulaTa 68%-s. intervali (μ_2σ)-dan (μ+2σ)-mde daaxloebiT moicavs qulaTa 95%-s. intervali (μ_3σ)-dan (μ+3σ)-mde daaxloebiT moicavs qulaTa 99%-s.

SeamowmeT Tqveni codna VI.1

1. ganmarteT: _ asimptoturi, uwyveti, μ, normaluri ganawileba, parametri, populacia, s, SerCeva, SerCeva, σ, statistika, .

2. daasaxeleT normaluri ganawilebis sami Tviseba.3. ratom ver iqneba adamianTa realuri populaciisgan miRebuli qulebi zustad

normalurad ganawilebuli? miniSneba: gaixsneT normaluri ganawilebis Tvisebebi.

4. davuSvaT, gaqvT qulaTa normalurad ganawilebuli populacia, romlis μ = 200 da σ = 20. miuTiTeT, qulaTa ra wili Seva qvemoT mocemul TiToeul intervalSi (gamoiyeneT suraTi N6.2).

intervalia) 220-220;b) 200-240;g) 200-260;e) 180-200;f) 160-200;v) 140-200;

z) 180-260;T) 160-240;i) 140-260;k) 220-240;l) 220-260;m) 140-160.

standartuli normaluri ganawileba

6


normaluri ganawilebis saSualosa da standartuli gadaxris codniT viRebT zust informacias konkretul intervalebSi qulaTa ganawilebis Sesaxeb. vinaidan normaluri ganawileba maTematikuri gantolebiT ganisazRvreba, Cven SegviZlia zusti fardobiTi sixSiris gansazRvra ganawilebis qulaTa nebismieri intervalisTvis; Tumca, aq Semdegi problemis winaSe vdgebiT: _ Cven unda gamoviyenoT rTuli maTematikuri gamosaxuleba, sadac Seva qulebis CvenTvis saintereso ganawilebis μ-sa da σ-is mniSvnelobebi. sabednierod, arsebobs am problemisTvis gverdis avlis martivi gza: _ normaluri ganawileba, romlis saSualoa μ da standartuli gadaxra _ σ (magaliTad, μ = 20, σ

= 2) unda gardavqmnaT standartuli formis normalur ganawilebad, romlis μ = 0, σ = 1.

amis Semdeg, Tu mxolod erT standartul normalur ganawilebas gamoviyenebT (romlis μ = 0, σ = 1), Cven SevZlebT gantolebis amoxsnas qulis mocemuli mniSvnelobisTvis da misi Sesabamisi fardobiTi sixSiris tabulirebas cxrilSi. SemdgomSi, Cven normaluri ganawilebis gantolebis gamoyenebis nacvlad SegviZlia, ubralod cxrilSi movZebnoT mocemuli qulis fardobiTi sixSire, an dagrovili fardobiTi sixSire.

standartuli normaluri ganawilebis saSualo nulis (μ = 0), standatruli gadaxra ki _ 1-is (σ = 1) tolia. nebismieri μ-sa da σ-is mqone normalurad ganawilebuli cvladisTvis konkretuli

qula (magaliTad, X) SeiZleba gardaiqmnas standartuli normaluri ganawilebis qulad, Tu mas SevitanT gamosaxulebaSi

,sadac z _ qulidan z-is realurad miRebuli1 mniSvneloba;

X _ sawyisi anu originaluri qula;μ _ saSualo;σ _ standartuli gadaxra im normaluri ganawilebisa, saidanac aRebulia X qula. mocemul formulaSi ricxviTi mniSvnelobebis Setanis Semdeg X-is

mniSvnelobisgan miRebuli z-is mniSvnelobas z realurs vuwodebT da zrealuri-iT aRvniSnavT. am terminologias imis

naTelsayofad viyenebT, rom z-is mniSvneloba formulaSi (zrealuri) ricxviTi mniSvnelobebis SetaniTaa gamoTvlili Tu a.1. cxrilSia napovni. aRniSnuli cxrilis Sesaxeb qvemoT visaubrebT. z-s xSirad standartul normalur gadaxras uwodeben, radgan misi mniSvneloba standartul normalur ganawilebaSi qulis mniSvnelobas gamoxatavs. standartuli normaluri gadaxra warmodgenilia suraTze N6-3. suraTze naCvenebia qulaTa ganawileba mocemul z qulebs Soris. yuradReba miaqcieT, rom vinaidan z = +1 miuTiTebs saSualodan erTi standartuli gadaxriT zemoT mdebare

1 SemdgomSi mTel teqstSi „realurad miRebulis“ nacvlad SemoklebiT gamoiyeneba „realuri“ (mTargmnelis SeniSvna).

standartuli normaluri ganawileba (Standard normal distribution): _ normaluri ganawileba, romlis μ = 0, σ = 1.

standartuli normaluri gadaxra (Standard

normal deviate): _ normaluri ganawileba, romlis μ = 0, σ = 1.

7


qulaze, z = 0-dan z = +1-mde SualedSi xvdeba qulaTa 3413 nawili, anu igive raodenoba, rac μ-sa da (μ+1)-s Soris suraTze 6.2.

აკლია სურათი 6-3suraTi N6.3. standartuli normaluri ganawileba, romlis μ = 0 da σ = 1.0. naCvenebia z-is or dadgenil mniSvnelobas Soris moxvedril qulaTa wili.

standartuli normaluri ganawilebis gamoyenebis sailustraciod ganvixiloT magaliTi. davuSvaT, 115 qula (anu X = 115) aviReT qulaTa normalurad ganawilebuli erTobliobidan, romlis μ = 100, σ = 15. es qula gadaviyvanoT zrealuri-Si formuliT

sawyisi 115 qulis gardaqmniT miRebuli standartuli normaluri qulaa +1.0. zrealuri = +1.0 niSnavs, rom sawyisi qula (anu 115) aris im normaluri ganawilebis erTi standartuli gadaxriT saSualos (anu 100) zemoT, saidanac is aviReT.

qulis gadayvana standartuli ganawilebis z qulaSi gviCvenebs, Tu ramdeni standartuli gadaxriT aris sawyisi qula (anu X) daSorebuli saSualosgan. magaliTad, aRebuli 115 qula +1.0σ erTeuliT aris daSorebuli ganawilebis saSualodan. ganvixiloT cxrilSi N6.2 mocemuli qulebi da maTi Sesabamisi zrealuri-is mniSvnelobebi. es qulebi miRebulia normaluri ganawilebidan, romlis μ = 75 da σ =10. SeecadeT, gamoTvaloT zogierTi mocemuli qula. yuradReba miaqcieT, rom zrealuri-is mniSvneloba nebismieri

qulisTvis ubralod warmoadgens imas, Tu standartuli gadaxris ramdeni erTeuliT aris daSorebuli qula standartuli normaluri ganawilebis saSualosgan (romelic nulis tolia) zemoT an qvemoT.

aqve unda gagafrTxiloT, rom X qulis gadayvana z qulaSi mxolod im SemTxvevaSi gviCvenebs sawyisi qulis

zust mdebareobas standartul normalur ganawilebaSi, Tu es sawyisi qula aRebulia qulaTa normalurad ganawilebuli simravlidan. Tu qula aRebulia ara normaluri ganawilebidan, aramed iseTi ganawilebidan, rogoricaa, magaliTad, marTkuTxa an

bimodaluri ganawileba, qulis gadayvana z qulaSi ver aqcevs sawyis qulebs normalurad ganawilebuls.

1051009590858075706560555045

+3.0+2.5+2.0+1.5+1.0+0.50.0-0.5-1.0-1.5-2.0-2.5-3.0

cxrili N6.2. normaluri ganawilebis (μ =75 da σ

=10) qulebis gadayvana standartuli ganawilebis z qulebSi.

qula (X) z

8


standartuli ganawilebis cxrilis gamoyenebaqulebis standartuli normaluri ganawilebis z qulebSi gadayvanis upiratesoba isaa,

rom z-is mniSvnelobebs Soris moxvedrili qulebis Tanafardoba cnobilia da mocemulia cxrilSi. es Tanafardobebi mocemulia danarTis cxrilSi a-1. am cxrilis nawili warmodgenilia N6.3 cxrilis saxiT. (a) svetSi mocemulia z-is mniSvnelobebi, romlebic miRebulia ricxviTi mniSvnelobebis SetaniT formulaSi

yuradReba miaqcieT, rom Cven viyenebT aRniSvnas zrealuri, rodesac vsaubrobT z-is nedli qulidan (magaliTad, X-dan) miRebul mniSvnelobaze, xolo aRniSvna z, indeqsis „realuri“-is gareSe, Seesabameba a-1 cxrilis (a) svetSi mocemul z-is mniSvnelobas. (b) svetSi mocemulia farTobi z = 0-sa da z-is (a) svetSi mocemul mniSvnelobas Soris. svetis Tavze warmodgenil grafikze es farTobi gamuqebulia. (g) svetSi mocemulia farTobi z-is (a) svetSi mocemuli mniSvnelobidan usasrulobamde. aqac svetis Tavze mocemul grafikze gamuqebulia es farTobi. vinaidan normaluri ganawileba simetriulia, z-is uaryofiTi mniSvnelobebiT SemosazRvruli farTobebi z-is dadebiTi mniSvnelobebis identuria. amrigad, cxrilSi ar aris warmodgenili z-is uaryofiTi mniSvnelobebi.

cxrilis gamoyenebis sailustraciod, ganvixiloT qulaTa normalurad ganawilebuli simravle, romlis μ =100 da σ =15. davuSvaT, am ganawilebidan aviReT qula 115. ganawilebis qulaTa ra wili iqneba 115-is toli an masze naklebi? am amocanis amosaxsnelad, Cven 115 qula gadagvyavs z qulaSi, anu

(a) (b) (g)0.95 .3289 .1711

cxrili N6.3. farTobebi normaluri ganawilebis qveS z = 0.95-dan z = 1.10-mde.

(a)

(b)

(g)

0

0

0

0

+z +z

-z -z -z

z-is mniSvneloba

9


0.960.970.980.991.001.011.021.031.041.051.061.071.081.091.10

.3315

.3340

.3365

.3389

.3413

.3438

.3461

.3485

.3508

.3531

.3554

.3577

.3599

.3621

.3643

.1685

.1660

.1635

.1611

.1587

.1562

.1539

.1515

.1492

.1469

.1446

.1423

.1401

.1379

.1357nawyveti amoRebulia II.1 cxrilidan, normaluri ganawileba da masTan dakavSirebuli ganawilebebi. CRC Handbook of Tables for Probability and Statistics (meore gamocema). COpyright 1968, CRCPress, Inc., Boca Raton, Florida. gamoiyeneba nebarTviT.

yuradReba miaqcieT, rom vinaidan z-is mniSvneloba nedli 115 qulidan miiReba, mas zrealuri-s vuwodebT, z-is cxrilSi N6-3 mocemuli mniSvnelobisgan gansasxvaveblad. saZiebeli qulis TvalsaCinod gamosaxvisTvis sasargebloa standartuli normaluri ganawilebis daxazva da masze nedli qulebisa da am qulebis Sesabamisi z qulebis miTiTeba. ganawilebis nedli qulebi da maTi Sesabamisi z qulebi naCvenebia suraTze N6-4. amocanis amosaxsnelad saWiroa, rom miviRoT ganawilebis farTobi z = _ ∞-dan z

= +1.0-mde. ganawilebis qulaTa erTi naxevari (anu 0.5000) z = 0-s qvemoT aris moTavsebuli. cxrilis N6.3 (b) sveti gviCvenebs, rom ganawilebis qulaTa .3413 moTavsebulia z = 0-sa da z = +1.0-s Soris (ix. suraTi N6.4, (g) RerZi). amrigad, amocanis pasuxia qulaTa z = _ ∞-dan z = 0-mde moTavsebul wils (.5000) plius qulaTa z

= 0-dan da z = +1.0-mde moTavsebuli wili (.3413 a-1 cxrilis (b) svetidan), anu .8413. 115-is toli an masze nakleb qulaTa wilia .8413 anu 84.13%.

აკლია სურათი 6-4suraTi N6.4. normalur ganawilebaSi, romlis μ = 100 da σ =15, Semavali qulis (115) adgilmdebareoba standartul normalur ganawilebaSi. X RerZis warwerebi gviCvenebs (a) nedl qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _∞-dan z = 0-mde da z

= 0-dan z = +1-mde.

a-1 cxrilis wakiTxvisas Zalze advilia Secdomis daSveba, amitom yovelTvis sasargebloa, rom amocanis amoxsniT miRebuli paxusi SeadaroT a-1 cxrilSi mocemul mniSvnelobebs. Secdomis Tavidan asacileblad, SegiZliaT daaxloebiT gansazRvroT pasuxi imis mixedviT, rom normalurad ganawilebul qulaTa erT mesamedze odnav meti (anu 33%-ze odnav meti) moTavsebulia ganawilebis saSualosa da erT standartul gadaxras Soris. amiT gansazRvravT, rom Tqveni pasuxi (50+33)%-ze anu 83%-ze odnav meti unda iyos. miRebuli pasuxi 84.13% Seesabameba aseT Sefasebas.

davuSvaT, axla gvsurs, rom vicodeT, qulaTa ra nawilia 115-is toli an masze meti. amis gamosaTvleli ori gza arsebobs:

10


(1) vinaidan Cven ukve viciT, rom qulaTa 84.13% tolia an naklebia 115-ze, normaluri ganawilebis qveS darCenili farTobi unda migviTiTebdes im qulaTa raodenobaze, romelic 115-is toli an masze meti iqneba. amitom saZiebel pasuxs miviRebT, Tu ganawilebis mTlian farTobs 1.000-s gamovaklebT .8413-s, anu qulaTa 1.000_.8413=.1587 nawili tolia an metia 115-ze.

(2) Tu winaswar ar gvaqvs gamoTvlili, rom qulaTa .8413 nawili tolia an naklebia 115-ze, maSin amocanis amosaxsnelad unda ganvsazRvroT, rom saZiebeli nawili aris ganawilebis farTobi z = +1.0-is Semdeg. cxrilis N6-3 (a) svetSi vpoulobT z = 1.00-is mniSvnelobas, Semdeg vnaxulobT mis gaswvriv (g) svetSi mocemul mniSvnelobas, sadac warmoadgenilia z-is zemoT qulaTa wili. z = +1.0-isTvis es mniSvnelobaa .1587. amrigad, qulaTa .1587 nawili anu 15.87% tolia an metia 115-ze.

amave ganawilebis magaliTze ganvixiloT kidev erTi amocana. qulaTa ra nawilia 82 qulis toli an masze naklebi? am SemTxvevaSic 82 gadagvyavs z qulaSi: zrealuri = (82_100)/15 = _1.2. es zrealuri gviCvenebs, rom 82 qula mdebareobs ganawilebis saSualodan 1.2 standartuli gadaxriT qvemoT. ganawilebis nedli qulebi da zrealuri = _1.2 naCvenebia suraTze N6-5. Semdeg gamovTvliT z = _1.2-is qvemoT mdebare qulaTa wils. yuradReba miaqcieT, rom naxazis mixedviT mxolod imis gansazRvra SegviZlia, rom es nawili Zalze mcire iqneba. vinaidan standartuli normaluri ganawileba simetriulia, imdeni qula iqneba moTavsebuli z = _1.2-is qvemoT, ramdenic z = +1.2-is zemoT; amitom viTvliT z = +1.2-is zemoT; Tumca, es mniSvneloba ar aris mocemuli cxrilSi N6-3. amitom viyenebT normaluri ganawilebis qveS mdebare farTobis wilebis srul cxrils (ix. cxrili a-1). (a) svetSi vpoulobT z = 1.2-s da vnaxulobT mis gaswvriv (g) cxrilSi mocemul mniSvnelobas. amrigad, qulaTa .1151 nawili anu 11.51% tolia an naklebia 82-ze. standartuli normaluri ganawilebis gamoyenebis kidev erTi magaliTi warmodgenilia qveTavSi „magaliTi N6.1“.

აკლია სურათი 6-5suraTi N6.5. normalur ganawilebaSi, romlis μ = 100 da σ =15, Semavali qulis (82) adgilmdebareoba standartul normalur ganawilebaSi. X RerZis warwerebi gviCvenebs (a) nedl qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _∞-dan z = _1.2-mde.

magaliTi N6.1

standartuli normaluri ganawilebis gamoyeneba

vTqvaT, gvaqvs normalurad ganawilebuli populacia, romlis μ = 80, da σ =5.

amocanaqulaTa ra nawilia 88 qulis toli an masze meti?

11


saWiro statistika:

amoxsna88 qula unda gadaviyvanoT zrealuri-Si. Semdeg cxrilis Na-1 gamoyenebiT vipoviT

zrealuri-is tol an masze met qulaTa wils. z-is formulaSi CavsvaT ricxviTi mniSvnelobebi:

.

a-1 cxrilis (g) sveti gviCvenebs, rom standartuli normaluri ganawilebis z = +1.6-is toli an masze meti farTobia .0548. aqedan gamomdinare, 88-is tol an masze met qulaTa wili iqneba .0548 anu 5.48%. amrigad, populaciis yoveli 1000 qulidan 54.8 tolia an metia 88-ze.

amocanaganxiluli ganawilebis qulaTa ra nawilia moTavsebuli 83-sa da 87-s Soris?

amoxsnaorive qula _ 83 da 87 unda gadaviyvanoT zrealuri qulebSi. Semdeg viTvliT standartuli

normaluri ganawilebis farTobs zrealuri-is or mniSvnelobas Soris. ricxviTi mniSvnelobebis Casmis Semdeg miviRebT:

zrealuri 83-isTvis ;

zrealuri 87-isTvis .

nedli qulebi da maTi Sesabamisi zrealuri qulebi mocemulia suraTze N6-6. zrealuri = +0.6-isTvis a-1 cxrilis (b) svetSi vnaxulobT, rom z = 0-sa da z = 0.6-s Soris mdebare qulaTa wilia .2257. zrealuri= +1.4-isTvis z = 0-sa da z = +1.4-s Soris mdebare qulaTa wilia .4192. ganawilebis farTobi, romelic z-is am or mniSvnelobas Soris mdebareobs, gamoxatavs 83 da 87 qulebs Soris moTavsebul qulaTa wils. am farTobis gamosaTvlelad .4192-s vaklebT .2257-s. miRebuli pasuxia .1935 (ix. suraTi N6.6, (g) RerZi). amrigad, qulebis wili, romelic Tavsdeba 83-sa da 87-s Soris intervalSi, aris .1935 anu 19.35%. yoveli 1000 qulidan 193.5 moTavsebulia 83 qulasa da 87 qulas Soris.

აკლია სურათი 6-6suraTi N6.6. normalur ganawilebaSi, romlis μ = 80 da σ =5, Semavali qulebis (83 da 87) adgilmdebareoba standartul normalur ganawilebaSi. X RerZis warwerebi gviCvenebs (a) nedl qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _0-dan z = +0.6-mde da z = 0-dan z = +1.4-mde.

amocanaqulaTa ra nawilia moTavsebuli 65-sa da 74-s Soris?

amoxsna

12


es amocana wina amocanis msgavsad ixsneba. qulebi 65 da 74 unda gadaviyvanoT zrealuri qulebSi. Semdeg viTvliT standartuli normaluri ganawilebis farTobs zrealuri-is or mniSvnelobas Soris. ricxviTi mniSvnelobebis Casmis Semdeg miviRebT:



nedli qulebi da maTi Sesabamisi zrealuri qulebi mocemulia suraTze N6.7. a-1 cxrilis (b) svetSi vnaxulobT, rom qulaTa .4987 moTavsebulia z = 0-sa da z = _3.0-s Soris. zrealuri = _1.2-isTvis qulaTa .3849 nawili moTavsebulia z = 0-sa da z = _1.0-s Soris. z = _3.0-sa da z = _1.2-s Soris moTavsebul qulaTa wilis gamosaTvlelad .4987-s gamovaklebT .3849-s da miviRebT .1138-s (ix. suraTi N6.7, (g) RerZi). qulaTa wili 65-dan 74-mde intervalSi aris .1138 anu 11.38%. populaciaSi yoveli 1000 qulidan 113.8 moTavsebulia 65-sa da 74-s Soris.

აკლია სურათი 6-7suraTi N6.7. normalur ganawilebaSi, romlis μ = 80 da σ =5, Semavali qulebis (65 da 74) adgilmdebareoba standartul normalur ganawilebaSi. X RerZis warwerebi gviCvenebs (a) nedl qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _0-dan z = -3.0-mde da z = 0-dan z = -1.2-mde.

amocanaqulaTa ra wili Seadgens ganawilebis Sua 80%-s?

amoxsnaamocanis amosaxsnelad a-1 cxrilSi unda vnaxoT (b) sveti, sadac warmodgenilia

saSualos orive mxares mdebare qulaTa .40 nawili anu 40%; am qulebisTvis ganvsazRvravT z-is mniSvnelobas da Semdeg z-is formulis amoxsniT viRebT X-is mniSvnelobebs. suraTze N6-8 naCvenebia qulaTa saZiebeli wilebi. z-is sapovnelad, romelic moicavs z = 0-sa da mis mniSvnelobas Soris moTavsebul qulaTa .40 wils, (b) svetSi veZebT .40-Tan yvelaze miaxloebul wils. es aris .3997, romelic Seesabameba z

= 1.28-s. amrigad, z = 0-sa da z = + 1.28-s Soris moTavsebuli qulebis Sexvedris fardobiTi sixSirea .3997 da z = 0-sa da z = _ 1.28-s Soris moTavsebuli qulebic igive fardobiTi sixSiriT (.3997) gvxvdeba. z = _ 1.28-sa da z = + 1.28-s Soris moTavsebuli qulebis saerTo raodenobaa .3997+.3997=.7994 anu daaxloebiT .80. bolo nabijia X-is mniSvnelobebis gamoTvla, romlebic z = _ 1.28-sa da z = + 1.28-s Seesabameba. am mniSvnelobebis gamosaTvlelad z-is, μ-sa da σ-s ricxviTi mniSvnelobebi CavsvaT formulaSi zrealuri=(X_μ)/σ da gamovTvaloT X-is mniSvnelobebi.

აკლია სურათი 6-8

13


suraTi N6.8. normalur ganawilebaSi, romlis μ = 80 da σ =5, Semaval qulaTa Sua 80%-is povna standartul normalur ganawilebaSi. X RerZis warwerebi gviCvenebs (a) nedl qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _0-dan z = -1.28-mde da z = 0-dan z

= +1.28-mde.

zrealuri= +(1.28)-isTvis

gantoleba amovxsnaT X-is mimarT

zrealuri= _1.28-isTvis

gantoleba amovxsnaT X-is mimarT

Sesabamisad, qulaTa 80% moTavsebulia 73.6-sa da 86.4-s Soris.

z qulebis gamoyenebis magaliTixSirad iTvleba, rom didi raodenobiT adamianebze Catarebul standartizebul testebSi

miRebuli qulebi daaxloebiT normalurad aris ganawilebuli da z-is mniSvnelobebiT gamoixateba. amitom cnobilia xolme, ganawilebis danarCen qulebTan mimarTebaSi zustad sad mdebareobs mocemuli qula. magaliTad, iarnoldma, grimma da miuserma (Yarnold, Grimm & Mueser, 1986) ikvlies, Tu ramdenad ukavSirdeba A da B tipis qcevis paternebi socialur konformulobas. adamianebis azriT, A tipis qceva gulisxmobs Sejibrianobas, agresiulobas, mouTmenlobasa da swrafad gaRizianebadobas. B tipis individebi xasiaTdebian rogorc araSejibrianebi, araagresiulebi, momTmenebi, romlebic mSvidad xvdebian imedgacruebebsa da frustraciebs. A da B tipis qcevis paternebis Sefasebis erT-erTi sazomia jenkinsis aqtiurobis kiTxvari (Jenkins, Zyzanski & Rosenman,

1979), romelic tipur aqtivobebs Seexeba. masSi Sedis iseTi kiTxvebi, rogoricaa, magaliTad:

dagisignalebiaT Tu ara frustraciis gamosaxatad avtomanqaniT sacobSi moxvedrisas?

arasodes xandaxan xSirad TiTqmis yovelTvis

14


A tipis individebi pasuxoben „xSirad“ an „TiTqmis yovelTvis“, xolo B tipis individebi _ „xandaxan“ an „arasodes“.

iarnoldma da sxvebma gamoTqves hipoTeza, rom xazis sigrZis Sefasebisas A tipis individebi ufro naklebad daeqvemdebarebodnen jgufis gavlenas da wavidodnen kompromisze, vidre _ B tipis individebi. kvlevis am hipoTezis Sesamowmeblad maT SearCies individTa ori jgufi: _ jenkinsis aqtiurobis kiTxvaris qulebis mixedviT, ukiduresad A tipis individebi da ukiduresad B tipis individebi. maTi monacemebiT, jenkinsis aqtiurobis kiTxvaris saSualo z qula A tipis individebiTvis iyo +1.22 (dadebiTi z

qula gviCvenebs A tipis qcevas), xolo B tipis individebisTvis _ _1.38 (uaryofiTi z qula gviCvenebs B tipis qcevas). TiToeuli jgufis saSualo z qulebis Sesaxeb informacia saSualebas gvaZlevs, zustad ganvsazRvroT TiToeuli jgufis mdebareoba jenkinsis aqtiurobis kiTxvaris A _ B tipis wrfeze. a-1 cxrilis (g) svetSi vnaxavT, rom standartul normalur ganawilebaSi z-is mniSvnelobaTa mxolod .11 nawili tolia z = +1.22-is, an aRemateba mas. amrigad, z = +1.22 A tipis qcevis jgufis saSualos qulaTa 89-e procentilze ayenebs. analogiurad, z = _1.38-is SemTxvevaSi vxedavT, rom B tipis wrfeze qulaTa mxolod .08 nawilia moTavsebuli am mniSvnelobis qvemoT. ukiduresad A tipis qcevis da B tipis qcevis mqone individebi sxvadasxva jgufSi Tavsdebodenen. miRebuli Sedegebi adasturebs hipoTezas: _ B tipis individebi 2-jer metad eqvemdebarebian jgufis gavlenas da midian konsensusze, vidre A tipis cdis pirebi.


1. ganmarteT: _ standartuli normaluri gadaxra, standartuli normaluri ganawileba, z, zrealuri.

2. X-is, μ-sa da σ-s TiToeuli mniSvnelobisTvis gamoTvaleT zrealuri. Semdeg cxrilis Na-1 gamoyenebiT ipoveT farTobi: _ (1) z = 0-sa da z-s Soris; (2) z-is zemoT.

3. amoxseniT Semdegi amocanebi „magaliTi N6-1“-Si warmodgenili midgomis gamoyenebiT. davuSvaT, Tqven CaatareT farTomasStabiani kvleva studentebSi da ikvlieT verbaluri analogiebiT msjelobis unari. vTqvaT, testis qulebi normalurad aris ganawilebuli, ganawilebis μ = 72 da σ = 8.a) ganawilebis qulaTa ra wilia 86-is toli an

masze meti?b) ganawilebis qulaTa ra wilia

moTavsebuli 80-sa da 90-s Soris?

μ σ X

a.

73 4.2 81

b.

127 11.6 121

g.

17 1.3 18

d.

50 3.0 39

e.

5 0.4 5.7

v.

256 35.0 212

15


g) ganawilebis qulaTa ra wilia moTavsebuli 60-sa da 70-s Soris?d) qulaTa ra diapazons moicavs testSi miRebul qulaTa ganawilebis Sua 90%?

albaToba

Cven ganvixileT normaluri ganawileba, rogorc fardobiT sixSireTa Teoriuli ganawileba, Tumca, is, amavdroulad, uwyveti cvladis albaTur ganawilebasac warmoadgens. es cnebebi statistikis mniSvnelovan komponents _ albaTobas _ ukavSirdeba; Tumca, saxelmZRvanelos Semdgom TavebSi albaTobis mxolod yvelaze mniSvnelovani aspeqtebia warmodgenili. am qveTavSi gavecnobiT albaTobis cnebas, romelic saWiroa albaTuri ganawilebis gagebisa da albaTobis statistikaSi gamoyenebisaTvis.

yoveldRiur cxovrebaSi ganxorcielebuli qmedebebi albaTobis intuitur wvdomas uzrunvelyofs. magaliTad, migiRiaT rodesme qvemoT ganxilul magaliTSi aRwerilis msgavsi werili?

saswrafo Setyobineba _ upasuxeT dauyovnebliv. Tqven moigeT qvemoT CamoTvlili prizebidan erT-erTi:

principulad axali modelis manqana, fasi $36.000.orkviriani ufaso mogzauroba havaize, fasi $6.000.farTo ekraniani stereo televizori, fasi $1995.mikrotalRovani Rumeli, fasi $375.umaRlesi xarisxis fotoaparati, fasi $75.kompiuteri, fasi $749.

mogebuli prizis wasaRebad Tqven ubralod unda mobrZandeT CvenTan da waiRoT kuTvnili prizebi.

mdidruli manqanisa Tu orkviriani dasvenebis perspeqtivam SeiZleba dagafiqroT, Tumca, lamazi ocneba gagiqrebaT, Tu waikiTxavT meore gverdze wvrilad dabeWdil teqsts, sadac naxavT TiToeuli saCuqris mogebis Sanss, romelic ase gamoiyureba:

davuSvaT, TiToeul monawiles erTi prizis mogebis toli Sansi aqvs. rogor fiqrobT, romeli prizis mogeba gendomebodaT yvelaze metad? romeli prizis mogeba gendomebodaT yvelaze naklebad? albaTobis Teoria saerTod rom ar gqondeT naswavli, Tqven, albaT, mainc ityodiT, rom manqanaze fiqrs azri ar aqvs da isev fotoaparatis arCeva sjobia. milioni

prizi

saCuqrebis raodenoba (1,000,000 monawile)

1. manqana2. dasveneba3. televizori4. mikrotalRovani Rumeli5. fotoaparati6. kompiuteri

1203466

999,84534

16


monawilidan mxolod erTi moigebs manqanas maSin, roca milionidan 999,845-s Sexvdeba fotoaparati. cxadia, rom, manqanasTan SedarebiT, ufro metia fotoaparatis mogebis Sansi.

davuSvaT, gekiTxebian, Tu ras udris albaToba imisa, rom televizori moigoT. mosalodnelia, rom daaxloebiT ase upasuxebT: _ „34 Sansi milionidan“. faqtobrivad, Tqven am pasuxiT miuTiTeT, rom konkretuli prizis mogebis albaToba gamoiTvleba am prizis raodenobis gayofiT mosageb prizTa saerTo raodenobaze, anu

.

Tu am formulas rigiT mesame prizis mimarT gamoviyenebT, miviRebT televizoris mogebis albaTobas anu 34/1,000,000=.000034. Tu am formulas TiToeuli prizis mimarT gamoviyenebT, maTTvis albaTobebis Semdeg ganawilebas miviRebT:

Tqven, albaT, SeamCnevT, rom prizis mogebis albaTobis formula Zalian hgavs fardobiTi sixSiris formulas, romelic me-3 TavSi ganvixileT. SegviZlia vTqvaT, rom es formula fardobiTi sixSiris formulis gamoyenebis specifikuri SemTxvevaa.

Tu prizebis ganawilebaSi TiToeul SesaZlo xdomilobas Sedegs vuwodebT, maSin am ganawilebaSi 1,000,000 SesaZlo Sedegi gveqneba. TiToeuli Sedegi Tanabrad albaTuria

da TiToeulis ganxorcielebis albaToba 1/1,000,000 anu .000,001 iqneba. CvenTvis saintereso Sedegebis erTobliobis ganxorcielebas xdomilobas uwodeben. magaliTad, Tu ganvixilavT me-5 prizis mogebis xdomilobas, mosalodnelia, rom es xdomiloba 999,845 SemTxvevaSi ganxorcieldes. Tu me-3 an me-4 prizis mogebis xdomiloba gvainteresebs, maSin vnaxavT, rom me-3 prizisTvis 34 SemTxvevaa, me-4 prizisTvis ki _ 66, rac jamSi am xdomilobis ganxorcielebis 100 SemTxvevas, SesaZlo Sedegs Seadgens. yuradReba miaqcieT, rom me-5 prizis mogebis xdomiloba 1,000,000 SesaZlo SemTxvevidan 999,845 Sedegia, magram imave 1,000,000 mxolod 20 SemTxvevaSia mosalodneli me-2 prizis mogeba. ufro zogadad, Tu populaciaSi xdomilobis ganxorcieleba ar aris damokidebuli sxva xdomilobebis ganxircielebaze, maSin xdomilobis ganxorcielebis albaToba (p) ganisazRvreba, rogorc

prizi

saCuqrebis raodenoba (1,000,000 monawile)

mogebis albaToba

123456

1203466

999,84534

.000001

.000020

.000034

.000066

.999845

.000034

17


simbolo p gamoiyeneba xdomilobis ganxorcielebis albaTobis aRsaniSnad. am formulis gamoyenebiT warmoebuli gamoTvlis Sedegi, rogorc wesi, aTwiladiT gamoixateba. magaliTad,

p(prizi 4) = .000066

miuxedavad amisa, albaTobebi procentebiT an wiliTac SeiZleba gamoisaxos. es formula gansazRvravs diskretuli xdomilebebis Sedegebs, anu iseTi xdomilebebis albaTobas,

romelTac aqvT mniSvnelobaTa sasruli raodenoba, rogoricaa, magaliTad, milion monawileze prizis saxiT gacemul manqanaTa raodenoba. amrigad, TiToeuli prizis miRebis albaTobebi qmnian diskretul albaTur ganawilebas.

diskretuli albaTuri ganawilebis kidev erTi magaliTis saxiT cxrilSi N3.3 mocemuli gamocdasTan dakavSirebuli SfoTvis 100 qula ganvixiloT. davuSvaT, gvainteresebs, ramdenia albaToba imisa, rom am qulebidan SemTxveviT arCeuli erTi qula moTavsebuli iqneba 45-sa da 49-s Soris SualedSi. es sakiTxi 45-sa da 49-s Soris qulis moxvedris xdomilobis albaTobis gansazRvras gulisxmobs. amrigad, am xdomilobaSi moiazreba is Sedegebi, rodesac aRebuli qula iqneba 45, 46, 47, 48 da 49. cxomilobis am albaTobis gamosaTvlelad Cven unda vipovoT am qulebTan dakavSirebuli Sedegebis saerTo raodenoba, risi gakeTebac am qulebis cxrilSi N3.4 mocemuli ganawilebis mixedviT SegviZlia. cxrili gviCvenebs, rom 45-49 intervalSi 15 qulaa. vinaidan cnobilia, rom mocemul ganawilebaSi sul 100 qulaa, albaToba imisa, rom SemTxveviT aRebuli qula moTavsebuli iqneba 45-sa da 49-s Soris, aris

aseve, SeiZleba vTqvaT, rom es albaToba:

sadac _ niSnavs „naklebia an tolia“. simbolos win mdgomi ricxvi naklebia an tolia ricxvis, romelic mosdevs am simbolos.

X _ SemTxveviT arCeuli qula.mocemuli gamosaxuleba ase ikiTxeba: „albaToba imisa, rom X tolia an metia 45-ze

(anu 45X) da tolia an naklebia 49-ze (anu X49) aris .15.“ es albaToba pirdapir cxrilSi N3.4 SeiZleba moviZioT am qulebisTvis v svetSi mocemuli fardobiTi sixSireebis gamoyenebiT, romelic gviCvenebs, rom 45-sa da 49-s Soris moTavsebuli qulebis fardobiTi sixSire .15-ia. yuradReba miaqcieT, rom es albaToba gamocdasTan dakavSirebuli SfoTvis kiTxvarSi miRebuli qulebis fardobiTi sixSiris histogramidan da fardobiTi sixSiris poligonidanac (Sesabamisad, suraTi 3.5 da 3.6) SeiZleba vipovoT.

Sedegi (outcome) _ albaTobis ganawilebaSi TiToeuli SesaZlo varianti.

xdomiloba (event) _ albaTobis ganawilebaSi Sedegebis konkretuli simravlis ganxorcieleba.

xdomilobis ganxorcielebis albaToba (probability of occurrence of en event – p) _

diskretuli Sedegebi (discrete outcomes) _ mniSvnelobebaTa sasruli raodenobis mqone Sedegebi ganawilebaSi.

18


diskretuli albaTuri ganawilebis Tvisebebi

albaTobebis ganawilebas aqvs Tvisebebi, romelTac Tqven, albaT, ukve icnobT. pirveli Tviseba isaa, rom xdomilebis albaToba meryeobs 0.00-dan +1.00-mde. xdomileba, romlis albaToba udris nuls, mosalodnelia, rom namdvilad ar ganxorcieldeba da xdomileba, romlis albaToba udris erTs, mosalodnelia, rom namdvilad ganxorcieldeba. suraTze N6.9 warmodgenilia albaTobis skala da individis cxovrebaSi zogierTi xdomilebis ganxorcielebis albaTobebi. zogierTi xdomileba sruliad mosalodnelia cxovrebaSi da misi albaToba 1.00-is tolia, zogierTi ki _ sruliad SeuZlebeli da maTi albaToba 0.00-is tolia. albaTobis diapazoni 0.00-dan 1.00-mde Seesabameba fardobiTi sixSireebis ganawilebis Tvisebas _ dagrovili fardobiTi sixSireebi (crƒ) icvleba 0-dan 1.0-mde diapazonSi (ix. cxrili N3.4, sveti (i)).

meore Tviseba isaa, rom albaTur ganawilebaSi Sedegebis saerTo raodenobasTan dakavSirebuli calkeuli albaTobebis jami 1.00-s Seadgens. albaTuri ganawilebis es maxasiaTebeli iolad SeiZleba davinaxoT cxrilSi N3.4: _ gamocdasTan dakavSirebuli SfoTvis kiTxvaris qulebis dagrovili fardobiTi sixSireebi 1.00-s Seadgenen (ix. i sveti). Tu Tqven Lazy Acres Vacation Resort-s ewveviT, prizis mogebis labaToba 1.00 iqneba.

19


suraTi 6.9. albaTobebis cxrili da zogierTi xdomilebis ganxorcielebis albaToba.

albaToba

1.0

.9

.8

.7

.6

.5

.4

.3

.2

.1

0

albaToba imisa, rom Tqven mokvdebiT.

.86 albaToba imisa, rom qali cocxali iqneba 60 wlis asakSi (yoveli 100 dabadebuli qalidan 86 cocxalia 60 wlis asakSi, p=86/100=.86).1

.81 albaToba imisa, rom mamakaci cocxali iqneba 60 wlis asakSi (yoveli 100 dabadebuli mamakacidan 81 cocxalia 60 wlis asakSi, p=81/100=.81).1

.64 albaToba imisa, rom aSS-is SemTxveviT SearCeul zrdasrul moqalaqes eqneba, sul mcire, erTi sakredito baraTi.2

albaToba imisa, rom agdebisas 50-TeTriani borjRalze dajdeba (p=1/2=.5)

.077 banqos dastidan valetis amoRebis albaToba (p=4/52=.077).

.000000005 totalizatorSi mogebis miaxloebiTi albaToba.3

1 1980 Commissioners Standard Ordinary (CSO) Mortality Table (The Wizards, 1989).2 The Way, 1985.3 Artis, 1993

20


mesame Tviseba isaa, rom urTierTgamomricxavi Sedegebis momcveli xdomilobis albaToba SeiZleba am

xdomilobis Semadgeneli Sedegebis albaTobebis SekrebiT vipovoT. urTierTgamomricxavi Sedegebi is Sedegebia, romlebic ver ganxorcieldeba erTdroulad. zemoT ganxilul magaliTSi individis mier me-2 da me-5 prizebis mogeba urTierTgamomricxavi xdomilobebia. mas SeuZlia moigos erTi an meore prizi, magram ara _ orive erTdroulad. analogiurad, gamocdasTan dakavSirebuli SfoTvis kiTxvarSi individi erTdroulad ver miiRebs 53 da 67 qulas. es Sedegebi urTierTgamomricxavia. magaliTad, davuSvaT, rom gvainteresebs, ramdenia albaToba imisa, rom gamocdasTan dakavSirebuli SfoTvis kiTxvaris qulebis ganawilebidan SemTxveviT aRebuli erTi qula 45 an masze meti iqneba? am kiTxvaze pasuxis gasacemad cxrilis N3.4 e sveti unda vnaxoT, sadac martiv sixSireTa ganawilebaa mocemuli. yuradReba miaqcieT, rom am svetSi 45-is toli an masze meti 28 qulaa. es ricxvi e svetSi mocemuli, 45-is toli an masze meti qulebis sixSireebis dajamebiTaa miRebuli (anu 15+5+3+2+1+2=28). amrigad, 45-is toli an masze meti qulis SemTxveviT arCevis albaTobaa 28/100 anu .28. albaTobis tolobis gamoyenebiT:

es albaToba N3.4 cxrilidanac SeiZleba miviRoT: _ 45-is toli anda masze meti qulebisTvis v svetSi mocemuli fardobiTi sixSireebis SekrebiT (anu .15+.05+.03+.02+.01+.02=.28).

es magaliTi gviCvenebs, rom Tu ori an meti Sedegi urTierTgamomricxavia, maSin erT-erTi SedegiT xdomilobis ganxorcielebis albaToba TiToeuli Sedegis albaTobebis dajamebiT gamoiTvleba. albaTobis terminebiT Tu vityviT, Tu A da B albaTuri ganawilebidan aRebuli ori urTierTgamomricxavi Sedegia, maSin erTis an meoris ganxorcielebis albaToba A-s ganxorcielebis albaTobisa da B-s ganxorcielebis albaTobis jamis tolia, anu:

es damokidebuleba cnobilia, rogorc aditiurobis wesi (anu damatebis, Sevsebis) urTierTgamomricxavi Sedegebis albaTobisTvis.

am formulis gamoyenebis kidev erTi magaliTi: _ davuSvaT, isev SemTxveviTad virCevT gamocdasTan dakavSirebuli SfoTvis kiTxvarSi miRebul erT qulas da gvainteresebs, ramdenia albaToba imisa, rom moTavsebuli iqneba 20-sa da 24-s, an 70-sa da 74-s Soris. cxrilSi N3.4 mocemuli sixSireTa ganawilebis gamoyenebiT vnaxavT, rom 20-sa da 24-s Soris, an 70-sa da 74-s Soris SemTxveviT qulis arCevis albaToba 20-24 intervalSi moTavsebuli qulebis fardobiTi sixSirisa (anu .01-is) da 70-74 intervalSi moTavsebuli qulebis fardobiTi sixSiris (anu .02-is) jamia anu sruli albaToba .03-s Seadgens. albaTobis gamosaxulebiT:

urTierTgamomricxavi Sedegebi (Mutually exclusive outcomes) _ Sedegebi, romelic ver ganxorcieldeba erTdroulad.

21


aditiurobis es wesi Zalze mniSvnelovani gaxdeba SemdgomSi _ am Tavis bolosa da Semdeg TavebSi, sadac hipoTezebis Semowmebaze iqneba saubari.

Teoriuli (klasikuri) da empiriuli albaTobaxdomilebis albaTobis gansazRvris ori gza arsebobs: _ Teoriuli da empiriuli.

zogierTi xdomilobisTvis Cven SegviZlia davaskvnaT misi xelSemwyob SemTxvevaTa albaTobis Sesaxeb. magaliTad, 50-TeTriani monetis agdebisas borjRalis gamosvlis Sesaxeb SeiZleba msjelobiT davaskvnaT. 50-TeTriani monetis agdebisas ori SesaZlo Sedegi gvaqvs: _ borjRalis an safasuris gamoCena. Tu moneta sworia, TiToeuli maTgani erTnairad mosalodnelia. Sesabamisad, borjRalis mosvliT dasrulebuli erTi agdebis albaToba, ori SesaZlo Sedegidan, erT-erTia, anu p(borjRalis mosvla) = 1/2 = .5. Teoriuli albaTobis modelebi xSirad am magaliTze gacilebiT rTulia. Semdeg qveTavSi ganvixilavT standartul normalur ganawilebas, rogorc albaTur Teoriul ganawilebas.

albaTuri Teoriuli ganawileba (Theoretical probability distribution) _ Teoriuli albaTobis modelis gamoyenebiT miRebuli albaTuri ganawileba.

albaTuri klasikuri ganawileba (Empirical probability distribution) _ xdomilobis realurad ganxorcielebis sixSiris daTvliT miRebuli albaTuri ganawileba.

zogierTi xdomilebis SemTxvevaSi Teoriulad ver vimsjelebT misi ganxorcielebis albaTobis Sesaxeb; magaliTad, Teoriuli msjelobiT ver davaskvniT, Tu ras udris albaToba imisa, rom Tu qalbatoni xarT, Tqven cocxali iqnebiT 60 wlis asakSi. am albaTobis gamosaTvlelad gamoviyenebT xdomilobis ganxorcielebis xangrZlivi SeswavliT miRebul empiriul albaTobebs. magaliTad, sicocxlis dazRvevis industriaSi aSS-is moqalaqeTa sicocxlis xangrZliobis sixSiris gansasazRvrad specialuri komisiis mier SemuSavebul bunebrivi sikvdilianobis standartul cxrilebs iyeneben (The Wizards,

1989). es cxrili empiriulad aris Sedgenili xangrZlivi drois manZilze sxvadasxva asakis mamakacebisa da qalebis sikvdilianobis sixSiris Canawerebis warmoebis Sedegad. cxrili gviCvenebs, rom yoveli 10,000,000 cocxali qalidan 8,603,801 jer kidev cocxalia 60 wlis asakSi. amrigad, dabadebisas albaToba imisa, rom aSS-is moqalaqe qali jer kidev cocxali iqneba 60 wlis asakSi, iqneba

p(60 wlis asakis cocxali qali) .

cxadia, p-is es mniSvneloba Seicvleba, Tu Seicvleba sikvdilianobis sixSire janmrTelobis dacvisa da kvebis gaumjobesebis Sedegad.

bihevioristul mecnierebaSi aris orive (Teoriuli da empiriuli) albaTobis gamoyenebis SemTxvevebi, Tumca, winamdebare saxlmZRvaneloSi aqcenti keTdeba iseT Teoriul

22


ganawilebaze, rogoricaa standartuli normaluri ganawileba.Semdgom TavebSi statistikuri daskvnis iseT problemebs SevexebiT rogoricaa

populaciidan aRebuli SerCevidan miRebul sazomebze dayrdnobiT am populaciis Sesaxeb daskvnebis gakeTeba; Tumca, aseTi daskvnebis gamotanisas verasodes viqnebiT absoluturad darwmunebulni maT sisworeSi; amrigad, Cveni daskvnebi albaTobis terminebiT iqneba formulirebuli.

standartuli normaluri ganawileba, rogorc albaTuri ganawileba

normaluri ganawileba fardobiTi sixSireebis Teoriuli ganawilebaa. vinaidan normalur ganawilebaSi warmodgenili fardobiTi sixSireebi populacias efuZneba, normaluri ganawileba albaTobebis ganawilebac aris; Tumca, is odnav gansxvavebuli albaTuri ganawilebaa, vidre axlaxans ganxiluli diskretuli albaTuri ganawileba. normalur ganawilebaSi Semavali X qula uwyveti cvladia, romelmac SeiZleba miiRos mniSvnelobaTa usasrulo raodenoba Tavisi cvalebadobis diapazonSi. magaliTad, davuSvaT, rom gvaqvs sxeulis wonis amsaxveli qulebis Teoriuli normaluri ganawileba. ras udris albaToba imisa, rom vipoviT individs, romelic zustad 123 funts (58 kg-s) iwonis? wona uwyveti cvladia da zogierTi adamiani SeiZleba iwonides 123.3 funts, 123.36 funts, an 123.364 funtsac ki. Teoriulad, Cven SeiZleba iseTi individic vipovoT, romelic 123.3642193 funts iwonis, Tu SevZlebT ase zustad gazomvas. SeiZleba, rom 123.3 funti ganvixiloT, rogorc 123 funti? am TvalsazrisiT, ra SeiZleba iTqvas 123.00001 funtze? principSi, Cven verasodes miviRebT zustad 123 funts, Tu yovelTvis ufro zustad gazomvas vcdiT da yovel jerze vnaxavT, rom gazomvis Sedegi ar aris zustad 123 funts.

Tu am mimarTulebiT vimsjelebT, uwyveti cvladis zusti mniSvnelobis miRebis albaToba nulis toli iqneba, magram garkveul intervalSi monacemis moxvedris albaToba ar aris nulis toli. miuxedavad imisa, rom ar SegviZlia zustad 123 funtis miReba, SegviZlia miviRoT wona 122.5 funtsa da 123.5 funts Soris intervalSi. magaliTad, ganurCevlad imisa, aTwiladSi mZimis Semdeg ramdeni Tanrigis sizustiT SegviZlia gazomva, 123.36 funti (anda 123.3642193 funtic ki) yovelTvis moxvdeba 122.5 funtsa da 123.5 funts Soris intervalSi. amrigad, normaluri ganawilebis saxiT mocemuli albaTobis funqcia uzrunvelyofs ara X-is diskretuli mniSvnelobebis albaTobebs, aramed imis albaTobas, rom X-is mocemuli mniSvneloba moxvdeba konkretul intervalSi. albaToba gamoisaxeba ganawilebis qveS mdebare farTobiT, romelic SemosazRvrulia intervaliT.

normaluri ganawilebis albaTuri ganawilebis saxiT gamoyenebis sailustraciod ganvixiloT Semdegi amocana. davuSvaT, iciT, rom qulaTa erToblioba normalurad aris ganawilebuli da ganawilebis μ = 150 da σ = 8. Tu Tqven ganawilebidan SemTxveviT airCevT qulas, ras udris albaToba imisa, rom aRebuli qula moTavsebuli iqneba 150-sa da 158-s Soris intervalSi? am kiTxvaze pasuxis gasacemad, pirvel rigSi, mocemuli

23


ganawileba unda gardavqmnaT standartul normalur ganawilebad, romlis μ = 0 da σ = 1; amisaTvis viyenebT formulas z = (X_ μ)/ σ. amrigad, 150 qula gaxdeba z = 0, xolo 158 qula _ z = +1.0. Sesabamisad, dasmuli kiTxva gadaiqceva kiTxvad imis Sesaxeb, Tu ramdenia z = 0-sa da z = +1.0-s Soris intervalSi moTavsebuli qulis miRebis albaToba. am kiTxvis pasuxia standartuli normaluri ganawilebis qveS mdebare da z = 0-sa da z = +1.0-s Soris intervaliT SemosazRvruli farTobi anu .3413 (ix. a-1 cxrilis (b) sveti). Sesabamisad, 150-sa da 158-s Soris intervalSi moTavsebuli qulis miRebis albaTobaa .3413.

magaliTi N6.2

standartuli normaluri ganawilebis rogorc albaTuri ganawilebis gamoyeneba

davuSvaT, Tqven gaqvT magaliTSi N6.1 aRwerili normalurad ganawilebuli populacia, romlis μ = 80 da σ = 5. qvemoT mocemuli amocana N6.1 magaliTSi mocemuli amocanis msgavsia, Tumca, aq mogeTxovebaT ara qulaTa wilis, aramed maTi albaTobebis gamoTvla. amocanis amoxsna gagiadvildebaT, Tu magaliTis N6.1 Tanmxleb suraTs mimarTavT.

amocanaras udris albaToba imisa, rom mocemul ganawilebaSi Semavali qula 88-is toli an

masze meti iqneba?

saWiro statistika:

amoxsna88 qula unda gadaviyvanoT zrealuri-Si. Semdeg a-1 cxrilis gamoyenebiT vipoviT

qulaTa wils, romelic tolia an metia zrealuri-ze. es wili warmoadgens saZiebel albaTobas. zrealuri-is formulaSi ricxviTi mniSvnelobebis Setanis Semdeg miviRebT:

.

a-1 cxrilis (g) sveti gviCvenebs, rom standartuli normaluri ganawilebis qveS mdebare farTobi, romelic tolia an metia z = +1.6-ze, aris .0548. amrigad, albaToba imisa, rom qula toli an meti iyos 88-ze, aris .0548 anu, Tu albaTobis saxiT CavwerT, p(X≥88) = .0548.

amocanaras udris albaToba imisa, rom mocemul ganawilebaSi Semavali qula moTavsebuli

iqneba 83-sa da 87-s Soris?

24


amoxsnaorive qula _ 83 da 87 unda gadaviyvanoT zrealuri qulebSi. Semdeg vpoulobT

standartuli normaluri ganawilebis qveS farTobs zrealuri-is or mniSvnelobas Soris (ix. suraTi N6.6). ricxviTi mniSvnelobebis Setanis Semdeg miviRebT:



cxrilis Na-1 (b) svetidan vxedavT, rom zrealuri = +0.6-isTvis z = 0-sa da z = +0.6-s Soris intervalSi moTavsebul qulaTa wilia .2257. zrealuri = +1.4-isTvis z = 0-sa da z = +0.6-s Soris intervalSi moTavsebul qulaTa wilia .4192. ganawilebis farTobi z-is am or mniSvnelobas Soris gvaZlevs 83-sa 87-s Soris moTavsebul qulaTa wils. am farTobis misaRebad .4192-s vaklebT .2257-s da viRebT .1935-s. miRebuli wili warmoadgens saZiebel albaTobas. amrigad, 83-sa da 87-s Soris intervalSi qulis moxvedris albaTobaa .1935; albaTobis terminebiT, p(83≤X≤87)=.1935.

amocanaras udris albaToba imisa, rom mocemul ganawilebaSi Semavali qula moTavsebuli

iqneba 65-sa da 74-s Soris?

amoxsnaorive qula _ 65 da 74 unda gadaviyvanoT zrealuri qulebSi. Semdeg vpoulobT

standartuli normaluri ganawilebis farTobs zrealuri-is or mniSvnelobas Soris (ix. suraTi N6.7). ricxviTi mniSvnelobebis Setanis Semdeg miviRebT:



Tu vnaxavT a-1 cxrilis (b) svets da gavixsenebT, rom standartuli normaluri ganawilebis simetriulobidan gamomdinare, erTnairi mimarTebaa z-is rogorc dadebiTi, ise _ uaryofiTi qulebis SemTxvevaSi, vnaxavT, rom z = 0-sa da z = _3.0-s Soris SualedSi moTavsebulia qulaTa .4987 nawili. zrealuri = _1.2-isTvis ki qulaTa .3849 nawili moTavsebulia z = 0-sa da z = _1.2-s Soris SualedSi. Sesabamisad, Cven miviRebT z = _3.0-sa da z = _1.2-s Soris moTavsebul qulaTa wils, Tu .4987-s gamovaklebT .3849-s, rac .1138-is tolia. es wili warmoadgens saZiebel albaTobas; aqedan gamomdinare, p(65≤X≤74)=.1138.


1. ganmarteT: _ diskretuli xdomileba, Sedegi, empiriuli albaToba, xdomiloba, urTierTgamomricxavi Sedegebi, xdomilobis albaToba, Teoriuli albaToba.

25


2. romeli formuliT gamoiTvleba diskretuli xdomilobis ganxorcielebis albaToba?3. velosipedis mosagebad SeiZineT latareis bileTi. bileTidan SeityveT, rom yovel

700 gayidul bileTze gaTamaSdeba mxolod erTi velosipedi. ras udris albaToba imisa, rom Tqven pirvelive gaTamaSebisas moigoT velosipedi?

4. X-is, μ-sa da σ-s qvemoT mocemuli TiToeuli mniSvnelobisTvis gamoTvaleT z-is mniSvneloba. Semdeg a-1 cxrilis gamoyenebiT moZebneT albaToba imisa,

rom qula toli an meti iqneba mocemul qulaze.5. Tqven CaatareT farTomasStabiani kvleva

studentebSi da ikvlevdiT verbaluri analogiebiT msjelobis unars. vTqvaT, testis qulebi normalurad aris ganawilebuli, ganawilebis μ = 72 da σ = 8.a) ras udris albaToba imisa, rom

ganawilebaSi Semavali qula 84-is toli an masze meti iqneba?

b) ras udris albaToba imisa, rom ganawilebaSi Semavali qula moTavsebuli iqneba 78-sa da 86-s Soris?

g) ras udris albaToba imisa, rom ganawilebaSi Semavali qula moTavsebuli iqneba 64-sa da 870-s Soris?

standartuli qulebi

Cven vnaxeT, rom SegviZlia normalurad ganawilebuli nedli qulis aReba da misi gadayvana standartuli normaluri ganawilebis qulad Semdegi formulis gamoyenebiT:

.

z aris standartuli qula, romlis gamoyeneba SesaZlebelia ori sxvadasxva normaluri ganawilebidan aRebuli qulebis Sesadareblad. magaliTad, vTqvaT, individma erT testSi miiRo 75 qula, romlis μ = 70 da σ = 5, xolo meore testSi, romlis μ = 100 da σ = 10, miiRo 110 qula. davuSvaT, rom orive testSi miRebuli qulebi normalurad aris ganawilebuli. TiToeuli qulisTvis (75 da 110) zrealuri aris +1. orive testSi individma miiRo saSualoze erTi standartuli gadaxriT meti qula. Sesabamisad, misi mdebareoba sxvebis mier miRebul qulebTan mimarTebaSi erTnairi iqneba orive testis SemTxvevaSi: _ man miiRo saSualoze erTi standartuli gadaxriT meti qula.

SerCevidan miRebuli qulebis msgavsi gardaqmnis ganxorcieleba maSinac gamosadegia, rodesac qulebi ar aris normalurad ganawilebuli, rac Semdegi formulis gamoyenebiT aris SesaZlebeli:

μ σ X

a.

73 4.2 81

b.

127 11.6 121

g.

17 1.3 18

d.

50 3.0 39

e.

5 0.4 5.7

v.

256 35.0 212

26


,

sadac X = saZiebeli qula.= qulaTa SerCevis saSualo.

S = SerCevis standartuli gadaxra, romelic, Tavis mxriv,

.

am formuliT gamoTvlil z-is mniSvnelobas xSirad standartul qulas uwodeben. es formula qulas gardaqmnis

ricxvad, romelic gviCvenebs, Tu ramdeni standartuli gadaxriT aris daSorebuli mocemuli qula SerCevis saSualodan. amis sailustraciod, davuSvaT, rom xuT students miveciT ori _ statistikis da istoriis _ testi da miviReT N6.4 cxrilSi mocemuli nedli da z

qulebi. standartuli qulebi miRebul iqna zemoT mocemuli formulis gamoyenebiT. magaliTad, anas mier statistikaSi miRebuli nedli qulaa 91. testis saSualo iyo 72.0, xolo S _ 14.2. amitom anas z qula iqneba

rodesac viciT standartuli qula, warmodgena gvaqvs klasis saSualosTan anas qulis mimarTebaze. es aris 1.34 standartuli gadaxra saSualos zemoT.

standartuli qulebis gamoyenebiT Cven martivad SevadarebT erTmaneTs or sxvadasxva testSi miRebul qulebs, risi gakeTebac SeuZlebelia nedli qulebis SemTxvevaSi. magaliTad, joni statistikis testis mixedviT,

moTavsebulia saSualodan 1.06 standartuli gadaxriT zemoT, xolo istoriis testis mixedviT _ 0.50 standartuli gadaxriT qvemoT SerCevis saSualodan. dadebiTi standartuli qulebi gviCvenebs SerCevis saSualos zemoT mdebare qulebs, nulis toli standartuli qulis (rogoricaa daviTis mier istoriaSi miRebuli qula) SemTxvevaSi qula SerCevis saSualos tolia, xolo uaryofiTi standartuli qulebi migviTiTebs SerCevis saSualos qvemoT mdebare qulebze; Tumca, yuradReba unda miaqcioT, rom vinaidan es z SerCevis qulebidan aris miRebuli, mas ar eqneba igive Tvisebebi normaluri

standartuli qula (Standard score): _ qula, romelic miiReba formuliT .

cxrili 6.4. xuTi studentis mier statistikisa da istoriis testebSi miRebuli nedli qulebi da standartuli qulebi.

testistatistika istoria

pirovneba

nedli qula z qulanedli qula

z qula

amaliadenisijefihansimaTTi

9162875664

+1.34-0.70+1.06-1.13-0.56

9379739060

+1.180.00

+0.50+0.92-1.60

27


ganawilebis TvalsazrisiT, rac z = (X_μ)/σ formuliT miRebul z-s axasiaTebs. miuxedavad amisa, standartuli qulebi xSirad gamoiyeneba testis Sedegebis warmodgenisas, rodesac aRiwereba individis fardobiTi mdgomareoba im sxva individebTan mimarTebaSi, vinc igive testi gaiara. unda aRiniSnos, rom standartul qulebSi gadayvana ar cvlis qulaTa ganawilebis formas. Tu nedli qulebis ganawileba asimetriulia, maSin standartuli qulebis ganawilebac asimetriuli iqneba.

Sejameba

normaluri ganawileba fardobiTi sixSireebis Teoriuli ganawilebaa. gazomvis Sedegad miRebul qulaTa arc erTi erToblioba ar aris zustad normalurad ganawilebuli, Tumca, xSirad gamosadegia imis daSveba, rom qcevis gazomvis zogierTi Sedegi daaxloebiT normalurad aris ganawilebuli.

normaluri ganawilebidan aRebuli qula SeiZleba standartuli normaluri ganawilebis, romlis µ=0 da σ=1, qulad gardaiqmnas. amisaTvis gamoiyeneba formula

z = (X_μ)/σ.

diskretuli xdomilebis albaToba gamoiTvleba formuliT

nebismier albaTuri ganawilebisTvis Sedegis albaToba .00-sa da 1.00-s Soris unda iyos moTavsebuli.

calkeuli albaTobebis jami, romlebic albaTur ganawilebaSi Sedegebis saerTo raodenobas ukavSirdeba, 1.00-s Seadgens.

Tu A da B urTierTgamomricxavi Sedegebia, maSin standartuli normaluri ganawileba albaTuri ganawilebaa, romelic gviCvenebs

ganawilebis garkveul intervalSi qulebis moxvedris albaTobebs. misi gamoyeneba SesaZlebelia normalurad ganawilebuli qulebis SemTxvevaSi rogorc konkretul intervalSi moxvedrili qulebis wilis, ise _ intervalebSi maTi moxvedris albaTobebis gansasazRvrad.

SerCevidan miRebuli nedli qulebis standartul qulebSi gadayvana SesaZlebelia formuliT

.

es formula qulas gardaqmnis ricxvad, romelic gviCvenebs, Tu ramdeni standartuli gadaxriT aris is daSorebuli saSualosgan. am dros ar icvleba nedli qulebis ganawilebis forma.

ZiriTadi terminebi da simboloebi

28


asimptoturi ganawilebaalbaToba diskretuli xdomilobadiskretuli xdomilobis empiriuli

albaTobaempiriuli albaTuri ganawilebaTeoriuli albaTobaTeoriuli albaTuri ganawilebanormaluri ganawilebaparametri

populaciasimetriuli ganawilebastandartuli normaluri gadaxrastandartuli normaluri ganawilebastandartuli qulastatistikaurTierTgamomricxavi Sedegebiuwyveti ganawilebaSedegebiSerCeva

xdomilobaxdomilobis

ganxorcielebis albaToba

μ

σ

s

z

zrealuri

kiTxvebi

1. inteleqtis testebSi xSirad Sedis ricxvebis Tanmimdevrobis dalagebis, instruqciebis Sesrulebis, leqsikonis, analogiebis danaxvis, suraTis dasrulebis, logikuri msjelobisa da maTematikuri gamoTvlebis warmoebis unarebis Sefaseba. axseniT, ratom aris mosalodneli, rom inteleqtis testis qulebi populaciaSi daaxloebiT normalurad iqneba ganawilebuli.

2. reaqciis (anu gamRizianeblis miwodebasa da masze reaqciis ganxorcielebas Soris) drois gazomvis Sedegebi, rogorc wesi, ar aris normalurad ganawilebuli. metic, miRebul qulaTa ganawilebas xSirad Sebrunebuli J-is forma aqvs, sadac qulaTa umravlesoba SedarebiT mcirea, umciresoba ki _ didi. axseniT, ratom aris mosalodneli, rom qulebi swored ase iqneba ganawilebuli.

3. wlebis manZilze dagrovda zrdasrulTa veqsleris inteleqtis skalis monacemebi. davuSvaT, miRebuli qulebi normalurad aris ganawilebuli da ganawilebis μ = 100 da σ = 15.

a) inteleqtis testis qulaTa ra nawili iqneba 115-is toli an masze meti?b) inteleqtis testis qulaTa ra wili iqneba 70-ze naklebi?g) inteleqtis testis qulaTa ra wili iqneba 60-ze naklebi?d) inteleqtis testis qulaTa ra wili iqneba moTavsebuli 100-sa da 120-s

Soris?e) inteleqtis testis qulaTa ra wili iqneba moTavsebuli 75-sa da 100-s

Soris?v) romel intervalSi iqneba moTavsebuli testis qulaTa Sua 80%?z) Tu SemTxveviT airCevT cdis pirs, ramdeni iqneba albaToba imisa,

rom misi inteleqtis qula moTavsebuli iyos 90-sa da 120-s Soris?

29


T) Tu SemTxveviT airCevT cdis pirs, ramdeni iqneba albaToba imisa, rom misi inteleqtis qula 80-is toli an masze naklebi iyos?

i) Tu SemTxveviT airCevT cdis pirs, ramdeni iqneba albaToba imisa, rom misi inteleqtis qula 125-is toli an masze meti iyos?

k) Tu SemTxveviT airCevT cdis pirs, ramdeni iqneba albaToba imisa, rom misi inteleqtis qula 140-is toli an masze meti iyos?

4. Tqven sxva 64 adamianTan erTad gamogiZaxes sasamarTloSi, rogorc nafici msajulobis erT-erTi kandidati. nafic msajulTa SesarCevad furclebze dawerili saxelebi yuTSi Cayares da saTiTaod amoiRes.

a) ras udris albaToba imisa, rom pirvelive jerze Tqveni saxeli amoiRon?b) Tu Tqveni saxeli ar aRmoCnda pirvel rva amoRebul furcelze, ras

udris albaToba imisa, rom mecxre amoRebul furcelze Tqveni saxeli eweros, Tu gaviTvaliswinebT, rom pirveli rva amoRebuli furceli aRar daabrunes yuTSi?

5. nacionalurma gazeTma USA Today-m gamoaqveyna statia saTauriT „Sansebi“ (31 dekemberi, 1985), sadac mocemuli iyo Semdegi wlis manZilze individis cxovrebaSi garkveuli xdomilebebis ganxorcielebis Sansebi. qvemoT CamoTvlili xdomilebebisTvis gamoTvaleT TiToeuli maTganis ganxorcielebis albaToba.

a) doRis dros mravali firma iRebs fsonebs. erT-erTi garbenze yovel 3000 ganacxadze erTi prizi gaicemoda. ras udrida albaToba imisa, rom am garbenze nebismieri erTi monawile moigebda prizs?

b) bolo wlebi mZime gamodga fermerebisTvis. weliwadSi yoveli 5314 fermeridan erTi tovebda Tavis bizness. ras udris albaToba imisa, rom mimdinare wels fermerma miatovos biznesi?

g) mraval Cvengans uocnebia popularuli romanis daweraze, Tumca, gamomcemlobaSi erTi wlis manZilze Sesuli yoveli 4637 wignis moculobis mqone xelnaweridan mxolod erTi ibeWdeba. Tqven dawereT romani. ras udris albaToba imisa, rom is daibeWdeba mimdinare wels?

6. 6-1 magaliTSi Cven ganvixileT iarnoldisa da sxvebis (Yarnold et al., 1986) kvleva, A da B tipis individebis Sedarebis Sesaxeb. A da B tipis individebis misaRebad iarnoldma da misma kolegebma cdis pirebi jenkinsis aqtiurobis kiTxvaris qulebis mixedviT SearCies. davuSvaT, rom jenkinsis aqtiurobis kiTxvarSi populaciis saSualoa 0 da populaciis standartuli gadaxra _ 10. vTqvaT, Tqven dainteresdiT da ganagrZeT iarnoldis kvleva. A tipis cdis pirebidan Tqven gsurT iseTi cdis pirebis SerCeva, romelTac jenkinsis aqtiurobis kiTxvarSi miRebuli aqvT +13-ze meti qula, xolo B tipis

30


cdis pirebidan _ visac miRebuli aqvT -13-ze naklebi qula. davuSvaT, jenkinsis aqtiurobis kiTxvaris qulebi normalurad aris ganawilebuli.

a) ras udris albaToba imisa, rom SemTxveviT arCeul individs jenkinsis aqtiurobis kiTxvarSi eqneba +13-ze meti qula?

b) ras udris albaToba imisa, rom SemTxveviT arCeul individs jenkinsis aqtiurobis kiTxvarSi eqneba -13-ze naklebi qula?

g) ras udris albaToba imisa, rom SemTxveviT arCeul individs jenkinsis aqtiurobis kiTxvarSi eqneba -13-sa da+13-s Soris intervalSi moTavsebuli qula?

7. davuSvaT, amerikeli 21-30 wlis mamakacebis saSualo wonaa 156 funti (70.36 kg), standartuli gadaxra ki _ 8 funti (4.63 kg).

a) 21-30 wlis mamakacTa ra nawili iwonis 144 funts (65 kg) an ufro naklebs?

b) 21-30 wlis mamakacTa ra nawili iwonis 170 funts (77 kg) an ufro mets?

g) 21-30 wlis mamakacTa ra nawilis wona iqneba 140-dan (63.5 kg) da 172 funtamde (78 kg)?

d) Tu SemTxveviT airCevT 21-30 wlis mamakacs, ras udris albaToba imisa, rom misi wona iyos 144-dan (65 kg) 168 funtamde (76 kg)?

e) Tu SemTxveviT airCevT 21-30 wlis mamakacs, ras udris albaToba imisa, rom misi wona 150 funti (68 kg) an ufro naklebi iyos?

v) Tu SemTxveviT airCevT 21-30 wlis mamakacs, ras udris albaToba imisa, rom misi wona 160 funti (72.6 kg) an ufro meti iyos?

8. davuSvaT, ikvlieT eqvsi studenti; gainteresebdaT kviraSi video TamaSebze saSualod daxarjuli dro (saaTebSi) da semestruli niSnebis saSualo. vTqvaT, miiReT Semdegi monacemebi:

a) gamoTvaleT saSualo da SerCevis standartuli gadaxra qulaTa TiToeuli jgufisTvis da Semdeg TiToeuli miRebuli qula gadaiyvaneT standartul qulaSi.

b) lelas video TamaSebisTvis daxarjuli drois standartuli qulaa nuli. ras geubnebaT es qula yoveli kviris manZilze video TamaSebisTvis daxarjuli drois Sesaxeb?

g) ra informacias gawvdiT ias semestruli niSnebis saSualos standartuli qula misi semestruli niSnebis saSualos Sesaxeb?

studenti

video TamaSebz

e daxarjuli

dro

semestruli niSnebis saSualo

aleqsiloreniivonalavaliboniiasoni

680

153

16

2.52.03.93.43.11.3

31


d) ras gviCvenebs semestruli niSnebis saSualos uaryofiTi standartuli qula?

e) ras gviCvenebs video TamaSebze daxarjuli drois dadebiTi standartuli qula?

9. cota xnis win literaturasa da istoriaSi gamocda CaabareT. Tqveni niSnebi da klasis saSualo da standartuli gadaxra (S) aseTia:

literatura istoriaTqveni qulaklasis saSualo klasis S

88836

70655

TanaklaselebTan SedarebiT romel gamocdaSi gaqvT ukeTesi Sedegi? ratom?

10.cxrilSi warmodgenilia aSS-is Census Bureau-is (marti, 1998) monacemebi, romlis mixedviTac, ojaxSi wevrebis raodenobis fardobiTi sixSireebi Semdegnairad nawildeba:

a)Tu SemTxveviT avarCevT erT ojaxs, ramdenia albaToba imisa, rom masSi zustad oTxi wevri iqneba?

b) Tu SemTxveviT avarCevT erT ojaxs, ramdenia albaToba imisa, rom masSi eqvsi

an meti wevri iqneba?

11.qvemoT mocemulia aSS-Si 1998 wel amerikeli qmrebis asakis sixSireTa ganawileba (U.S. Census Bureau, 1998, marti).

a)Tu SemTxveviT avarCevT erT ojaxs, ramdenia albaToba imisa, rom qmari 44 wlis an ufro axalgazrda iqneba?

b)Tu SemTxveviT avarCevT erT ojaxs, ramdenia albaToba imisa, rom qmari 34 wlis an ufri axalgazrda, anda 75 wlis da ufro xnovani iqneba?

asakifardobiTi sixSire

24 an naklebi25-3435-4445-5455-5960-6162-6465-6970-7475-7980-84

85 an meti

.023

.178

.257

.217

.079

.029

.040

.062

.049

.039

.019

.008

ojaxis wevrTa raodenoba

fardobiTi sixSire

23456

7 da meti

.427

.230

.206

.092

.029

.016

32

Documents

Tavi 1 - TSU Learn · Web viewmocemul formulaSi ricxviTi mniSvnelobebis Setanis Semdeg X-is mniSvnelobisgan miRebuli z-is mniSvnelobas z realurs vuwodebT da zrealuri-iT aRvniSnavT