pijan'othtes kai statistik'h january 18, 2011sf lmata 1 epilègoume kai kajorÐzoume to a...
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![Page 1: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/1.jpg)
Pijanìthtec kaiStatistik
S. Malef�kh
M�jhma 90
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 2: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/2.jpg)
'Elegqoi Upojèsewn
Up�rqei èna f�rmako me to opoÐo an gÐnei jerapeÐa emfanÐzontai tapr¸ta apotelèsmata mèsa se 10 mèrec. AnakalÔptetai èna kainoÔriof�rmako gia thn Ðdia asjèneia kai h farmakeutik etaireÐa isqurÐzetai ìtito nèo f�rmako fèrnei apotelèsmata se ligìtero qronikì di�sthma. Anloipìn eÐnai gnwstì ìti o mèsoc qrìnoc m gia thn emf�nish twn pr¸twnapotelesm�twn eÐnai 10 mèrec (dhlad µ = 10), qrei�zetai na elegqjeÐ anpragmatik� to kainoÔrio f�rmako dra suntomìtera (dhlad µ < 10).
H diadikasÐa aut onom�zetai èlegqoc upìjeshc statistikìc èlegqoc.
H upìjesh ìti µ = 10 lègetai mhdenik upìjesh
H upìjesh ìti µ < 10 lègetai enallaktik upìjesh
H apìfash an ja apodeqtoÔme ja aporrÐyoume th mhdenik upìjesh,basÐzetai se mÐa sun�rthsh twn dedomènwn mac.
H perioq pou aporrÐptetai h mhdenik upìjesh onom�zetai perioq
apìrriyhc krÐsimh perioq .
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 3: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/3.jpg)
'Elegqoi Upojèsewn
Up�rqei èna f�rmako me to opoÐo an gÐnei jerapeÐa emfanÐzontai tapr¸ta apotelèsmata mèsa se 10 mèrec. AnakalÔptetai èna kainoÔriof�rmako gia thn Ðdia asjèneia kai h farmakeutik etaireÐa isqurÐzetai ìtito nèo f�rmako fèrnei apotelèsmata se ligìtero qronikì di�sthma. Anloipìn eÐnai gnwstì ìti o mèsoc qrìnoc m gia thn emf�nish twn pr¸twnapotelesm�twn eÐnai 10 mèrec (dhlad µ = 10), qrei�zetai na elegqjeÐ anpragmatik� to kainoÔrio f�rmako dra suntomìtera (dhlad µ < 10).
H diadikasÐa aut onom�zetai èlegqoc upìjeshc statistikìc èlegqoc.
H upìjesh ìti µ = 10 lègetai mhdenik upìjesh
H upìjesh ìti µ < 10 lègetai enallaktik upìjesh
H apìfash an ja apodeqtoÔme ja aporrÐyoume th mhdenik upìjesh,basÐzetai se mÐa sun�rthsh twn dedomènwn mac.
H perioq pou aporrÐptetai h mhdenik upìjesh onom�zetai perioq
apìrriyhc krÐsimh perioq .
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 4: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/4.jpg)
'Elegqoi Upojèsewn
Up�rqei èna f�rmako me to opoÐo an gÐnei jerapeÐa emfanÐzontai tapr¸ta apotelèsmata mèsa se 10 mèrec. AnakalÔptetai èna kainoÔriof�rmako gia thn Ðdia asjèneia kai h farmakeutik etaireÐa isqurÐzetai ìtito nèo f�rmako fèrnei apotelèsmata se ligìtero qronikì di�sthma. Anloipìn eÐnai gnwstì ìti o mèsoc qrìnoc m gia thn emf�nish twn pr¸twnapotelesm�twn eÐnai 10 mèrec (dhlad µ = 10), qrei�zetai na elegqjeÐ anpragmatik� to kainoÔrio f�rmako dra suntomìtera (dhlad µ < 10).
H diadikasÐa aut onom�zetai èlegqoc upìjeshc statistikìc èlegqoc.
H upìjesh ìti µ = 10 lègetai mhdenik upìjesh
H upìjesh ìti µ < 10 lègetai enallaktik upìjesh
H apìfash an ja apodeqtoÔme ja aporrÐyoume th mhdenik upìjesh,basÐzetai se mÐa sun�rthsh twn dedomènwn mac.
H perioq pou aporrÐptetai h mhdenik upìjesh onom�zetai perioq
apìrriyhc krÐsimh perioq .
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 5: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/5.jpg)
'Elegqoi Upojèsewn
Up�rqei èna f�rmako me to opoÐo an gÐnei jerapeÐa emfanÐzontai tapr¸ta apotelèsmata mèsa se 10 mèrec. AnakalÔptetai èna kainoÔriof�rmako gia thn Ðdia asjèneia kai h farmakeutik etaireÐa isqurÐzetai ìtito nèo f�rmako fèrnei apotelèsmata se ligìtero qronikì di�sthma. Anloipìn eÐnai gnwstì ìti o mèsoc qrìnoc m gia thn emf�nish twn pr¸twnapotelesm�twn eÐnai 10 mèrec (dhlad µ = 10), qrei�zetai na elegqjeÐ anpragmatik� to kainoÔrio f�rmako dra suntomìtera (dhlad µ < 10).
H diadikasÐa aut onom�zetai èlegqoc upìjeshc statistikìc èlegqoc.
H upìjesh ìti µ = 10 lègetai mhdenik upìjesh
H upìjesh ìti µ < 10 lègetai enallaktik upìjesh
H apìfash an ja apodeqtoÔme ja aporrÐyoume th mhdenik upìjesh,basÐzetai se mÐa sun�rthsh twn dedomènwn mac.
H perioq pou aporrÐptetai h mhdenik upìjesh onom�zetai perioq
apìrriyhc krÐsimh perioq .
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 6: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/6.jpg)
'Elegqoi Upojèsewn
Up�rqei èna f�rmako me to opoÐo an gÐnei jerapeÐa emfanÐzontai tapr¸ta apotelèsmata mèsa se 10 mèrec. AnakalÔptetai èna kainoÔriof�rmako gia thn Ðdia asjèneia kai h farmakeutik etaireÐa isqurÐzetai ìtito nèo f�rmako fèrnei apotelèsmata se ligìtero qronikì di�sthma. Anloipìn eÐnai gnwstì ìti o mèsoc qrìnoc m gia thn emf�nish twn pr¸twnapotelesm�twn eÐnai 10 mèrec (dhlad µ = 10), qrei�zetai na elegqjeÐ anpragmatik� to kainoÔrio f�rmako dra suntomìtera (dhlad µ < 10).
H diadikasÐa aut onom�zetai èlegqoc upìjeshc statistikìc èlegqoc.
H upìjesh ìti µ = 10 lègetai mhdenik upìjesh
H upìjesh ìti µ < 10 lègetai enallaktik upìjesh
H apìfash an ja apodeqtoÔme ja aporrÐyoume th mhdenik upìjesh,basÐzetai se mÐa sun�rthsh twn dedomènwn mac.
H perioq pou aporrÐptetai h mhdenik upìjesh onom�zetai perioq
apìrriyhc krÐsimh perioq .
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 7: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/7.jpg)
'Elegqoi Upojèsewn
H diadikasÐa tou statistikoÔ elègqou
1 OrÐzoume th mhdenik upìjesh (H0 : µ = µ0)
2 OrÐzoume thn enallaktik upìjesh (H1 : µ 6= µ0) (H0 : µ < µ0) (H0 : µ > µ0)
3 UpologÐzoume thn tim thc statistik c sun�rthshc sto deÐgma mac
4 OrÐzoume thn perioq apìrriyhc thc H0
5 Ex�goume ta sumper�smata mac.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 8: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/8.jpg)
'Elegqoi Upojèsewn
H diadikasÐa tou statistikoÔ elègqou
1 OrÐzoume th mhdenik upìjesh (H0 : µ = µ0)
2 OrÐzoume thn enallaktik upìjesh (H1 : µ 6= µ0) (H0 : µ < µ0) (H0 : µ > µ0)
3 UpologÐzoume thn tim thc statistik c sun�rthshc sto deÐgma mac
4 OrÐzoume thn perioq apìrriyhc thc H0
5 Ex�goume ta sumper�smata mac.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 9: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/9.jpg)
'Elegqoi Upojèsewn
H diadikasÐa tou statistikoÔ elègqou
1 OrÐzoume th mhdenik upìjesh (H0 : µ = µ0)
2 OrÐzoume thn enallaktik upìjesh (H1 : µ 6= µ0) (H0 : µ < µ0) (H0 : µ > µ0)
3 UpologÐzoume thn tim thc statistik c sun�rthshc sto deÐgma mac
4 OrÐzoume thn perioq apìrriyhc thc H0
5 Ex�goume ta sumper�smata mac.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 10: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/10.jpg)
'Elegqoi Upojèsewn
H diadikasÐa tou statistikoÔ elègqou
1 OrÐzoume th mhdenik upìjesh (H0 : µ = µ0)
2 OrÐzoume thn enallaktik upìjesh (H1 : µ 6= µ0) (H0 : µ < µ0) (H0 : µ > µ0)
3 UpologÐzoume thn tim thc statistik c sun�rthshc sto deÐgma mac
4 OrÐzoume thn perioq apìrriyhc thc H0
5 Ex�goume ta sumper�smata mac.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 11: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/11.jpg)
'Elegqoi Upojèsewn
H diadikasÐa tou statistikoÔ elègqou
1 OrÐzoume th mhdenik upìjesh (H0 : µ = µ0)
2 OrÐzoume thn enallaktik upìjesh (H1 : µ 6= µ0) (H0 : µ < µ0) (H0 : µ > µ0)
3 UpologÐzoume thn tim thc statistik c sun�rthshc sto deÐgma mac
4 OrÐzoume thn perioq apìrriyhc thc H0
5 Ex�goume ta sumper�smata mac.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 12: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/12.jpg)
Sf�lmata
1 Sf�lma tÔpou I (na aporrÐyoume th mhdenik upìjesh en¸ eÐnaiswst )
a = P(apìrriyh H0|H0 swst )
2 Sf�lma tÔpou II (na apodeqtoÔme th mhdenik upìjesh en¸ eÐnail�joc)
β = P(apodoq H0|H0 l�joc)
IsqÔc enìc elègqou onom�zetai h pijanìthta apìrriyhc thc H0 ìtan h H0
eÐnai pragmatik� l�joc.
γ = P(apìrriyh H0|H0 l�joc) = 1− β
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 13: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/13.jpg)
Sf�lmata
1 Sf�lma tÔpou I (na aporrÐyoume th mhdenik upìjesh en¸ eÐnaiswst )
a = P(apìrriyh H0|H0 swst )
2 Sf�lma tÔpou II (na apodeqtoÔme th mhdenik upìjesh en¸ eÐnail�joc)
β = P(apodoq H0|H0 l�joc)
IsqÔc enìc elègqou onom�zetai h pijanìthta apìrriyhc thc H0 ìtan h H0
eÐnai pragmatik� l�joc.
γ = P(apìrriyh H0|H0 l�joc) = 1− β
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 14: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/14.jpg)
Sf�lmata
1 Sf�lma tÔpou I (na aporrÐyoume th mhdenik upìjesh en¸ eÐnaiswst )
a = P(apìrriyh H0|H0 swst )
2 Sf�lma tÔpou II (na apodeqtoÔme th mhdenik upìjesh en¸ eÐnail�joc)
β = P(apodoq H0|H0 l�joc)
IsqÔc enìc elègqou onom�zetai h pijanìthta apìrriyhc thc H0 ìtan h H0
eÐnai pragmatik� l�joc.
γ = P(apìrriyh H0|H0 l�joc) = 1− β
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 15: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/15.jpg)
Sf�lmata
1 Sf�lma tÔpou I (na aporrÐyoume th mhdenik upìjesh en¸ eÐnaiswst )
a = P(apìrriyh H0|H0 swst )
2 Sf�lma tÔpou II (na apodeqtoÔme th mhdenik upìjesh en¸ eÐnail�joc)
β = P(apodoq H0|H0 l�joc)
IsqÔc enìc elègqou onom�zetai h pijanìthta apìrriyhc thc H0 ìtan h H0
eÐnai pragmatik� l�joc.
γ = P(apìrriyh H0|H0 l�joc) = 1− β
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 16: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/16.jpg)
Sf�lmata
1 Epilègoume kai kajorÐzoume to a kai b�sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a eÐnai sun jwc polÔ mikrì(p.q. 0.05, 0.01, 0.1)
2 ParathroÔmeno epÐpedo shmantikìthtac apl� shmantikìthta touelègqou onom�zetai h pijanìthta na parathrhjeÐ mÐa tim thcstatistik c sun�rths c megalÔterh apì aut pou èdwse tosugkekrimèno deÐgma
P(Z > |z ||H0 swst )
ìpou z h tim tou statistikoÔ gia to sugkekrimèno deÐgma.
H parap�nw pijanìthta anafèretai se monìpleurouc elègqouc en¸gia dÐpleurouc elègqouc diplasi�zetai
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
![Page 17: Pijan'othtes kai Statistik'h January 18, 2011Sf lmata 1 Epilègoume kai kajorÐzoume to a kai b sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a naieÐ sun jwc](https://reader034.vdocuments.net/reader034/viewer/2022042203/5ea43784020c6739382511f2/html5/thumbnails/17.jpg)
Sf�lmata
1 Epilègoume kai kajorÐzoume to a kai b�sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a eÐnai sun jwc polÔ mikrì(p.q. 0.05, 0.01, 0.1)
2 ParathroÔmeno epÐpedo shmantikìthtac apl� shmantikìthta touelègqou onom�zetai h pijanìthta na parathrhjeÐ mÐa tim thcstatistik c sun�rths c megalÔterh apì aut pou èdwse tosugkekrimèno deÐgma
P(Z > |z ||H0 swst )
ìpou z h tim tou statistikoÔ gia to sugkekrimèno deÐgma.
H parap�nw pijanìthta anafèretai se monìpleurouc elègqouc en¸gia dÐpleurouc elègqouc diplasi�zetai
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Sf�lmata
1 Epilègoume kai kajorÐzoume to a kai b�sh autoÔ orÐzetai h perioq apìrriyhc thc mhdenik c upìjeshc. To a eÐnai sun jwc polÔ mikrì(p.q. 0.05, 0.01, 0.1)
2 ParathroÔmeno epÐpedo shmantikìthtac apl� shmantikìthta touelègqou onom�zetai h pijanìthta na parathrhjeÐ mÐa tim thcstatistik c sun�rths c megalÔterh apì aut pou èdwse tosugkekrimèno deÐgma
P(Z > |z ||H0 swst )
ìpou z h tim tou statistikoÔ gia to sugkekrimèno deÐgma.
H parap�nw pijanìthta anafèretai se monìpleurouc elègqouc en¸gia dÐpleurouc elègqouc diplasi�zetai
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Genik idèa
'Estw ìti jèloume na elègxoume thn upìjesh
H0 : θ = θ0 kat� H1 : θ 6= θ0
O plhjusmìc me b�sh k�poion kanìna qwrÐzetai se dÔo perioqèc, sthnperioq apodoq c thc H0 kai sthn perioq apìrriyhc thc. Katìpin, gÐnetaio èlegqoc b�sh tou krithrÐou pou orÐsthke kai apofasÐzetai an h H0
gÐnetai deqt ìqi
'Estw θ ektimht c tou θ
Xèroume ìti θ ∼ N (θ, τ 2)Upì thn mhdenik upìjesh èqoume
θ ∼ N (θ0, τ2)⇒ Z =
θ − θ0
τ∼ N (0, 1)
'Ara an h H0 eÐnai swst , dhlad an to deÐgma mac proèrqetai apì ènan
plhjusmì me mèsh tim θ0 tìte h metablht Z ja paÐrnei timèc me meg�lh
pijanìthta (1− a) gÔrw apì to 0.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Genik idèa
'Estw ìti jèloume na elègxoume thn upìjesh
H0 : θ = θ0 kat� H1 : θ 6= θ0
O plhjusmìc me b�sh k�poion kanìna qwrÐzetai se dÔo perioqèc, sthnperioq apodoq c thc H0 kai sthn perioq apìrriyhc thc. Katìpin, gÐnetaio èlegqoc b�sh tou krithrÐou pou orÐsthke kai apofasÐzetai an h H0
gÐnetai deqt ìqi
'Estw θ ektimht c tou θXèroume ìti θ ∼ N (θ, τ 2)
Upì thn mhdenik upìjesh èqoume
θ ∼ N (θ0, τ2)⇒ Z =
θ − θ0
τ∼ N (0, 1)
'Ara an h H0 eÐnai swst , dhlad an to deÐgma mac proèrqetai apì ènan
plhjusmì me mèsh tim θ0 tìte h metablht Z ja paÐrnei timèc me meg�lh
pijanìthta (1− a) gÔrw apì to 0.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Genik idèa
'Estw ìti jèloume na elègxoume thn upìjesh
H0 : θ = θ0 kat� H1 : θ 6= θ0
O plhjusmìc me b�sh k�poion kanìna qwrÐzetai se dÔo perioqèc, sthnperioq apodoq c thc H0 kai sthn perioq apìrriyhc thc. Katìpin, gÐnetaio èlegqoc b�sh tou krithrÐou pou orÐsthke kai apofasÐzetai an h H0
gÐnetai deqt ìqi
'Estw θ ektimht c tou θXèroume ìti θ ∼ N (θ, τ 2)Upì thn mhdenik upìjesh èqoume
θ ∼ N (θ0, τ2)⇒ Z =
θ − θ0
τ∼ N (0, 1)
'Ara an h H0 eÐnai swst , dhlad an to deÐgma mac proèrqetai apì ènan
plhjusmì me mèsh tim θ0 tìte h metablht Z ja paÐrnei timèc me meg�lh
pijanìthta (1− a) gÔrw apì to 0.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Genik idèa
'Estw ìti jèloume na elègxoume thn upìjesh
H0 : θ = θ0 kat� H1 : θ 6= θ0
O plhjusmìc me b�sh k�poion kanìna qwrÐzetai se dÔo perioqèc, sthnperioq apodoq c thc H0 kai sthn perioq apìrriyhc thc. Katìpin, gÐnetaio èlegqoc b�sh tou krithrÐou pou orÐsthke kai apofasÐzetai an h H0
gÐnetai deqt ìqi
'Estw θ ektimht c tou θXèroume ìti θ ∼ N (θ, τ 2)Upì thn mhdenik upìjesh èqoume
θ ∼ N (θ0, τ2)⇒ Z =
θ − θ0
τ∼ N (0, 1)
'Ara an h H0 eÐnai swst , dhlad an to deÐgma mac proèrqetai apì ènan
plhjusmì me mèsh tim θ0 tìte h metablht Z ja paÐrnei timèc me meg�lh
pijanìthta (1− a) gÔrw apì to 0.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Genik Idèa
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Genik Idèa
Poia eÐnai h perioq apìrriy c ìtan h enallaktik upìjesh
eÐnai
H1 : θ > θ0
Poia eÐnai h perioq apìrriy c ìtan h enallaktik upìjesh
eÐnai
H1 : θ < θ0
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Genik Idèa
Poia eÐnai h perioq apìrriy c ìtan h enallaktik upìjesh
eÐnai
H1 : θ > θ0
Poia eÐnai h perioq apìrriy c ìtan h enallaktik upìjesh
eÐnai
H1 : θ < θ0
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to mèso m enìc plhjusmoÔ
me gnwst diaspor�
H0 : µ = µ0 kat� H1 : µ 6= µ0 µ < µ0 µ > µ0
X ∼ N (µ, σ2/n) isodÔnama X−µσ/√n∼ N (0, 1)
Z =X − µ0
σ/√n
1 an H1 : µ 6= µ0 tìte aporrÐptw th mhdenik upìjesh an
Z < −za/2 Z > za/2
2 an H1 : µ > µ0 tìte aporrÐptw th mhdenik upìjesh an
Z > za3 an H1 : µ < µ0 tìte aporrÐptw th mhdenik upìjesh an
Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to mèso m enìc plhjusmoÔ
me gnwst diaspor�
H0 : µ = µ0 kat� H1 : µ 6= µ0 µ < µ0 µ > µ0
X ∼ N (µ, σ2/n) isodÔnama X−µσ/√n∼ N (0, 1)
Z =X − µ0
σ/√n
1 an H1 : µ 6= µ0 tìte aporrÐptw th mhdenik upìjesh an
Z < −za/2 Z > za/2
2 an H1 : µ > µ0 tìte aporrÐptw th mhdenik upìjesh an
Z > za3 an H1 : µ < µ0 tìte aporrÐptw th mhdenik upìjesh an
Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to mèso m enìc plhjusmoÔ
me gnwst diaspor�
H0 : µ = µ0 kat� H1 : µ 6= µ0 µ < µ0 µ > µ0
X ∼ N (µ, σ2/n) isodÔnama X−µσ/√n∼ N (0, 1)
Z =X − µ0
σ/√n
1 an H1 : µ 6= µ0 tìte aporrÐptw th mhdenik upìjesh an
Z < −za/2 Z > za/2
2 an H1 : µ > µ0 tìte aporrÐptw th mhdenik upìjesh an
Z > za
3 an H1 : µ < µ0 tìte aporrÐptw th mhdenik upìjesh an
Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to mèso m enìc plhjusmoÔ
me gnwst diaspor�
H0 : µ = µ0 kat� H1 : µ 6= µ0 µ < µ0 µ > µ0
X ∼ N (µ, σ2/n) isodÔnama X−µσ/√n∼ N (0, 1)
Z =X − µ0
σ/√n
1 an H1 : µ 6= µ0 tìte aporrÐptw th mhdenik upìjesh an
Z < −za/2 Z > za/2
2 an H1 : µ > µ0 tìte aporrÐptw th mhdenik upìjesh an
Z > za3 an H1 : µ < µ0 tìte aporrÐptw th mhdenik upìjesh an
Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to mèso m enìc plhjusmoÔ
me �gnwsth diaspor�
H0 : µ = µ0 kat� H1 : µ 6= µ0 µ < µ0 µ > µ0
X−µs/√n∼ tn−1
T =X − µ0
s/√n
1 an H1 : µ 6= µ0 tìte aporrÐptw th mhdenik upìjesh an
T < −tn−1,a/2, T > tn−1,a/2
2 an H1 : µ > µ0 tìte aporrÐptw th mhdenik upìjesh an
T > tn−1,a
3 an H1 : µ < µ0 tìte aporrÐptw th mhdenik upìjesh an
T < −tn−1,a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to mèso m enìc plhjusmoÔ
me �gnwsth diaspor�
H0 : µ = µ0 kat� H1 : µ 6= µ0 µ < µ0 µ > µ0
X−µs/√n∼ tn−1
T =X − µ0
s/√n
1 an H1 : µ 6= µ0 tìte aporrÐptw th mhdenik upìjesh an
T < −tn−1,a/2, T > tn−1,a/2
2 an H1 : µ > µ0 tìte aporrÐptw th mhdenik upìjesh an
T > tn−1,a
3 an H1 : µ < µ0 tìte aporrÐptw th mhdenik upìjesh an
T < −tn−1,a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to mèso m enìc plhjusmoÔ
me �gnwsth diaspor�
H0 : µ = µ0 kat� H1 : µ 6= µ0 µ < µ0 µ > µ0
X−µs/√n∼ tn−1
T =X − µ0
s/√n
1 an H1 : µ 6= µ0 tìte aporrÐptw th mhdenik upìjesh an
T < −tn−1,a/2, T > tn−1,a/2
2 an H1 : µ > µ0 tìte aporrÐptw th mhdenik upìjesh an
T > tn−1,a
3 an H1 : µ < µ0 tìte aporrÐptw th mhdenik upìjesh an
T < −tn−1,a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to mèso m enìc plhjusmoÔ
me �gnwsth diaspor�
H0 : µ = µ0 kat� H1 : µ 6= µ0 µ < µ0 µ > µ0
X−µs/√n∼ tn−1
T =X − µ0
s/√n
1 an H1 : µ 6= µ0 tìte aporrÐptw th mhdenik upìjesh an
T < −tn−1,a/2, T > tn−1,a/2
2 an H1 : µ > µ0 tìte aporrÐptw th mhdenik upìjesh an
T > tn−1,a
3 an H1 : µ < µ0 tìte aporrÐptw th mhdenik upìjesh an
T < −tn−1,a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to gia th diafor� twn mèswn
µ1 − µ2 dÔo anex�rthtwn plhjusm¸n
me gnwstèc diasporèc
H0 : µ1−µ2 = δ κατά H1 : µ1−µ2 6= δ ή µ1−µ2 < δ ή µ1−µ2 > δ
Z =X − Y − δ√σ2
1/n + σ22/m
1 an H1 : µ1 − µ2 6= δ tìte aporrÐptw th mhdenik upìjesh
an Z < −za/2 Z > za/2
2 an H1 : µ1 − µ2 > δ tìte aporrÐptw th mhdenik upìjesh
an Z > za3 an H1 : µ1 − µ2 < δ tìte aporrÐptw th mhdenik upìjesh
an Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to gia th diafor� twn mèswn
µ1 − µ2 dÔo anex�rthtwn plhjusm¸n
me gnwstèc diasporèc
H0 : µ1−µ2 = δ κατά H1 : µ1−µ2 6= δ ή µ1−µ2 < δ ή µ1−µ2 > δ
Z =X − Y − δ√σ2
1/n + σ22/m
1 an H1 : µ1 − µ2 6= δ tìte aporrÐptw th mhdenik upìjesh
an Z < −za/2 Z > za/2
2 an H1 : µ1 − µ2 > δ tìte aporrÐptw th mhdenik upìjesh
an Z > za3 an H1 : µ1 − µ2 < δ tìte aporrÐptw th mhdenik upìjesh
an Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to gia th diafor� twn mèswn
µ1 − µ2 dÔo anex�rthtwn plhjusm¸n
me gnwstèc diasporèc
H0 : µ1−µ2 = δ κατά H1 : µ1−µ2 6= δ ή µ1−µ2 < δ ή µ1−µ2 > δ
Z =X − Y − δ√σ2
1/n + σ22/m
1 an H1 : µ1 − µ2 6= δ tìte aporrÐptw th mhdenik upìjesh
an Z < −za/2 Z > za/2
2 an H1 : µ1 − µ2 > δ tìte aporrÐptw th mhdenik upìjesh
an Z > za
3 an H1 : µ1 − µ2 < δ tìte aporrÐptw th mhdenik upìjesh
an Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to gia th diafor� twn mèswn
µ1 − µ2 dÔo anex�rthtwn plhjusm¸n
me gnwstèc diasporèc
H0 : µ1−µ2 = δ κατά H1 : µ1−µ2 6= δ ή µ1−µ2 < δ ή µ1−µ2 > δ
Z =X − Y − δ√σ2
1/n + σ22/m
1 an H1 : µ1 − µ2 6= δ tìte aporrÐptw th mhdenik upìjesh
an Z < −za/2 Z > za/2
2 an H1 : µ1 − µ2 > δ tìte aporrÐptw th mhdenik upìjesh
an Z > za3 an H1 : µ1 − µ2 < δ tìte aporrÐptw th mhdenik upìjesh
an Z < −zaS. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to gia th diafor� twn mèswn
µ1 − µ2 dÔo anex�rthtwn plhjusm¸n
me �gnwstec diasporèc kai Ðsec
T =X − Y − δ
sp√
1/n + 1/m
ìpou s2p =
(n−1)s21 +(m−1)s2
2n+m−2
1 an H1 : µ1 − µ2 6= δ tìte aporrÐptw th mhdenik upìjesh
an T < −tn+m−2,a/2 T > tn+m−2,a/2
2 an H1 : µ1 − µ2 > δ tìte aporrÐptw th mhdenik upìjesh
an T > tn+m−2,a
3 an H1 : µ1 − µ2 < δ tìte aporrÐptw th mhdenik upìjesh
an T < −tn+m−2,a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to gia th diafor� twn mèswn
µ1 − µ2 dÔo anex�rthtwn plhjusm¸n
me �gnwstec diasporèc kai Ðsec
T =X − Y − δ
sp√
1/n + 1/m
ìpou s2p =
(n−1)s21 +(m−1)s2
2n+m−2
1 an H1 : µ1 − µ2 6= δ tìte aporrÐptw th mhdenik upìjesh
an T < −tn+m−2,a/2 T > tn+m−2,a/2
2 an H1 : µ1 − µ2 > δ tìte aporrÐptw th mhdenik upìjesh
an T > tn+m−2,a
3 an H1 : µ1 − µ2 < δ tìte aporrÐptw th mhdenik upìjesh
an T < −tn+m−2,a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to gia th diafor� twn mèswn
µ1 − µ2 dÔo anex�rthtwn plhjusm¸n
me �gnwstec diasporèc kai Ðsec
T =X − Y − δ
sp√
1/n + 1/m
ìpou s2p =
(n−1)s21 +(m−1)s2
2n+m−2
1 an H1 : µ1 − µ2 6= δ tìte aporrÐptw th mhdenik upìjesh
an T < −tn+m−2,a/2 T > tn+m−2,a/2
2 an H1 : µ1 − µ2 > δ tìte aporrÐptw th mhdenik upìjesh
an T > tn+m−2,a
3 an H1 : µ1 − µ2 < δ tìte aporrÐptw th mhdenik upìjesh
an T < −tn+m−2,a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to gia th diafor� twn mèswn
µ1 − µ2 dÔo anex�rthtwn plhjusm¸n
me �gnwstec diasporèc kai �nisec
T =X − Y − δ√s2
1/n + s22/m
1 αν H1 : µ1 − µ2 6= δ τότε απορρίπτω τη μηδενική υπόθεση ανT < −tv ,a/2 ή T > tv ,a/2
2 αν H1 : µ1 − µ2 > δ τότε απορρίπτω τη μηδενική υπόθεση ανT > tv ,a
3 αν H1 : µ1 − µ2 < δ τότε απορρίπτω τη μηδενική υπόθεση ανT < −tv ,a
− > n = m tìte v = 2(n − 1)
− > n 6= m tìte v =
(s21n
+s22m
)2
(s21n
)2
n−1+
(s22m
)2
m−1
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to gia th diafor� twn mèswn
µ1 − µ2 dÔo anex�rthtwn plhjusm¸n
me �gnwstec diasporèc kai �nisec
T =X − Y − δ√s2
1/n + s22/m
1 αν H1 : µ1 − µ2 6= δ τότε απορρίπτω τη μηδενική υπόθεση ανT < −tv ,a/2 ή T > tv ,a/2
2 αν H1 : µ1 − µ2 > δ τότε απορρίπτω τη μηδενική υπόθεση ανT > tv ,a
3 αν H1 : µ1 − µ2 < δ τότε απορρίπτω τη μηδενική υπόθεση ανT < −tv ,a
− > n = m tìte v = 2(n − 1)
− > n 6= m tìte v =
(s21n
+s22m
)2
(s21n
)2
n−1+
(s22m
)2
m−1
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to gia th diafor� twn mèswn
µ1 − µ2 dÔo anex�rthtwn plhjusm¸n
me �gnwstec diasporèc kai �nisec
T =X − Y − δ√s2
1/n + s22/m
1 αν H1 : µ1 − µ2 6= δ τότε απορρίπτω τη μηδενική υπόθεση ανT < −tv ,a/2 ή T > tv ,a/2
2 αν H1 : µ1 − µ2 > δ τότε απορρίπτω τη μηδενική υπόθεση ανT > tv ,a
3 αν H1 : µ1 − µ2 < δ τότε απορρίπτω τη μηδενική υπόθεση ανT < −tv ,a
− > n = m tìte v = 2(n − 1)
− > n 6= m tìte v =
(s21n
+s22m
)2
(s21n
)2
n−1+
(s22m
)2
m−1
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to gia th diafor� twn mèswn
µ1 − µ2 dÔo anex�rthtwn plhjusm¸n
me �gnwstec diasporèc kai �nisec
T =X − Y − δ√s2
1/n + s22/m
1 αν H1 : µ1 − µ2 6= δ τότε απορρίπτω τη μηδενική υπόθεση ανT < −tv ,a/2 ή T > tv ,a/2
2 αν H1 : µ1 − µ2 > δ τότε απορρίπτω τη μηδενική υπόθεση ανT > tv ,a
3 αν H1 : µ1 − µ2 < δ τότε απορρίπτω τη μηδενική υπόθεση ανT < −tv ,a
− > n = m tìte v = 2(n − 1)
− > n 6= m tìte v =
(s21n
+s22m
)2
(s21n
)2
n−1+
(s22m
)2
m−1S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diafor� twn mèswn µ1 − µ2
dÔo exarthmènwn plhjusm¸n
H0 : µ1−µ2 = δ κατά H1 : µ1−µ2 6= δ ή µ1−µ2 < δ ή µ1−µ2 > δ
Kataskeu�zw ta zi = xi − yi , i = 1, . . . , n.
H0 : z = δ kat� H1 : z 6= δ z < δ z > δ
T =z − δsz/√n
1 an H1 : z 6= δ tìte aporrÐptw th mhdenik upìjesh an
T > |tn−1,a/2|2 an H1 : z > δ tìte aporrÐptw th mhdenik upìjesh an
T > tn−1,a
3 an H1 : z < δ tìte aporrÐptw th mhdenik upìjesh an
T < −tn−1,a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diafor� twn mèswn µ1 − µ2
dÔo exarthmènwn plhjusm¸n
H0 : µ1−µ2 = δ κατά H1 : µ1−µ2 6= δ ή µ1−µ2 < δ ή µ1−µ2 > δ
Kataskeu�zw ta zi = xi − yi , i = 1, . . . , n.
H0 : z = δ kat� H1 : z 6= δ z < δ z > δ
T =z − δsz/√n
1 an H1 : z 6= δ tìte aporrÐptw th mhdenik upìjesh an
T > |tn−1,a/2|2 an H1 : z > δ tìte aporrÐptw th mhdenik upìjesh an
T > tn−1,a
3 an H1 : z < δ tìte aporrÐptw th mhdenik upìjesh an
T < −tn−1,a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diafor� twn mèswn µ1 − µ2
dÔo exarthmènwn plhjusm¸n
H0 : µ1−µ2 = δ κατά H1 : µ1−µ2 6= δ ή µ1−µ2 < δ ή µ1−µ2 > δ
Kataskeu�zw ta zi = xi − yi , i = 1, . . . , n.
H0 : z = δ kat� H1 : z 6= δ z < δ z > δ
T =z − δsz/√n
1 an H1 : z 6= δ tìte aporrÐptw th mhdenik upìjesh an
T > |tn−1,a/2|
2 an H1 : z > δ tìte aporrÐptw th mhdenik upìjesh an
T > tn−1,a
3 an H1 : z < δ tìte aporrÐptw th mhdenik upìjesh an
T < −tn−1,a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diafor� twn mèswn µ1 − µ2
dÔo exarthmènwn plhjusm¸n
H0 : µ1−µ2 = δ κατά H1 : µ1−µ2 6= δ ή µ1−µ2 < δ ή µ1−µ2 > δ
Kataskeu�zw ta zi = xi − yi , i = 1, . . . , n.
H0 : z = δ kat� H1 : z 6= δ z < δ z > δ
T =z − δsz/√n
1 an H1 : z 6= δ tìte aporrÐptw th mhdenik upìjesh an
T > |tn−1,a/2|2 an H1 : z > δ tìte aporrÐptw th mhdenik upìjesh an
T > tn−1,a
3 an H1 : z < δ tìte aporrÐptw th mhdenik upìjesh an
T < −tn−1,a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diafor� twn mèswn µ1 − µ2
dÔo exarthmènwn plhjusm¸n
H0 : µ1−µ2 = δ κατά H1 : µ1−µ2 6= δ ή µ1−µ2 < δ ή µ1−µ2 > δ
Kataskeu�zw ta zi = xi − yi , i = 1, . . . , n.
H0 : z = δ kat� H1 : z 6= δ z < δ z > δ
T =z − δsz/√n
1 an H1 : z 6= δ tìte aporrÐptw th mhdenik upìjesh an
T > |tn−1,a/2|2 an H1 : z > δ tìte aporrÐptw th mhdenik upìjesh an
T > tn−1,a
3 an H1 : z < δ tìte aporrÐptw th mhdenik upìjesh an
T < −tn−1,a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia thn analogÐa p enìc plhjusmoÔ
H0 : p = p0 kat� H1 : p 6= p0 p < p0 p > p0
Z =p − p0√p0(1−p0)
n
1 an H1 : p 6= p0 tìte aporrÐptw th mhdenik upìjesh an
Z < |za/2|2 an H1 : p > p0 tìte aporrÐptw th mhdenik upìjesh an
Z > za3 an H1 : p < p0 tìte aporrÐptw th mhdenik upìjesh an
Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia thn analogÐa p enìc plhjusmoÔ
H0 : p = p0 kat� H1 : p 6= p0 p < p0 p > p0
Z =p − p0√p0(1−p0)
n
1 an H1 : p 6= p0 tìte aporrÐptw th mhdenik upìjesh an
Z < |za/2|
2 an H1 : p > p0 tìte aporrÐptw th mhdenik upìjesh an
Z > za3 an H1 : p < p0 tìte aporrÐptw th mhdenik upìjesh an
Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia thn analogÐa p enìc plhjusmoÔ
H0 : p = p0 kat� H1 : p 6= p0 p < p0 p > p0
Z =p − p0√p0(1−p0)
n
1 an H1 : p 6= p0 tìte aporrÐptw th mhdenik upìjesh an
Z < |za/2|2 an H1 : p > p0 tìte aporrÐptw th mhdenik upìjesh an
Z > za
3 an H1 : p < p0 tìte aporrÐptw th mhdenik upìjesh an
Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia thn analogÐa p enìc plhjusmoÔ
H0 : p = p0 kat� H1 : p 6= p0 p < p0 p > p0
Z =p − p0√p0(1−p0)
n
1 an H1 : p 6= p0 tìte aporrÐptw th mhdenik upìjesh an
Z < |za/2|2 an H1 : p > p0 tìte aporrÐptw th mhdenik upìjesh an
Z > za3 an H1 : p < p0 tìte aporrÐptw th mhdenik upìjesh an
Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diafor� twn analogÐwn
p1 − p2 dÔo anex�rthtwn plhjusm¸n
H0 : p1 − p2 = δ kat� H1 : p1 − p2 6= δ p1 − p2 < δ p1 − p2 > δ
Se aut thn perÐptwsh diakrÐnoume peript¸seican δ 6= 0
Z =p1 − p2 − δ√
p1(1− p1)/n + p2(1− p2)/m
an δ = 0
Z =p1 − p2√
p(1− p)(1/n + 1/m)
ìpou p = x+yn+m
1 an H1 : p1 − p2 6= δ tìte aporrÐptw th mhdenik upìjesh an Z > |za/2|2 an H1 : p1 − p2 > δ tìte aporrÐptw th mhdenik upìjesh an Z > za
3 an H1 : p1 − p2 < δ tìte aporrÐptw th mhdenik upìjesh an Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diafor� twn analogÐwn
p1 − p2 dÔo anex�rthtwn plhjusm¸n
H0 : p1 − p2 = δ kat� H1 : p1 − p2 6= δ p1 − p2 < δ p1 − p2 > δ
Se aut thn perÐptwsh diakrÐnoume peript¸seic
an δ 6= 0
Z =p1 − p2 − δ√
p1(1− p1)/n + p2(1− p2)/m
an δ = 0
Z =p1 − p2√
p(1− p)(1/n + 1/m)
ìpou p = x+yn+m
1 an H1 : p1 − p2 6= δ tìte aporrÐptw th mhdenik upìjesh an Z > |za/2|2 an H1 : p1 − p2 > δ tìte aporrÐptw th mhdenik upìjesh an Z > za
3 an H1 : p1 − p2 < δ tìte aporrÐptw th mhdenik upìjesh an Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diafor� twn analogÐwn
p1 − p2 dÔo anex�rthtwn plhjusm¸n
H0 : p1 − p2 = δ kat� H1 : p1 − p2 6= δ p1 − p2 < δ p1 − p2 > δ
Se aut thn perÐptwsh diakrÐnoume peript¸seican δ 6= 0
Z =p1 − p2 − δ√
p1(1− p1)/n + p2(1− p2)/m
an δ = 0
Z =p1 − p2√
p(1− p)(1/n + 1/m)
ìpou p = x+yn+m
1 an H1 : p1 − p2 6= δ tìte aporrÐptw th mhdenik upìjesh an Z > |za/2|2 an H1 : p1 − p2 > δ tìte aporrÐptw th mhdenik upìjesh an Z > za
3 an H1 : p1 − p2 < δ tìte aporrÐptw th mhdenik upìjesh an Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diafor� twn analogÐwn
p1 − p2 dÔo anex�rthtwn plhjusm¸n
H0 : p1 − p2 = δ kat� H1 : p1 − p2 6= δ p1 − p2 < δ p1 − p2 > δ
Se aut thn perÐptwsh diakrÐnoume peript¸seican δ 6= 0
Z =p1 − p2 − δ√
p1(1− p1)/n + p2(1− p2)/m
an δ = 0
Z =p1 − p2√
p(1− p)(1/n + 1/m)
ìpou p = x+yn+m
1 an H1 : p1 − p2 6= δ tìte aporrÐptw th mhdenik upìjesh an Z > |za/2|
2 an H1 : p1 − p2 > δ tìte aporrÐptw th mhdenik upìjesh an Z > za
3 an H1 : p1 − p2 < δ tìte aporrÐptw th mhdenik upìjesh an Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diafor� twn analogÐwn
p1 − p2 dÔo anex�rthtwn plhjusm¸n
H0 : p1 − p2 = δ kat� H1 : p1 − p2 6= δ p1 − p2 < δ p1 − p2 > δ
Se aut thn perÐptwsh diakrÐnoume peript¸seican δ 6= 0
Z =p1 − p2 − δ√
p1(1− p1)/n + p2(1− p2)/m
an δ = 0
Z =p1 − p2√
p(1− p)(1/n + 1/m)
ìpou p = x+yn+m
1 an H1 : p1 − p2 6= δ tìte aporrÐptw th mhdenik upìjesh an Z > |za/2|2 an H1 : p1 − p2 > δ tìte aporrÐptw th mhdenik upìjesh an Z > za
3 an H1 : p1 − p2 < δ tìte aporrÐptw th mhdenik upìjesh an Z < −za
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diafor� twn analogÐwn
p1 − p2 dÔo anex�rthtwn plhjusm¸n
H0 : p1 − p2 = δ kat� H1 : p1 − p2 6= δ p1 − p2 < δ p1 − p2 > δ
Se aut thn perÐptwsh diakrÐnoume peript¸seican δ 6= 0
Z =p1 − p2 − δ√
p1(1− p1)/n + p2(1− p2)/m
an δ = 0
Z =p1 − p2√
p(1− p)(1/n + 1/m)
ìpou p = x+yn+m
1 an H1 : p1 − p2 6= δ tìte aporrÐptw th mhdenik upìjesh an Z > |za/2|2 an H1 : p1 − p2 > δ tìte aporrÐptw th mhdenik upìjesh an Z > za
3 an H1 : p1 − p2 < δ tìte aporrÐptw th mhdenik upìjesh an Z < −zaS. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diaspor� σ2 enìc plhjusmoÔ
H0 : σ2 = σ20 kat� H1 : σ2 6= σ2
0 σ2 < σ20 σ2 > σ2
0
X 2 =(n − 1)s2
σ20
(n − 1)s2 =∑
(xi − X )2 =∑
x2i − nX 2
1 an H1 : σ2 6= σ20 tìte aporrÐptw th mhdenik upìjesh an
X 2 < X 2n−1,1−a/2, X 2 > X 2
n−1,a/2
2 an H1 : σ2 > σ20 tìte aporrÐptw th mhdenik upìjesh an
X 2 > X 2n−1,a
3 an H1 : σ2 < σ20 tìte aporrÐptw th mhdenik upìjesh an
X 2 < X 2n−1,1−a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diaspor� σ2 enìc plhjusmoÔ
H0 : σ2 = σ20 kat� H1 : σ2 6= σ2
0 σ2 < σ20 σ2 > σ2
0
X 2 =(n − 1)s2
σ20
(n − 1)s2 =∑
(xi − X )2 =∑
x2i − nX 2
1 an H1 : σ2 6= σ20 tìte aporrÐptw th mhdenik upìjesh an
X 2 < X 2n−1,1−a/2, X 2 > X 2
n−1,a/2
2 an H1 : σ2 > σ20 tìte aporrÐptw th mhdenik upìjesh an
X 2 > X 2n−1,a
3 an H1 : σ2 < σ20 tìte aporrÐptw th mhdenik upìjesh an
X 2 < X 2n−1,1−a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diaspor� σ2 enìc plhjusmoÔ
H0 : σ2 = σ20 kat� H1 : σ2 6= σ2
0 σ2 < σ20 σ2 > σ2
0
X 2 =(n − 1)s2
σ20
(n − 1)s2 =∑
(xi − X )2 =∑
x2i − nX 2
1 an H1 : σ2 6= σ20 tìte aporrÐptw th mhdenik upìjesh an
X 2 < X 2n−1,1−a/2, X 2 > X 2
n−1,a/2
2 an H1 : σ2 > σ20 tìte aporrÐptw th mhdenik upìjesh an
X 2 > X 2n−1,a
3 an H1 : σ2 < σ20 tìte aporrÐptw th mhdenik upìjesh an
X 2 < X 2n−1,1−a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia th diaspor� σ2 enìc plhjusmoÔ
H0 : σ2 = σ20 kat� H1 : σ2 6= σ2
0 σ2 < σ20 σ2 > σ2
0
X 2 =(n − 1)s2
σ20
(n − 1)s2 =∑
(xi − X )2 =∑
x2i − nX 2
1 an H1 : σ2 6= σ20 tìte aporrÐptw th mhdenik upìjesh an
X 2 < X 2n−1,1−a/2, X 2 > X 2
n−1,a/2
2 an H1 : σ2 > σ20 tìte aporrÐptw th mhdenik upìjesh an
X 2 > X 2n−1,a
3 an H1 : σ2 < σ20 tìte aporrÐptw th mhdenik upìjesh an
X 2 < X 2n−1,1−a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to lìgo σ21/σ
22 twn diaspor¸n dÔo
anex�rthtwn plhjusm¸n
H0 : σ21/σ
22 = 1 κατά H1 : σ2
1/σ22 6= 1 ή σ2
1/σ22 < 1 ή σ2
1/σ22 > 0
F =s2
1
s22
1 an H1 : σ21/σ
22 6= 1 tìte aporrÐptw th mhdenik upìjesh an
F < Fn,m,1−a/2 F < Fn,m,a/2
2 an H1 : σ21/σ
22 > 1 tìte aporrÐptw th mhdenik upìjesh an
F < Fn,m,a3 an H1 : σ2
1/σ22 < 1 tìte aporrÐptw th mhdenik upìjesh an
F > Fn,m,1−a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to lìgo σ21/σ
22 twn diaspor¸n dÔo
anex�rthtwn plhjusm¸n
H0 : σ21/σ
22 = 1 κατά H1 : σ2
1/σ22 6= 1 ή σ2
1/σ22 < 1 ή σ2
1/σ22 > 0
F =s2
1
s22
1 an H1 : σ21/σ
22 6= 1 tìte aporrÐptw th mhdenik upìjesh an
F < Fn,m,1−a/2 F < Fn,m,a/2
2 an H1 : σ21/σ
22 > 1 tìte aporrÐptw th mhdenik upìjesh an
F < Fn,m,a3 an H1 : σ2
1/σ22 < 1 tìte aporrÐptw th mhdenik upìjesh an
F > Fn,m,1−a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to lìgo σ21/σ
22 twn diaspor¸n dÔo
anex�rthtwn plhjusm¸n
H0 : σ21/σ
22 = 1 κατά H1 : σ2
1/σ22 6= 1 ή σ2
1/σ22 < 1 ή σ2
1/σ22 > 0
F =s2
1
s22
1 an H1 : σ21/σ
22 6= 1 tìte aporrÐptw th mhdenik upìjesh an
F < Fn,m,1−a/2 F < Fn,m,a/2
2 an H1 : σ21/σ
22 > 1 tìte aporrÐptw th mhdenik upìjesh an
F < Fn,m,a
3 an H1 : σ21/σ
22 < 1 tìte aporrÐptw th mhdenik upìjesh an
F > Fn,m,1−a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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'Elegqoc upìjeshc gia to lìgo σ21/σ
22 twn diaspor¸n dÔo
anex�rthtwn plhjusm¸n
H0 : σ21/σ
22 = 1 κατά H1 : σ2
1/σ22 6= 1 ή σ2
1/σ22 < 1 ή σ2
1/σ22 > 0
F =s2
1
s22
1 an H1 : σ21/σ
22 6= 1 tìte aporrÐptw th mhdenik upìjesh an
F < Fn,m,1−a/2 F < Fn,m,a/2
2 an H1 : σ21/σ
22 > 1 tìte aporrÐptw th mhdenik upìjesh an
F < Fn,m,a3 an H1 : σ2
1/σ22 < 1 tìte aporrÐptw th mhdenik upìjesh an
F > Fn,m,1−a
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Sqèsh metaxÔ elègqwn upojèsewn kai diasthm�twn
empistosÔnhc
'Estw ìti èqoume ton èlegqo
H0 : µ = µ0 kat� H1 : µ 6= µ0
Pìte apodeqìmaste th mhdenik upìjesh?
−za/2 < Z < za/2
−za/2 <X − µ0
σ/√n< za/2
X −za/2σ√
n< µ0 < X +
za/2σ√n
H mhdenik upìjesh gÐnetai apodeqt se epÐpedo shmantikìthtac a ìtan htim µ0 an kei sto 100(1− a)% di�sthma empistosÔnhc gia th mèsh tim tou plhjusmoÔ.
O trìpoc autìc elègqei pollèc mhdenikèc upojèseic tautìqrona.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Sqèsh metaxÔ elègqwn upojèsewn kai diasthm�twn
empistosÔnhc
'Estw ìti èqoume ton èlegqo
H0 : µ = µ0 kat� H1 : µ 6= µ0
Pìte apodeqìmaste th mhdenik upìjesh?
−za/2 < Z < za/2
−za/2 <X − µ0
σ/√n< za/2
X −za/2σ√
n< µ0 < X +
za/2σ√n
H mhdenik upìjesh gÐnetai apodeqt se epÐpedo shmantikìthtac a ìtan htim µ0 an kei sto 100(1− a)% di�sthma empistosÔnhc gia th mèsh tim tou plhjusmoÔ.
O trìpoc autìc elègqei pollèc mhdenikèc upojèseic tautìqrona.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Sqèsh metaxÔ elègqwn upojèsewn kai diasthm�twn
empistosÔnhc
'Estw ìti èqoume ton èlegqo
H0 : µ = µ0 kat� H1 : µ 6= µ0
Pìte apodeqìmaste th mhdenik upìjesh?
−za/2 < Z < za/2
−za/2 <X − µ0
σ/√n< za/2
X −za/2σ√
n< µ0 < X +
za/2σ√n
H mhdenik upìjesh gÐnetai apodeqt se epÐpedo shmantikìthtac a ìtan htim µ0 an kei sto 100(1− a)% di�sthma empistosÔnhc gia th mèsh tim tou plhjusmoÔ.
O trìpoc autìc elègqei pollèc mhdenikèc upojèseic tautìqrona.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Sqèsh metaxÔ elègqwn upojèsewn kai diasthm�twn
empistosÔnhc
'Estw ìti èqoume ton èlegqo
H0 : µ = µ0 kat� H1 : µ 6= µ0
Pìte apodeqìmaste th mhdenik upìjesh?
−za/2 < Z < za/2
−za/2 <X − µ0
σ/√n< za/2
X −za/2σ√
n< µ0 < X +
za/2σ√n
H mhdenik upìjesh gÐnetai apodeqt se epÐpedo shmantikìthtac a ìtan htim µ0 an kei sto 100(1− a)% di�sthma empistosÔnhc gia th mèsh tim tou plhjusmoÔ.
O trìpoc autìc elègqei pollèc mhdenikèc upojèseic tautìqrona.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Sqèsh metaxÔ elègqwn upojèsewn kai diasthm�twn
empistosÔnhc
'Estw ìti èqoume ton èlegqo
H0 : µ = µ0 kat� H1 : µ 6= µ0
Pìte apodeqìmaste th mhdenik upìjesh?
−za/2 < Z < za/2
−za/2 <X − µ0
σ/√n< za/2
X −za/2σ√
n< µ0 < X +
za/2σ√n
H mhdenik upìjesh gÐnetai apodeqt se epÐpedo shmantikìthtac a ìtan htim µ0 an kei sto 100(1− a)% di�sthma empistosÔnhc gia th mèsh tim tou plhjusmoÔ.
O trìpoc autìc elègqei pollèc mhdenikèc upojèseic tautìqrona.
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011
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Mègejìc deÐgmatoc
Pìso prèpei na eÐnai to mègejoc tou deÐgmatoc ètsi ¸ste na
èqw di�sthma empistosÔnhc pl�touc d ?(X −
za/2σ√n, X +
za/2σ√n
)
d =2za/2σ√
n
S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 18 IanouarÐou 2011