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R*  H  0޽h ? ̙33H p( <(   0p+? P   ? Z* b     0@    ? \* b     6@ `P  @ Z* b     6@ `  @ \* b   H  0޽h ? ̙33l 0( ,   6Z?00>p` Contraste de Hipotesis. b(b(4      j  0`?z f  Para contrastar nos hace falta medir y para medir necesitamos una regla (una distribucion) que nos diga que es pequeo y que es grande.o 2M   -  H  0޽h ? 999MMMV  Pp(    B?fԔ   ? r S ?< ?  - 6n Eb(G- . 6?nd D&Z- / 6x?} A1$ * 68?` s  Gs*ZG9 + 6? (  @"P9 , 6d@i  =2  > <@;  Eb(G# ? < @Nj  Ab$G# @ <@ 6 B  B)=$C& A <t@  PE(2G&C& B <@  =1  C <@Y  A1$ D <@7L e @"Z# G <#@* Zb2<G(O( H <)@  B)=$G( I <$-@Ry) Eb(G( J 6\1@") PE(2G(C( L <6@Bc =2  M <(:@"| D&Z( 3 6L>@\ "  Eb(G- 4 6B@ d D&Z- 5 6F@ p  A2$X l 0 m 6J@ d eficientes   `   ~ n BA ?00)X?0 `  ~ o BA ?00)X?`  ` p c $A I??P  IH  0޽h ? ̙33  ~'h (    8 L  bp  ! 6@uL o  A($G9 " 6@9 L o  A)$G9 # 08 M }  As$G+3 $ 6w   Cb&xG3 % 6 M  Ab$G& & 6$bk  Ab$G& ' 6   ?2"Z3 ( 6  f  > 3 ) 6X    ?1"x3 * 6t  =1  + 6#w  A1$ , 6(0 | }  =, & - 6+W3p  =N & . 6\0 p  =~ & / 63J)  @"Z& 8  9 N  cp   0 68 .  A($G8 1 6< 9  A)$G8 2 0`A P  As$G+3 3 6 6uV A  @"Z&l  C Lw p  8 A e W <{zWz BP  Z <}A BZ  ] 6p` Zv 4. Supongamos que ui~N(0, s2) Y|X ~N(b1 + b2X , s2) L< b   "CKC"C C CKCKt         g 0e0eԔ p   H  0޽h ? ^WNff3  -U/(  ~ s *`fg   l C   8 l  U0 V 4 6l4 Eb(G  5 6h D&Z  6 6 I Y A1$ 7 6 i 5 Gb*nG  8 6  H C1&n 9 6|lQ Eb(G  : 6=3 D&P  ; 6ܶ #H  A2$ < 6( 3i f Eb(G  = 6x   A2$XB  @ 0D     B CDEdF,33pp`0ppppppp @0    PX   @        |< XB # 0D  J 6] A($G& K 6] A)$G& L 6,OYe Ab$G M 6;65 @"Z N 6R mh =f  O 6H| A2$  & <hMP P  XDensidad 2 dB + <D ll ^B , 6D <<  > 6 M Ed(G33 ? 6P L E-(G33 @ 6 y Eb(G33 A 6  D&Z33 B 6 A2$ 33 E 6 N Ed(G33 F 6 J E+(G33 G 6 w Eb(G33 H 6X  D&Z33 I 6 A2$ 33 C 6 '  Eb(G D 6Hk/ [  A2$ ^B 0 6D l  P 6( )5 Eb(G Q 6l D D&Z R 6| \ A2$ ^B 3 6D|   ^ S 6LY_0F T 6x0  ,Intervalo aleatorio (intervalo de confianza)B- +fd`N        H  0޽h ? f̙     ;F ( x x c $()p0    <*^' Ed(G"ff  <,' A-$G"  <3! Cb&ZG"  <X7< @"P"  <p;n =2  ! <?c] Ad$G"f " <Cc A+$G" # <G~ Ab$G" $ <Kh< @"Z" % <O` =2 ^r 4 60V  5 < T  j0.99 0.95 0.90, d`   7 0V `   f 8 s *A ?00? @  f 9 s *A ?00?` f : s *A ?00?  f ; s *A ?00?0 0 ` p  H  0޽h ? ^WNff33  22M\ R2( '^  r  S ^   r  S x_ @  4B " # [ R\  # 6at 7  A$G' $ 6e  A$G' % 6\iO  Es(G'33 & 6mq @4  A=$G' ' 6|qv   As$G'3 ( 0u =  ]donde" ) 6pzq 9'  Eb(G33 * 6~   ?x"' + 6 I  =n ' , 6`0F  =x ' - 6 Ua  =2  . 6p &  ?i" / 6ē  =2  0 6ܗ b  ?i" 1 6  =2 3 2 6 | :2  A2$33 3 60\ 6  D&Z33 4 6| 9iT  A1$ 33 C 6 3 A($G7 D 6԰ t  A)$G7 E 6  As$G,3 F 6 (  Cb&nG3 G 6 T Ab$G' H 64% G Gb*nG' I 6X 4  323 J 6 f 32 K 6pO 32 L 6   D&Z33 M 6 ]. ^  =2  3 N 6\* ` =, ' O 6* ~ =N ' P 6* 6 =~ ' Q 6 *] @"Z'  6 p  A($G7  64 E p  A)$G7  6Lw |  Es(G,33  6pK ~  Eb(G33  6 -  Ab$G'  6 G  Eb(G'  6 pv  ;23  6\ )B  31  6p q 8W  31  6,   B$33  6X  @  =1  3  68 <7  =, '  6 W7  =N '  6 7  =~ ' ! 6h#| *  @"Z'4B 5 #  J  6 6'  A$G' 7 6d+q @  Es(G'33 8 6/ s A=$G' 9 6x4 ;  Es(G'33 : 08P  ]donde" ; 6< X Eb(G33 < 60B$ ?x"' = 6C  ;2 > 6LH ;i ? 6L^ UD  323 @ 6XO m  323 A 6$S M D&Z33 B 6Wvd =2  3 R 6\* A($G7 S 6_ >  A)$G7 T 6d Es(G,33 U 6i* ;23 V 6,m 3u3 W 6Ppo ;i X 6t;@ =, ' Y 6(y;T5 =O '00 Z 6}0 =N ' [ 6t;- =~ ' \ 6 b =U 'H  0޽h ? f̙  %$O( s $r $S $`@    $ B CDEdF,pp`0ppppppp @0    PX   @        P ^B  $ 6Dp^B  $ 6DXB  $ 0D @ @ @ $ 6Z A($G& $ 6W A)$G& $ 6CAY Eb(G $ 6/*) @"Z $ 6F =f   $ 6@<p A2$  $ 6s = Eb(G $ 6_ XY D&Z $ 6p3 A2$  $ 6 f  A($G' $ 6| k  A)$G' $ 6 OP  o X b:GG $ 68O o B =$G $ 6tf O  Ab$G $ 6R 6 U D&Z $ 6i  =E  $ 6LJ  =2  $ 6 l  A2$ #$ 0hf  El valor estimado de b2 puede caer en cualquiera de estas regionesPE `flf,`                 ^B $$@ 6Df0  ^B %$@ 6Df0  H $ 0޽h ? f̙A  %((   ( ( B CDEdF,pp`0ppppppp @0    PX   @         p@` ^B ( 6Dp^B ( 6D 8   j (x < ( 6E  A($G% ( 6;  A)$G%<B ( # ` ( 6{ Eb(G ( 6 ?  Eb(G ( 69   A-$G ( 6X   Eb(G ( 6( g +a A=$G ( 6  @"Z ( 6n> >Se  ( 6tq   D&Z ( 64  y =Z  ( 6@j A2$ ( 6$!@o   A2$ ( 6%}  A2$  ( ZgֳgֳA# ? ?PS   ( ZgֳgֳA ? ?p 2 XB  ( 0D P P @ #( 0)f 0 gEstandarizacion `  ] %( B-m 7Construccion de intervalos de confianza para bi (cont.)L8-G  D    H ( 0޽h ? f̙u   %  0 ( V} 0r 0S 5 @  $ 0  BP CDEdF,33  zS[AnU8n:H "t  2 2 g|  P @        v@ ^B 0 6D `  dB 0 <D   0 ZgֳgֳA' ? ? z    0 ZgֳgֳA* ? ? `  dB  0 <D    0 ZgֳgֳA) ? ?   dB  0 <D    0 BpCPDEFP\` 0p0@  @ P  0 ZgֳgֳA+ ? ?    0 B@CDEF@ @8@   0p  0 ZgֳgֳA+ ? ? @   0 ZgֳgֳA, ? ?   0 BCDE(F 0 `p 0 @ @    @  0 0<0p6 Region de aceptacion0 2e3a Q 0 6? 7Construccion de intervalos de confianza para bi (cont.)L8-G  D    H 0 0޽h ? f̙.  ..G`47.(   4  4 6IK , A($G'  4 6HNK  A)$G'4B  4 #    4 6LR j >95 ! 4 6 W Wj =. ! 4 6[  j =0 ! 4 6_ j >96 ! 4 6Td wj =. ! 4 6h +j =1 ! 4 6m k 2  =) ! 4 6`r  > ! 4 6w 2  =( ! 4 6{ 2  >Se ! 4 6< G Y @"Z! 4 6` m% j >96 ! 4 6 j =. ! 4 6 j =1 ! 4 60 j >Pr ! 4 6H >95 ! 4 6`l =. ! 4 6x =0 ! 4 6>  >96 !  4 6  =. ! !4 6U p  =1 ! "4 696 ! $4 6d =. ! %4 6:U =1 ! &4 6L~ >Pr ! '4 6 r S  =2  (4 6  =2  )4 6L t  A2$ *4 6gm A=$G! 34 6 A m A<$G! 44 6p  =  Cb&nG! 54 6? ~ | Ab$G! 64 6T7 v  A-$G! 74 6Q [  Eb(G! 84 6 D m A<$G! 94 6!m A-$G! :4 6@m A$G! ;4 6| A=$G! <4 6  A<$G! =4 6  A<$G! >4 6$ t A-$G! 4 0  0  intervalo de confianza al 95 %$ 2 f $  4B A4 #    B4 6$S   >96 ! C4 6S P  =. ! D4 6S   =1 ! E4 6 P : x =) ! F4 6# M @"n! G4 6\(  x =( ! H4 6- U x >Se ! I4 61   @"Z! J4 66S !  >96 ! K4 6<;S   =. ! L4 6?S 4T  =1 ! M4 6D D  =2  N4 6Se ! #8 6,(FNe >96 ! $8 6(Ce =. ! %8 6(ve =1 ! &8 6\ + @"Z! '8 6S + @"Z! (8 6(e >Se ! )8 6(aie >96 ! *8 6<(^e =. ! +8 6(e =1 ! ,8 6;+ @"Z! -8 6<. A2$ .8 63 =2  /8 6A  A2$ 08 6   =2  18 6R A2$ 28 6T 9 A($G. 38 6 9 A)$G. 48 6 z9 A($G. 58 6Di 9 A)$G. 68 6(0  Ab$G! 78 0p@ C&nG! 88 6hLA Ab$G! 98 6$!( A$G! :8 6X%`  D&P! ;8 6)E >Se ! <8 6$.emE >96 ! =8 62aE =. ! >8 66E =1 ! ?8 6:D  @"Z! @8 6>s  = =2  A8 6BR A2$"  8 0 Gc  8En la practica, es desconocido. Lo estimaremos con9 29 P   Q8 0L|C  Es(G%33 S8 6PJ A2$33 C8 6U M  CSCR$!3333 D8 6W @  =2 ! E8 6] #  =n ! H8 6bb ,  A2$NN L8 6tf g l /  A$G!338  >=  V8c > <B B8 # o$ >%  F8 6(k0   =u !N G8 6oB   > !N I8 6sG   =i N J8 6Pxj 4  =2 3 K8 0| @   @"Z%3 M8 6 =  A-$G! N8 6(0   A$G! O8 6\@@ }  A=$G! P8 6@   As$G%3 8 07 }m 6(En lugar de usar la distribucion Normal, tenemos que usar la distribucion-t. No olvidad que cuando  n es grande estas dos distribuciones coinciden.. =fJ   Q Y8 6 7Construccion de intervalos de confianza para bi (cont.)L8-G  D    H 8 0޽h ? f̙   *C<:(   <4B < # uWv4B < # { < 6   A($G. < 6   A)$G.4B < #  , 4B < # i+ ,gp 4B < # S ip 4B < # S  T 4B < #    < 6 7  Es(G%33 < 6\L #  A$G! < 6B J  Ab$G! < 6: 2  A-$G! < 6PS c  Eb(G! < 6 o  A=$G! < 6>  A=$G! < 6 Ab$G!  < 6 k< Ab$G! !< 6|t A-$G! "< 6M Ab$G! #< 61 A=$G! $< 6 `  D&Z%33 %< 6p{   ?x"! &< 6> X[  @"Z! '< 6 n  =t ! ,< 05  !desviacion standard del estimador$"!!$  4 0< 633P2 <estimado  verdadero parametro8 !!!6   1< 6$=  =t ! 2< 6 =) ! 3< 6| {) D&Z! 4< 6 v =( ! 6< 6 D&Z! 7< 6H* =t ! 8< 6M = +  =2  9< 6L P  =2  :< 6# i  A2$ ;< 6 (J =2  << 6,V =2  =< 6T1o =2 ] B< B5- 7Construccion de intervalos de confianza para bi (cont.)L8-G  D    ` C< c $A ?? H < 0޽h ? f̙  0"D!(  Dl DC =P   D ZgֳgֳA ? ?p+ `  D c $A ?? H   !D 0@c eUsamos el estadistico t para construir un intervalo de confianza para b2 , de la forma siguiente: f 2  . G O          ` "D c $A J?? `(  JH D 0޽h ? f̙@  ??@atT?(  l C N0P    6QB U  A($G-  6VBf v U  A)$G-4B  #    6tZ5a >90 !  6|^a =. !  6$bXfa =0 !  6Xf# B ! ?t"P!  6pj   @"Z!  6@n3@P @"Z!  6 q#(! ?t"P!  6uZa >Pr !  6(zV | ?c"F  6~p p 0  =2   6m  =2   6Pg =2   6< ?c"P  6^^ =2   6x 8^ =n   60a ^ =,   6 ^ >05   6  ^ =.   6XW ^ =0   6^^ =2  ! 6|7^ =n  " 6`^ =,  # 6^ >05  $ 6ľ^ =.  % 6>~D =0  & 6xb A=$G! + 6 Tj A$G! , 6$jL A$G! - 6jw  A$G! . 6(j A$G! / 6 h b A$G! 0 6 !  Cb&nG! 1 606 ] s Eb(G! 2 6.V  A-$G! 3 6GW Eb(G! 4 6 @b A$G! 5 6Vb A-$G! 6 6` A-$G 7 6` A-$G 9 6db  _  A($G, : 6Hb  _  A)$G, ; 6 b _  A($G, < 6@P   A)$G, = 6| \W  A($G= > 6 p:  A)$G= ? 6  >90   @ 6 m =.   A 6# 0* =0   B 6( P  >   D 6, ;5  =t   E 60 u  >   F 64 `  >   H 68 z  ?t"F  I 6 = U  >   J 6l@ Y8  >Pr   K 6D `  =2  L 6DI n  =c  M 6M  `  =2  N 6hR ^  ?c"F O 6V ``  =2  P 6Z w >  A2$ Q 6(_ - -  =2  R 6dc K\  =2  S 6g \  =n  T 6k ,\  =,  U 6 p B\  >05  V 6$t z\  =.  W 605  \ 6 u1  =.  ] 6̐ T1  =0  ^ 6 Y A=$G  _ 6   Ab$G  ` 6 8  E*(G a 6 7  A+$G  b 6 }   Ab$G  c 6< A 7  A$G  d 6p R  Eb(G  e 6 :  A$G  f 6< 0   Ab$G  g 6 Q"  ;* h 6̿ :  A-$G  i 6 ]  Ab$G  j 6 u]  A-$G k 6y .2  A-$GXb q 00 ` r c $A ?? ` s c $A ??   ` t c $A ??   H  0޽h ? ^WNff3R,  ,+PCT+(  l C  P  P8 <- = J1   6-Z  A($G/  60 -  A)$G/  6`   Ab$G"  6<L C*&G  6pr A$G"  6< Eb(G"  6{W A-$G  64p   > "  6 "   6U ?t"P"  6[ D&Z"  6 pD : =2   6D % ?c"<  6\ s0= =2   6f  =2   6@%fi  =n   6tf  =,    6  f  >05  ! 6X#f  =.  " 6'f  =0 8 @ M Lp } 7 <(,( @  A($G/ 8 <0{ @  A)$G/ 9 <4 6  Eb(G" < 689 %  > " = <= "  ? <@1  M =2 4B # # g    $ 6D #  A($G/ % 60I E J  A)$G/ & 6Md f  Ab$G" ' 6`RJ   C*&G ( 6 WJ v  A$G" ) 6$[d l  Eb(G" * 6_s  A-$G + 6c   Ca&FG , 6LhN s  @"Z" . 6l Ro  ?t"F" / 6pN s  @"Z" 0 6t$ @  =1  1 6x\   ?c"< 2 6|$ G  A1$ 3 6p 0t# =2  4 6u p =n  5 6 @b# =,  6 6< c ?2"Z  6T M  Eb(G"  6xO   A1$F H s *T @ ,El intervalo de confianza al 90% para b2 es:L-& E  P  28 p  Op@` ; <j  A-$G > <̥ ?t"P" @ <G8 ?c"< A <yc =2  B <poy =n  C <y& =,  D < ky >05  E <lFyh =.  F <yA =0 ` M 0p X N 0@ ` R c $A ??0` h @ ` S c $A ??  ` T c $A ??` p  H  0޽h ? ^WNff3(  ''`<MP6'( de P P 6ff P`   r PS @`0   P 6O W  A($G/  P 6`h A)$G/  P 6}] e  A($G/  P 6}PX A)$G/  P 6O: Eb(G"  P 6Xu A<$G" P 0S L Eb(G" P 6 @ : Eb(G" P 6H u A>$G" P 6@L Ab$G" P 63`y Eb(G" P 6$+ A$G" P 6Ec  Ab$G" P 63 Z y Eb(G" P 6 +9  A$G" P 6hE^ Eb(G" P 6 =~ A2$ P 6H A =2  P 6r   =1  P 6 F ~ A2$ P 6\$E =2  P 6(o =1  P 6,N A2$ P 60 R A2$  P 65t   =0  !P 69`  =2  "P 6h>d A2$ #P 6B =0  $P 6Gt  D&Z" %P 6DK  s =: " &P 6O/ v T ?H"Z" 'P 6xT' @"Z" (P 60Y ?s =: " )P 6]/ T ?H"Z" *P 6|b/ T @"Z" +P 6(gJ  =: " ,P 6@kn h  ?H"Z" -P 6Xo/ET D&Z" .P 6sJ] =: " /P 6$xn  ?H"Z" 0P 6|  6  A($G/ 1P 6 0 8 6  A)$G/4B 2P # w  x  3P 6< c  Ab$G" 4P 6̉ 1 :  Eb(G" 5P 6 u  A-$G" 6P 6h@L  Ab$G" 7P 63  A=$G" 8P 6,~ y  @"Z" :P 6D'  @"Z" ;P 6$>1  =t " P 6DE  A2$ IP 6 ` 0P  iValor calculado `  ` KP c $A ??   ` LP c $A ?? h  H P 0޽h ? f̙6     pj (  l C     x c $      `gֳgֳAF ? ?`    `gֳgֳAF ? ?`    < A  Ct&P%  <d Q  Cc&Z  <$  =H    <PP ?0"Z  <p`  =H    < p  =0   <P `@  CH&x   <T gN  C0&   `gֳgֳAF ? ?    0f @`  s Cola derecha `&     0Hf   uCola Izquierda `&    H  0޽h ? ^WNff3K @(  S  6?0 }Algunas hipotesis de interes: 1) Viven menos los pacientes viejos que los jovenes despues de un transplante de corazon? 2) Predicen las notas de selectividad los resultados academicos universitarios? 3) Es la propension marginal a consumir >.95? 4) Son las preferencias en vivienda Cobb-Douglas? 5) Es el crecimiento de la tasa media anual de la oferta monetaria superior al 7%?  2` bK2fZf3Qb.f002bUf                                                                  H  0޽h ? 999MMM  <(  x c $ (    0| P   H  0޽h ? !2ffm   0(  x c $P?   ? x c $xQ? 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