Diferencia de Proporciones

EJERCICIOS RESUELTOS RESUELTOS SERGIO PALACIOS Diferencia de Proporciones

Los Los homb hombr rss ! m"# m"#r rss $"& $"&'o 'oss r$( r$()$ )$o oss * "*$ "*$ )(" )("$ $  +r$* +r$*  & & *or *or' ' (, (, o(*(o*s sobr &$ rom"&+$)(* rom"&+$)(*  &$ *$  m"r' $r$ rso*$s )"&$b&s S )r )r /" & & 12%  &os hombr hombrs s $"&'os $"&'os s'* s'* $ $or $or   &$ *$ *$  m"r' m"r' /" /" s s&o 10% 10%   &$s &$s m"#r "#rs s $"& $"&'$ '$ss &o &o s' s' *. S( s s r r+"* +"*'$ '$ $ o os m" m"s'r s'r$ $s $& $& 100 hombrs ! 100 m"#rs s" o(*(* sobr &$ rom"&+$)(*  &$ *$  'rm(* 'rm(* &$ &$ rob$b rob$b(&($ (&($   /" /" & or)*'$ or)*'$# #  hombr hombrs s $ $or $or s$ $& $& m* /" &  &$s m"#rs

Solución:

S r)"r$ r)"r$ /" s s' (*)&"!*o (*)&"!*o  )orr )orr) ))( )(* *   0.5 0.5 or or sr sr "*$ "*$ (s' (s'r( r(b" b")( )( s s' "'(&($*o &$ (s'r(b")(*

Datos: PH =

0.12

PM =

0.1

nH =

100 Z=

nM =

100 0.025

100-12%= 100-10%=

0.88 0.9

0.11

EJERCICIOS RESUELTOS RESUELTOS SERGIO PALACIOS

r* * s"s  $ss(*$'o. 3 m(*'r$s &$'or($s  m"r'3 s 4% m$!or

& $)'or  * b(*om($& ! orm$&.

U*$ *)"s'$ & ;os'o* Co&&+ )o*s'  420 'r$b$#$ors  <()h(+$* /" s(os *'r 199 ! 198:3 *)o*'r /" 20% h$b>$* s'$o s(* 'r$b$#o "r m*os os $?os. S"*+$s /" '"(r$ /" s&))(o*$r o'r$ m"s'r$ $&$'or( 'r$b$#$ors  *'r 'oos &os m&$os s(os *'r 199 ! 198:. @C" rob$b(&($  /" s" or)*'$# m"s'r$&  'r$b$#$ors s(* m&o "r$*'  os $?os3 (,r$ & or)*'$# ob'*(o * &$ *)"s'$  ;os'o* Co&&+3 * 5 So&")(*B E* s' #r)()(o s )"*'$ *()$m*' )o* "*$ ob&$)(*3  &$ )"$& s s'* D' m"s'r$s ! s /"(r s$br &$ rob$b(&($  &$ (r*)($  &os or)*'$#s  m"s'r$s3 or &o /" s b  "'(&($r &$ (s'r(b")(* m"s'r$&  roor)(o*s )o /" s "*$ m(sm$ ob&$)(*. O'r$  &$s s('"$)(o*s )o* &$ )"$& *os 'o$mos s /" s)o*o)mos &$ roo 'r$b$#$ors s(os *'r 199 ! 198: /" s'"(ro* sm&$os or "*  &o m*os os $?os3 s&o s )o*o) &$ 1= 0.20 !$ /" $& 'om$r "*$ m"s'r$  420 'r$b$#$ors s obsr s$ r E* &$ rm"&$  &$ (s'r(b")(* m"s'r$&  roor)(o*s $r$ & )&)"&o  rob *)s('$ s$br &$s roor)(o*s  &$s ob&$)(o*s3 &$s )"$&s * s' #r)( s)o*o)mos3 or &o /" s "'(&($r & $&or  0.20 )omo "*$ s'(m$)(* "*'" s(+"(*' 'm$ s $bor$r & 'm$  s'(m$)(* s'$>s'()$ ! s )omr*r s'$mos "'(&($*o  s$ m$*r$ & $'o.

Datos: p1 = 0.20

0.2

n1 = 320 trabajadores

420

n2 = 320 trabajadores

420

P1 = P2

100-20%=62 0.05-70.05420

0.8 0.0:8:

Z=

1.54

 

"ro* *' or &o  420 & sr>$ &$ r &o m*os % o ms

$!*o os s$s os * P1= P23 !$ r)(*  r>oo  or or)(*. b(&($ s (o &$s &  P. E* & & or/"

S s$b /" 4  )$$  ro")'os $br()$os or &$ m/"(*$ 1 so* )'"oso )$$ 5 ob#'os $br()$os or &$ m/"(*$ 2 so* )'"ososF s 'om$* m"s'r$ ob#'os  )$$ m/"(*$B @)"& s &$ rob$b(&($  /" &$ roor)(*  $r'>)"&os )'"osos  &$ rb$s $ &$ m/"(*$ 1 * or &o m*os 0.10 Datos: P1 = 3/6=

0.5

P2 = 2/5=

0.: Z=

n1 =

120

n2 =

120

0.10-70.5120 P2-P1= 61= 62=

0.0958 -0.1 0.5 0.

4.0

 ! /" 2   120 /"(*$ 2

S s$b /" 4  )$$  ro")'os $br()$os or &$ m/"(*$ 1 so* )'"osos ! / 5 ob#'os $br()$os or &$ m/"(*$ 2 so* )'"ososF s 'om$* m"s'r$s  120 o )$$ m/"(*$B @)"& s &$ rob$b(&($  /" &$ roor)(*  $r'>)"&os )'"osos  &$ m/"(* &$ m/"(*$ 2 * or &o m*os 0.15

Datos: P1 = 3/6=

0.5

P2 = 2/5=

0.: Z=

n1 =

120

n2 =

120

0.15-70.15120

0.1:58

P2-P1=

0.1

61= 62=

0.5 0.

0.2

 2  )$$ b#'os  1 rb$s $

E* "* s'"(o $r$ )om$r$r &os sos rom(o  *(?os ! *(?$s sD'o +r s)"&$ r(m$r($ s "s$r "*$ m"s'r$ $&$'or($ 20 *(?os ! o'r$  25 *(?$s. '$*'o $r$ *(?os )omo $r$*(?$s &os sos s(+"* "*$ (s'r(b")(* *orm$&. E& &os sos 'oos &os *(?os  sD'o +r$o  s$ s)"&$ s  100 &(br$s ! s s'*$r s  1:.1:23 m(*'r$s /" & rom(o  &os sos 'o$s &$s *(? +r$o  s$ s)"&$ s  85 &(br$s ! s"s($)(* s'*$r s  12.2: rrs*'$ & rom(o &os sos  20 *(?os ! s & rom(o  &os s m"s'r$ 25 *(?$s3 *)"*'r &$ rob$b(&($  /" & rom(o  &os s *(?os s$ $& m*os 20 &(br$s ms +r$* /" &  &$s 25 *(?$s.  $'osB H1= 100 &(br$s

100

H2= 85 &(br$s

85

1= 1:.1:2 &(br$s

1:.1:2

2= 12.2: &(br$s*

12.2:

*1= 20 *(?os

20

*2= 25 *(?$ 71-220

25 Z=

Por &o ' rob$b(&($ rom(o  &$ m"s'r$ m*os 20 +r$* / m"s'r$  0.1

o * "*$ S s$b /" rom(o  s($)(* s & sD'o &(br$s. S( s  "*$ s &os 20

1.25

*'o3 &$   /" & &os sos  *(?os s$ $& &(br$s ms  &  &$ &$s*(?$s s 5.

E& $r'$m*'o  r)"rsos h"m$*os  "*$ +r$* mrs$ s$ )omrob$r s"s (*()$ors r rom(o  r*(m(*'o * &$s *'$s  s" rso*$&. Por (*orm$)(* $s$$ s s$b /" & >* r*(m(*'o  &os m&$os /" r)(b* "* )"rso  )$$)('$)(* s m$!or $&  $/"&&os /" )"rso. Ams s s$b /" &$ $r($*$ & r*(m(*'o  &os /" $s(s'* $& )"rso  )$$)('$) 1.::3 m(*'$s /" * $/"&&os /" *o $s(s'* s (+"$& $ 2.25. P$r$ )omrob$r s( &$ $,rm$)(* s(+ s 'om "*$ m"s'r$ $&$'or($  0 *ors $(s'r$os ob'*(*os "* >*()  r*(m( o'r$ $r'3 s s&))(o*$ro* 80 *ors *o )$$)('$os rs"&'$*o "* >*()  .85. A "* *( " $,rm$r /" &$ (r*)($ s m$*'(*

So&")(* S$ M1B N*() rom(o  r*(m(*'o  &os m&$os /" $s(s'* $ "* ) N*() rom(o  r*(m(*'o  &os m&$os /" *o $s(s'* $ "* )  Q21 = 1.::F  *K1=0 1 = .45  Q22 = 2.25 n2 = 80 2

 = 6.85

1.:: 0 .45 2.25 80 .85

 $)"ro $& rob&m$ bmos rso& oB M1  M2 1B M1  M2

s $r($*$s ob&$)(o*$&s so* )o*o)($s3 "s$rmos &$ (s'r(b")(* *orm$& E& s' 

=

Como s )o&$ $ &$ r)h$3 & $&or )r>'()o srB Z71- = 1.:5  &$ )om$r$)(*  s'os $&ors omos )o*)&"(r /" s r)h$$ &$ h(' /" s(+*(,)$ /" s )(r'o /" & >*() m(o  r*(m(*'o  &os m&$o & )"rso s s"r(or $&  $/"&&os /" *o &o r)(b*.

s)'o $& >*() () rom(o  *o $s(s'* $& (* s (+"$& $  s(*o )(r'$3 *'o  .45. Por & & 5% @s

"rso  M2B "rso. r & mo&oB

>s'()o  &$ 2.190

s(s *"&$3 &o /" r)(b*

E& rso*s$b& & s)'or  'r$b$#o s$ 'rm(*$r &$ r)"*)($  sm&o * )("$sB I)$ ! Ar/"($. V& sos)h$ /" * I)$ & sm&o s h$ (*)rm*'$o  $?os ms /" * Ar/"($. P$r$ )omrob$r s'o3 s r$&($ "* m"s'ro s&))(o*$ rso*$s * )$$ "*$  s'$s )("$s ! s *)"*'r$ /" * I)$ h$! 45 sm& Ar/"($.

Co* & 5%  s(+*(,)$)(* s " $,rm$r /" & rso*s$b&  ()ho s)'or So&")(* S$ W1 ! W2 &$s roor)(o*s  sm&o * &$s )("$s  I)$ ! Ar/"($3 r *1 *2

= = 

500 1 500 2 = 0.94

= =

0.0 0.05 0.05

1-1= 1-2= L$s h('s(sB oB W1  W2 E& *(&  sm&o * I)$ *o s m$!or /" * Ar/"($ 1B W1  W2 E& *(&  sm&o * I)$ s m$!or /" * Ar/"($ Es'$>s'()o  &$ r"b$B

0.95

Z)= 1.44 E& $&or )r>'()o $r$ "* mo&o  )o&$ $ &$ r)h$ s Z1- = 1.:5 Como ZC *o s m$!or /" Z1- *'o*)s *o s r)h$$ o. Es'o s(+*(()$ /" * sos)h$ & rso*s$b& & s)'or3 *o '(* r$*.

os +r$*s &os &'(mos *o $ 500 os ! 25 *

'(* r$* s)'($m*'.

o s )(r'$ &$

S s$ &&$r $ )$bo "* m"s'ro * L(*) ! Jss <$r>$ )o* & ob#'o   roor)(*  $m(&($s  $&'os (*+rsos m*s"$&s /" $/"(r* "* $r'$ *"os )o*#"*'os h$b('$)(o*$&s * )$$ (s'r('o.  "*$ m"s'r$ $&$'or( $r'$m*'os * o)"$os * L(*) r& /" 150 s'$b$* o)"$$s or $ (*+rsos m*s"$&s m(*'r$s /" * "*$ m"s'r$  400 $r'$m*'os s& $&$'or($m*' * Jss <$r>$ (*()$b$ /" 5:  &&os s'$b$* o)"$os or $&'os (*+rsos m*s"$&s. Tom$*o * )"*'$ s'os rs"&'$os3 $ "* *(&  s & 5% s " $,rm$r /" *o D(s' (r*)($ s(+*(,)$'($ * &$ roor)(* /" $/"(r* $r'$m*'os * $mbos (s'r('os So&")(* S$ W1 ! W2 &$s roor)(o*s  $r'$m*'os o)"$os or $m(&($s  $&'os L(*) ! Jss <$r>$3 rs)'($m*'. *1= *2= 

00 1= 400 2= =

0.25 0.18 0.05

L$s h('s(s oB W1 = W2 1B W1 X W2 Z)=

2.:8

S(*o "*$ r"b$ b(&$'r$&3 &os $&ors )r>'()os so*B Z2 = -1.9 ! Z1-2 = 1.  $)"ro $& )r('r(o  )(s(* $r$ s' mo&o3 bmos r)h$$r &$ h(' /" s(+*(,)$ /" s> D(s' (r*)($ s(+*(,)$'($ * & *mro  $m(&($s  $& m*s"$&s /" $/"(r* $r'$m*'os * $mbos (s'r('os.

'rm(*$r &$ *'o * &os  00 (&($s  $&'os ))(o*$os $m(&($s  (+*(,)$)(*  $m(&($s

(*+rsos *

 s(s *"&$3 &o 'os (*+rsos

U* $br()$*'  bomb(&&os sos)h$ /" "*$  s"s &>*$s  ro"))(* s' r bomb(&&os )o* "*$ "r$)(* rom(o m*or /" &$  o'r$ &>*$. P$r$ rob$r s" so "*$ m"s'r$ $&$'or($  1 bomb(&&os  &$ &>*$ sos)hos$ 7s ! 18  &$ o'r$ ob'*(*o &os s(+"(*'s rs"&'$osB &(* sos)h$s$

Ds= D)= *s= *s-1=

So&")(*

ZD-D)=

1.:2 Ss2= 1.: S)2= 1 *)= 15 *)-1=

92000 4000 18 1

15.544

 se calculan los grados de libertad de la distribución t

Y=

 

o")(*o )h$ 'om$ &>*$ 7)3

0.022199

I*s'(+$r s( h$! (r*)($ * &$s o(*(o*s  os +r"os  $*$&(s'$s3 "*o  mrs$ ! o'ro  )orr"r>$s.

* r&$)(* )o* &$ ,)(*)($  &os mr)$os o s$ &$ $#"()$)(*  o*os $r$ "so Prob$r &$ h('s(s $ "* 5%  *(&  s(+*(,)$*)($ o rror  /"(o)$rs * r)h$$r

$&ors *1 1

Corredores

Empresas

A*$&(s'$s 5 O(*(o*  41 mr)$os ,)(*'s p1 0.413 P$so 1. Es'$b&)(m(*'o  h('s(s

*2 2

110 :2 .8

Alfa/2 1-Alfa/2

p2

0.382

[r$))(* 

 Ho : π  1

− π  2 =

0....otra. forma.... Ho : π  1

= π  2

 Ha : π  1

− π  2 ≠

0.......... ................ Ha : π  1

≠ π  2

Por tanto se trata de una

P$so 2. C&)"&o & s'$>s'()o  r"b$ Z)  Z c

 p1

=

 p1 (1 −  p1 )

0.042

−  p 2 +

n1

 p 2 (1 −  p 2 )

=  Z) = 0.:29054

0.044:12

n2

 Tom$mos $ 1 )omo & m$!or $r$ )om$r$r Z) )o*'r$ Z os('($ P$so 4. 'rm(*$r &$ Z  ED)& o  '$b&$s $r$ Z 70.95 = 1.9599498

1-A&$2

0.95

IST.\OR<.STA\.I\ 70.

P$so :. Com$r$*o &os $&ors Z) )$&)"&$o )o*'r$ ZD)& 70.95 o 0.025 * & )$so *

P(Z>= + 1.6 ! = a"#a/2

P(Z$= % 1.6 ! = a"#a/2

Z70.025 = -1.9599498 Z) = 0.:29054

Z70.95 = 1.9599498 $&or  $r$ P7-Z) =

Como Zc es menor que Zexcel, no cae en el área de reca!o, " por #an#o no a" suficien#e e$idencia para reca!ar %o &e conclu"e que la opini'n de los corredores " empresarios no difiere

P$so $()(o*$&. E& I*'r$&o  )o*,$*$ & 95% sobr &$ (r*)($  m($s )o* s(+m$s s)o*o)($s sB  s p1− p 2 =

( p1

 p1 (1 −  p1 ) n1

+

 p 2 (1 −  p 2 ) n2

−  p 2 ) ±  Z α  / 2 s  p1− p 2

= Error s'*$r ZD)&

0.044:12 1.9599498

= I*'r$&o  )o*,$*$ 7

\o s obsr$ (rr*)($ *'r &$s roor)(o*s



0.042

±

(0.112 )* P+ )*

s ro")'(o. o.

0.025 0.975

&$s m"s'r$s

rueba de dos colas

71- A&$2

5 +$'(o s '(*

) s (+"$& $ 0.412095   A&$

  ob&$)(o*$&s3

0.1:4:24

0.1-