X... ne ij =, i =1,p, j = 1,q T 2. Se calculează statistica testului: Se calculează valoarea critică a testului:
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1 Descrerea ue varable calave Prcpal dcaor care su calcula peru varablele calave su: - frecveţa absoluă care repreză uărul de dvz la care se regsrează o auă odalae - frecveţa relavă care repreza frecveţa absoluă raporaă la uărul de dvz. Legăura dre două varable calave. Tesul de depedeță χ Fe două dsrbuț = { T} ș Y = {y y y T}. Varabla are p odalăț ar varabla Y q odalăț. Frecvețele îcrucșae su eorae î abelul N: q q N. U elee oarecare repreză uărul de sațe la care p p pq varabla are odalaea ar varabla Y are odalaea. Frecvețele cuulae pe l ș coloaă su defe asfel: ; q p - uărul de sațe la care se îâleșe dalaea peru varabla p q - uărul de sațe la care se îâleșe dalaea peru varabla Y. Tesul χ ese ulza peru a sabl dacă esă o legăură îre cele două varable calave (oale). Ipoeza ulă specfcă fapul că u esă o relaţe îre cele două varable adcă: H0: Cele două varable su depedee H: Cele două varable su depedee Paș aplcăr esulu su:. Se calculează frecveţele ed esae: e = =p = q T. Se calculează sasca esulu: Calcula p q e e Crc ( ; r) 3. Se calculează valoarea crcă a esulu: ude α ese pragul de sefcaţe al esulu ar r ese uărul gradelor de lberae î aces caz r = (p-) (q-). 4. Decza asupra accepăr sau respger poeze H0 se a asfel: dacă Calcula Crc ( ; r ) auc se respge poeza ulă cu u vel de credere - α dec cele două varable se fluețează recproc. Î cele a ule suaț se ulzează frecvețe relave. Acesea se deeră pr raporarea frecvețelor absolue la uărul de sațe. Frecvețele relave se eorează f f f îr-u abel f f f F f p f p f Sasca esulu se va calcula asfel: p q f f f Calcula T f f ude f f f. T T T Descrerea dvzlor q q pq. U dvd ese descrs pr ulţea de valor luae de u grup de varable peru dvdul respecv. Se oează cu arcea valorlor luae de sațe (dvz) peru varable: Fecăru dvd se asocază o podere dvzlor l se asocază aceeaș podere auc p =. Sua poderlor ese. Dacă p. Mulţea pucelor dae de ce dvz care forează colecvaea sudaă corespude uu or de puce -desoale ( fd uărul varablelor).
2 Se poae def barcerul pucelor sau cerul de greuae al orulu vecorul edlor arece ale celor varable care descru colecvaea: p g pw p p ude w ese vecorul coloaă reprezeâd valorle luae de cele varable peru dvdul. Vo u erţe oală a orulu de puce eda poderaă a păraelor dsaţelor pucelor faţă de cerul de greuae: I g p ( w vecorulu coloaă w -g. g) M ( w g) ude (w -g) ese u vecor le raspusul Ierţa orulu î rapor cu u puc oarecare d spaţu ese: I h p ( w h) M ( w h) Relaţa dre cele doua valor ale erţe ese: I I ( g h) M( g h). (Relaţa lu Huyges) h g Ierţa î rapor cu cerul de greuae ese ă. Tabele de dae. U abel de dae ese o arce care se cosrueşe d asablul de dvz ş varable. - Tabelul de observaț. U asfel de abel se obţe auc câd eleeele arce su valor uerce oarecare. Pe l su aşezaţ dvz ar pe coloae su aşezae varablele urăre. Fe uărul de sațe ş uărul de varable. Tabelul de observaț ese de obce oa: = - Tabelul de cogeţă. Peru dae calave. ş Y două varable calave f f fq Z = f f fq cu frecveţe relave ude p ese uărul de f p f p f pq odalăț al varable ş q ese uărul de odalăţ al varable Y. O ala foră a abelulu de cogeţă ese urăoarea: Modalae a a varable Modalae a a varable Y f Modalae a a varable Modalae a a varable Y f Z = Modalae a p a varable Modalae a a varable Y f p. Modalae a a varable Modalae a a varable Y f Modalae a p a varable Modalae a q a varable Y f pq Aces abel are pq l ş 3 coloae. Prele două coloae cuprd valor oale peru varablele calave ar a rea coloaă cuprde frecveţele. - Tabelul dsucv cople. Ese ulza peru varable calave. 0 0 D = 0 0. Are l ş p coloae. 0 0 uărul de sațe p uărul de odalăț peru varabla calavă. D D ese arcea dagoală a frecveţelor absolue. Peru v varable calave:
3 D = - Tabelul de preferţe. Ese u p de abel peru varable calave ş epră preferţele uu grup de dvz faţă de valorle ue caracersc. Foloseşe o scală de p ordal. Tabelele de aces p su frecve ulzae î sud de areg. De eeplu î abelul urăor su prezeae preferţele a paru persoae peru cc ărc de parfu: Persoae Mărc de parfuur erogae M M M 3 M 4 M 5 w w w w Preferţele su eprae pr oe de la loa 5. - Tabelul bar. Ese abelul care u coţe decâ valor 0 sau. Ese folos ca ş abelul de preferţe î achee sasce. Î eeplul de a os se găsesc răspusurle uor dvz la îrebarea Cţ zarul Z?. Idvz v Zare Z Z Z 3 Z 4 w w 0 0 w Răspusurle po f Da sau Nu ş su codfcae cu respecv 0. w 3 w 3 3 w Tabelul de proăţ. Auc câd se evaluează aseăărle sau dfereţele îre fecare cuplu de dvz se cosrueşe u abel de proăţ. De eeplu pue cosdera paru ărc de auourse pe care să le copară uele cu alele. Aproperea dre o arcă ş o arcă poae f făcuă prr-o oă de la la 0 sau eda uor oe de la la 0 dae eveual de u grup de specalş î doeu. Se poae observa î Tabelul urăor că se foloseşe o scală de p rapor. Mărc Mărc auourse auourse M M M3 M4 M M M M Tabele eerogee. Su abele î care varablele su de dfere pur. U eeplu de abel eeroge ese urăorul: Produse Varable Preţ Puc de lucru Mod de raspor p p 4 3 p 3 0 Paraerulu preţ s-a asoca o varablă caavă ăsurablă paraerulu Puc de lucru s-a asoca o varablă calavă ordală ar paraerulu Mod de raspor s-a asoca o o varablă calavă ordală. - Tabele rdesoale. Po f orcare d purle de abele dea prezeae la care se adaugă o a rea desue de eeplu pul. Asfel peru abelul eeroge de a sus dacă evaluarea se face la uue ervale de p se obţe u abel cu re desu cofor fgur: - Tabelul de odalăţ. Auc câd fecare îrebare a ue achee sasce presupue a ule răspusur e găs î faţa uu abel de odalăţ. Asfel dacă la îrebarea d eeplul aeror s-ar puea da re răspusur: Ncodaă - răspus codfca cu valoarea aurală Câeodaă - codfca cu Deseor - codfca cu 3 abelul de odalăţ asoca ar puea f urăorul: Idvz Zare Z Z Z 3 Z 4
4 p Idvz Evoluţe î Varable Măsur de aseăare Se ueşe ăsură de aseăare (slarae/dslarae) orce aplcaţe cu valor uerce care pere să se epre o legăură îre dvz sau îre varable. Dacă se oează cu ulțea dvzlor sau varablelor u dce de slarae pe ulţea ese o aplcaţe s care verfcă urăoarele re propreăţ:. s ese o aplcaţe a lu î R + ;. s ese sercă: (ww') : s(ww') = s(w'w); 3. (ww' ) cu w w' : s(ww) = s(w'w') > s(ww'). U dce de dslarae ese o aplcaţe s' care sasface prele două codţ d defţa dcelu de slarae ar codţa 3 ese îlocuă pr cerţa ca: 3'. w: s'(ww) = 0. O dsaţă oaă cu d ese u dce de dslarae care verfcă î plus urăoarele două propreăţ: 4. d (ww') = 0 w = w'; 5. d (ww') d (ww'' ) + d (w''w') ww'w'' - egalaea rughulu. Dacă lua î cosderare do dvz d abelul de observaț w ş w ( ese uărul de varable) dsaţa dre ce do dvz se poae def sub fora: d (w w ) = ( w w ) Qw w ude Q ese o arce sercă pozv defă uă ercă. Ma des ulzae su uraoarele pur de dsaţe:. Dsaţa eucldaă splă. Q ese arcea uae.. Dsaţa lu Mahalaobs aplcaă î aalza dscraă ude Q repreză arcea de covaraţă. ) ) ) ) ) ) ude = V = ) ) su îregsrărle peru varabla la ce dvz. 3. Dsaţa ese î od parcular be adapaă abelelor de frecveţe. Se aplcă î aalza facorală a corespodeţelor ar Q ese o arce a dcaorlor calculaț peru fecare pereche de varable. Schbarea de varablă Necesae: a) Câd daele su grupae î abele eerogee ş se doreşe eprarea ue varable descrpve cu auorul alea asfel îcâ oae varablele să devă de acelaş p. b) Peru a puea aplca o auă eodă de aalză a daelor î suaţa î care aceasa ese copablă cu pul daelor. De eeplu pr schbare de varablă se poae rasfora u abel de dae calave îr-u abel de odalăţ puâd asfel aplca îro aeră a efceă aalza facorală a corespodeţelor ulple. c) Peru a seza foraţa coţuă îr-u abel de dae reducâdu- asfel ărea. Se poae de eeplu îlocu ulţea de varable pr care se descre o colecvae prr-o cobaţe lară de acese varable Modalăţ de schbare a varablelor: Schbarea de varablă pr sadardzare Schbarea de varabla pr oralzare Schbarea de varablă pr codfcare Codfcarea ue varable caave pr grupare Gruparea pe efecve egale Gruparea î ervale egale Gruparea pr zarea erţe oale Codfcarea varablelor calave )
5 Codfcarea cu srucură de orde Codfcarea fără srucură de orde Codfcarea pr ragul edu Codfcarea dsucvă copleă a varablelor oale
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