INFERENCIA ESTADÍSTICA. INTERVALOS DE CONFIANZA

1. 5(/$&,Ï1 (VWLPDFLyQ
/D FRQFHQWUDFLyQ PHGLD GH GLy[LGR GH FDUERQR HQ HO DLUH HQ XQD FLHUWD ]RQD QR HV KDELWXDOPHQWH
PDRU TXH S S P Y SDUWHV SRU PLOOyQ HQ YROXPHQ 6H VRVSHFKD TXH HVWD FRQFHQWUDFLyQ HV
PDRU HQ OD FDSD GH DLUH PiV SUy[LPD D OD VXSHUILFLH 6H KD DQDOL]DGR HO DLUH HQ SXQWRV HOHJLGRV
DOHDWRULDPHQWH D XQD PLVPD DOWXUD FHUFD GHO VXHOR UHVXOWDQGR XQD PHGLD PXHVWUDO GH S S P Y
XQD FXDVLGHVYLDFLyQ WtSLFD GH 6XSRQLHQGR QRUPDOLGDG SDUD ODV PHGLFLRQHV GDU XQD HVWLPDFLyQ
SXQWXDO SDUD OD FRQFHQWUDFLyQ PHGLD GH GLy[LGR GH FDUERQR FHUFD GHO VXHOR FDOFXODU XQ LQWHUYDOR GH
FRQILDQ]D DO QLYHO GH FRQILDQ]D
6ROXFLyQ [ ], & =
$QWH OD VRVSHFKD GH XQD GLIHUHQFLD VLVWHPiWLFD HQWUH GRV ODERUDWRULRV $ % D OD KRUD GH GHWHUPLQDU
OD FDQWLGDG GH DOE~PLQD VpULFD H[SUHVDGD HQ JU PO VH KD UHDOL]DGR XQD H[SHULHQFLD FRQVLVWHQWH
HQ OD H[WUDFFLyQ GH VDQJUH D SDFLHQWHV 3DUD FDGD PXHVWUD GH VDQJUH VH PLGLy WDO SURWHtQD HQ
DPERV ODERUDWRULRV ODV GLIHUHQFLDV HQWUH ODERUDWRULRV $±% IXHURQ ODV VLJXLHQWHV
±
&RQVLGHUDQGR QRUPDOLGDG FDOFXODU XQ LQWHUYDOR GH FRQILDQ]D SDUD OD GLIHUHQFLD PHGLD GH PHGLFLyQ DO
QLYHO GH FRQILDQ]D FRQVLGHUDQGR TXH OD GHVYLDFLyQ WtSLFD GH ODV GLIHUHQFLDV SREODFLRQDOHV HV
$ HVWH QLYHO GH FRQILDQ]D ¢TXp WDPDxR PtQLPR GH PXHVWUD GHEHUtDPRV WRPDU SDUD TXH OD DPSOLWXG
GHO LQWHUYDOR IXHVH PHQRU R LJXDO TXH OD PLWDG GHO DQWHULRU"
6ROXFLyQ [ ], & = Q ≥
3DUD HYDOXDU OD YLDELOLGDG GH XQ SURHFWR GH UHIRUHVWDFLyQ GH XQD ]RQD VRPHWLGD D VWUHVV WXUtVWLFR
SDUD HO TXH VH KD VROLFLWDGR XQD VXEYHQFLyQ S~EOLFD VH DQDOL]D OD FRPSRVLFLyQ HQ PJ SRU FP GH
GHVHFKRV RUJiQLFRV GHO WHUULWRULR /RV GDWRV TXH VH REWLHQHQ VRQ
&RQVLGHUDQGR QRUPDOLGDG HVWLPDU PHGLDQWH XQ LQWHUYDOR GH FRQILDQ]D OD GLVSHUVLyQ GH OD
GLVWULEXFLyQ GH ORV GDWRV FRQVLGHUDU XQ QLYHO GH FRQILDQ]D GH
6ROXFLyQ [ ], & =
&RQ HO ILQ GH HVWXGLDU HO HIHFWR GH ORV UDRV ; VREUH OD YLDELOLGDG KXHYR ODUYD HQ ³7ULEROLXP
FDVWDQHXP´ VH LUUDGLDURQ KXHYRV GH ORV TXH UHVXOWDURQ ODUYDV +DOODU XQ LQWHUYDOR GH
FRQILDQ]D SDUD OD SURSRUFLyQ GH ODUYDV HQ KXHYRV LUUDGLDGRV DO QLYHO GH FRQILDQ]D GHO
6ROXFLyQ [ ], & =

2. /D VLJXLHQWH WDEOD SURSRUFLRQD GDWRV VREUH OD SUHFLSLWDFLyQ WRWDO UHJLVWUDGD HQ HVWDFLRQHV
PHWHRUROyJLFDV GH GRV SURYLQFLDV HVSDxRODV
3URYLQFLD $
3URYLQFLD %
6XSRQLHQGR LQGHSHQGHQFLD QRUPDOLGDG GDU XQD HVWLPDFLyQ PHGLDQWH XQ LQWHUYDOR GH FRQILDQ]D
DO QLYHO GH FRQILDQ]D GH SDUD
D &RFLHQWH GH YDULDQ]DV GH OD SOXYLRVLGDG HQWUH ODV GRV SURYLQFLDV
E 'LIHUHQFLD GH ODV PHGLDV GH OD SOXYLRVLGDG HQWUH ODV GRV SURYLQFLDV VL VH VDEH SRU H[SHULHQFLD
SUHYLD TXH OD YDULDQ]D GH ODV SUHFLSLWDFLRQHV HQ OD SURYLQFLD $ HQ OD SURYLQFLD % VRQ LJXDOHV
F 'LIHUHQFLD GH ODV PHGLDV GH SOXYLRVLGDG VL VH VDEH SRU H[SHULHQFLD SUHYLD TXH OD YDULDQ]D GH ODV
SUHFLSLWDFLRQHV HQ OD SURYLQFLD $ HV GH HQ OD SURYLQFLD % GH
6ROXFLyQ D [ ], & = E [ ], & = − − F , & > @
0XFKRV DXWRUHV DILUPDQ TXH ORV SDFLHQWHV FRQ GHSUHVLyQ WLHQHQ XQD IXQFLyQ FRUWLFDO LQIHULRU D OD
QRUPDO GHELGR D XQ ULHJR VDQJXtQHR SRU GHEDMR GH ORV QLYHOHV KDELWXDOHV $ GRV PXHVWUDV GH
LQGLYLGXRV XQRV FRQ GHSUHVLyQ RWURV QRUPDOHV VH OHV PLGLy XQ tQGLFH GHO IOXMR VDQJXtQHR FHUHEUDO
GDGR HQ PJ J PLQ REWHQLpQGRVH
7DPDxR PXHVWUDO 0HGLD PXHVWUDO 'HVYLDFLyQ WtSLFD PXHVWUDO
'HSUHVLYRV
1RUPDOHV
+DOODU XQ LQWHUYDOR GH FRQILDQ]D DO QLYHO GH FRQILDQ]D GH SDUD OD GLIHUHQFLD GHO tQGLFH PHGLR
GH IOXMR VDQJXtQHR FHUHEUDO FRQVLGHUDQGR QRUPDOLGDG H LQGHSHQGHQFLD
6ROXFLyQ [ ], & = − −
(Q XQD SLVFLIDFWRUtD VH GHVHD FRPSDUDU HO SRUFHQWDMH GH SHFHV DGXOWRV TXH PLGHQ PHQRV GH FP
FRQ ORV TXH PLGHQ PiV GH FP 3DUD HOOR VH WRPD XQD PXHVWUD GH SHFHV REVHUYDQGR TXH GH
HOORV PLGHQ PHQRV GH FP PiV GH FP +DOODU XQ LQWHUYDOR GH FRQILDQ]D SDUD OD GLIHUHQFLD
GH SURSRUFLRQHV DO QLYHO GH FRQILDQ]D GHO
6ROXFLyQ [ ], & = − −
&RQWHVWD UD]RQDGDPHQWH D HVWDV SUHJXQWDV
D ¢&XiO WLHQH PDRU DPSOLWXG XQ LQWHUYDOR GH FRQILDQ]D DO QLYHO y "
E ¢&XiO HV HO HIHFWR GHO WDPDxR PXHVWUDO VREUH OD DPSOLWXG GH XQ LQWHUYDOR GH FRQILDQ]D"
6ROXFLyQ D (O TXH WLHQH XQ QLYHO GH FRQILDQ]D GH E 0DRU WDPDxR PXHVWUDO PHQRU DPSOLWXG
FRQ HO PLVPR QLYHO GH FRQILDQ]D
(Q XQD SLVFLIDFWRUtD KD XQD SURSRUFLyQ GHVFRQRFLGD GH SHFHV GH XQD HVSHFLH $ 3DUD REWHQHU
LQIRUPDFLyQ VREUH HVD SURSRUFLyQ VH VDFDQ SHFHV GH ORV FXDOHV VRQ WLSR $ (VWLPDU GLFKD
SURSRUFLyQ PHGLDQWH XQ LQWHUYDOR GH FRQILDQ]D DO QLYHO GH FRQILDQ]D
6ROXFLyQ [ ], & =

3. (Q XQD H[SHULHQFLD JHQpWLFD VH H[WUDHQ PRVFDV GH XQD FDMD H[SHULPHQWDO 0HGLGD OD ORQJLWXG GHO
DOD HQ FDGD PRVFD VH REWXYLHURQ ORV VLJXLHQWHV YDORUHV
6XSRQLHQGR TXH OD ORQJLWXG GHO DOD VLJXH XQD GLVWULEXFLyQ QRUPDO KDOODU XQ LQWHUYDOR GH
FRQILDQ]D GH QLYHO SDUD ORV SDUiPHWURV ȝ ı
6ROXFLyQ [ ], & µ = ( ) [ ], & σ =
4XHUHPRV FRPSDUDU GRV PpWRGRV UiSLGRV SDUD HVWLPDU OD FRQFHQWUDFLyQ GH XQD KRUPRQD HQ XQD
VROXFLyQ 7HQHPRV GRVLV SUHSDUDGDV HQ HO ODERUDWRULR YDPRV D PHGLU OD FRQFHQWUDFLyQ GH FDGD
XQD FRQ ORV GRV PpWRGRV 6H REWLHQHQ ORV VLJXLHQWHV UHVXOWDGRV
'RVLV
0pWRGR $
0pWRGR %
D &DOFXODU XQ LQWHUYDOR GH FRQILDQ]D DO QLYHO SDUD HO FRFLHQWH GH YDULDQ]DV
E 6XSRQLHQGR TXH ODV YDULDEOHV VRQ LQGHSHQGLHQWHV VLJXHQ GLVWULEXFLRQHV 1RUPDOHV ODV
YDULDQ]DV SREODFLRQDOHV VRQ LJXDOHV FDOFXODU XQ LQWHUYDOR GH FRQILDQ]D DO QLYHO SDUD OD GLIHUHQFLD
GH FRQFHQWUDFLRQHV PHGLDV
6ROXFLyQ D [ ]YDU, & &RFLHQWH GH LDQ]DV =
E [ ], & 'LIHUHQFLD GH PHGLDV = −
(Q HO DQiOLVLV GH XQ SLJPHQWR FRQWHQLGR HQ XQD FLHUWD IORU GH XQD SODQWD YHJHWDO VH REWXYLHURQ ORV
VLJXLHQWHV UHVXOWDGRV H[SHULPHQWDOHV H[SUHVDGRV FRPR PJ GH SLJPHQWR SRU JUDPR GH IORU
&RQVLGHUDQGR QRUPDOLGDG FDOFXODU XQ LQWHUYDOR GH
FRQILDQ]D SDUD HO Q~PHUR PHGLR GH PJ GH SLJPHQWR SRU JUDPR GH IORU DVt FRPR SDUD VX YDULDQ]D
QLYHO GH FRQILDQ]D
6ROXFLyQ [ ], & µ = ( ) [ ], & σ =
8Q JUXSR GH LQYHVWLJDGRUHV HVWi LQWHUHVDGR HQ FRQRFHU OD FRQFHQWUDFLyQ PHGLD GH XQD HQ]LPD HQ
FLHUWD SREODFLyQ GH DOJDV 6H VDEH SRU H[SHULHQFLDV SUHYLDV TXH OD YDULDQ]D GH OD FRQFHQWUDFLyQ GH
HVWD HQ]LPD HV GH 6L VH REWLHQH XQD PXHVWUD GH WDPDxR GiQGRQRV XQ QLYHO GH FRQFHQWUDFLyQ
PHGLD GH FDOFXODU XQ LQWHUYDOR GH FRQILDQ]D DO QLYHO VH VXSRQH QRUPDOLGDG
6ROXFLyQ [ ], & =
6H FRQVLGHUD XQ H[SHULPHQWR SDUD HVWXGLDU VL OD WHUDSLD FRJQLWLYD HV PiV HIHFWLYD SDUD OD GHSUHVLyQ
TXH OD SVLFRWHUDSLD SVLFRGLQiPLFD 6H FRQVLGHUDQ GRV PXHVWUDV GH SHUVRQDV FDGD XQD VREUH ODV TXH
VH UHDOL]D FDGD WHUDSLD 7UDV VHPDQDV GH WHUDSLD OD PHMRUtD HQ FDGD SDFLHQWH VH FRPSUXHED (VWD
PHMRUtD HV PDUFDGD SDUD FDGD SDFLHQWH FRPR VLJXH
&RJQLWLYD
3VLFRGLQiPLFD
&RQVLGHUDQGR QRUPDOLGDG H LQGHSHQGHQFLD VXSRQLHQGR ODV YDULDQ]DV LJXDOHV HVWLPDU OD GLIHUHQFLD
GH PHGLDV HQ HIHFWLYLGDG FDOFXODU XQ LQWHUYDOR GH FRQILDQ]D DO QLYHO
6ROXFLyQ [ ], & =

4. /RV LQYHVWLJDGRUHV GH OD (QYLURQPHQWDO 3URWHFWLRQ $JHQF (3$ VH LQWHUHVDQ SRU OD FDOLGDG GHO
DLUH 8QR GH ORV LQGLFDGRUHV GH OD FDOLGDG GHO DLUH HV HO Q~PHUR PHGLR GH PLFURRUJDQLVPRV GH
SDUWtFXODV HQ VXVSHQVLyQ SRU PHWUR F~ELFR GH DLUH (V GHFLU HO LQWHUpV VH FHQWUD HQ µ OD PHGLD GH OD
YDULDEOH DOHDWRULD ; Q~PHUR GH PLFURRUJDQLVPRV GH SDUWtFXODV HQ VXVSHQVLyQ SRU PHWUR F~ELFR GH
DLUH 3DUD FRQWURODU OD VLWXDFLyQ VH KDFH XQD OHFWXUD FDGD VHLV GtDV H[WUDHQGR XQ PHWUR F~ELFR GH
DLUH D WUDYpV GH XQ ILOWUR GHWHUPLQDGR HO Q~PHUR GH PLFURRUJDQLVPRV GH SDUWtFXODV HQ VXVSHQVLyQ
FRQFHQWUDGDV HQ pO 'HVSXpV GH XQ SHUtRGR GH WUHLQWD GtDV VH KD JHQHUDGR XQD PXHVWUD DOHDWRULD GH
WDPDxR /RV YDORUHV REVHUYDGRV GH HVWDV YDULDEOHV SDUD HO SHUtRGR GDGR GH GtDV VRQ
+DOODU XQ LQWHUYDOR GH FRQILDQ]D DO QLYHO GH FRQILDQ]D SDUD HO Q~PHUR PHGLR GH
PLFURRUJDQLVPRV GH SDUWtFXODV HQ VXVSHQVLyQ SRU PHWUR F~ELFR GH DLUH SDUD ORV VLJXLHQWHV VXSXHVWRV
D (V FRQRFLGR SRU H[SHULHQFLDV SUHYLDV TXH OD YDULDEOH HQ HVWXGLR HVWi QRUPDOPHQWH GLVWULEXLGD FRQ
YDULDQ]D SREODFLRQDO LJXDO D
E (V FRQRFLGR SRU H[SHULHQFLDV SUHYLDV TXH OD YDULDEOH HQ HVWXGLR HVWi QRUPDOPHQWH GLVWULEXLGD
6ROXFLyQ D [ ], & = E [ ], & =