a b c d e f g h i j k
Observación 1 2 3 4 ! " # $ 1% 11 12
H & B es una empresa familiar que durante muchos años se ha estado vendiendo Se anuncian extensamente en radio y televisión, haciendo hincapié en sus baos "l propietario le #ustar$a estudiar la relación entre los in#resos por ventas %n #astadas en publicidad %n unidades de '((((). *a tab tabla la si# si#ui uien ente te res resum umee info inform rmac ació iónn de in#r in#res esos os por por ven venta tass y #ast #astos os de de pub publi li ealice un dia#rama de dispersión. stable-ca la variable dependiente y la inde etermine los coeficientes de re#resión a y b. /nterprete el el si#nificado de cad ealic ealicee el an!li an!lisis sis de de varian varian-a -a y deter determin minee la valid validee- del del modelo. modelo. %eal %ealice ice la la pru 0alcul 0alculee el coefi coeficie ciente nte de de correl correlaci ación ón y de deter determin minaci ación. ón. /nter /nterpre prete te el si#ni si#nific fic eaalice e lice el an!li n!lisi siss resi residu duaal. %1o %1orma rmalida lidad, d, Homo omoceda cedast stic icid idaad, /nde /ndeppende ndenci ncia) st st nula y alternativa, el valor del estad$stico, el valor 2 y la conclusión. 2lantee el modelo lineal encontrado. 3s v!lido este modelo4 xplique. Halle intervalos de confian-a del 657 para α y β 32uede considerarse que α > (4 2ara un #asto de publicidad de 5(((( de pesos mensuales mensuales pronostique el el in#res 667 para los in#resos y el intervalo de confian-a del 667 para los in#resos pro 3Se#+n su punto de vista, son apropiados los medios utili-ados por esta empres empres basado en los resultados anteriores. 30onsi 30onsider deraa uste usted, d, como como prof profesi esiona onall d dee su su campo campo,, que que se deb debee se#u se#uir ir implem implement ent en cuenta factores adicionales4 xplique .
X (Gastos) Y (Ingresos) 89.: ;(.85 <( ;= <9 6(.5 <;.: 68 :8 68.(8= :<.5 68.(8= :; ;6.5 :; 69.<9; :;.6 69.<9; :6.8 66.85 :6.8 66.85 :6.5 '(8.<
X^2 969.69 6(( '869 ':=:.59 '=9: ';68.85 8<(: 8<(: 8<6'.8' 8:8(.9: 8:8(.9: 8:5(.85
Y^2 9::(.' =596.( ;'6(.< ;:9:.( ;:96.( ;:96.( ;('(.< 68;9.; 68;9.; 6;5(.9 6;5(.9 '(:95.<
XY 8'';.9( 89'(.(( <85;.(( <5<8.;( <;95.'< :((<.'= :869.(( :985.99 :='8.:( :;;<.'( :;;<.'( 5(9<.;5
Ygorro ;'.;6 ;:.:5 ;;.=: 6(.:5 6<.(8 6:.(6 6=.<( 6=.<( 6=.6: 6;.'5 6;.'5 6;.<=
13 14 1 1! 1" 1# 1$ 2% 21 22 23 24 2 2! 2" 2# 2$ 3% 31 32 33 34 3 3! 3" 3# 3$ 4% 41 42 43 44 4 4! 4" 4#
55.8 59 59.5 5;.< 56.' 9< 9< 9; =(.; =(.; =5.< =5.9 =9 =9.; =6 =6.' ;'.8 ;< ;9.< ;;.; 6( 68 68.' 69 6=.9 6; 6;.5 6;.9 '(8.< '(: '(:.< '(:.: '(5 ''9.: '8<.9 '86.9
69.85 '(8.'59 6;.8 '(:.= '(' '89 '': '8( ''(.5 ''(.5 ''5.6(85 ''=.=5 ''5.6(85 '(=.5 '';.=695 ''5.6(85 '';.=695 ':9 '';.=695 '8<.5 '89.(<'5 '8: '8:.5;:5 '<'.(69 '<'.(69 '<8.5:< '<8.5:< '<'.(69 ':< ':' '<9.8 ':8 '<=.9(=5 '5'.5 '5< ':5.5
<(:=.(: <'<9 <'68.85 <<6;.;6 <:68.;' <696 <696 :98: 5('8.9: 5('8.9: 59=(.(6 5='5.<9 5==9 5;6;.8: 98:' 9859.;' 956<.:: 9;;6 =::=.96 =;;5.:: ;'(( ;:9: ;:;8.:' 68'9 6585.=9 69(: 6=(8.85 6=8'.69 '(:95.86 '(;'9 '(;=;.:6 '(;66.<9 ''(85 '<5:;.69 '58=9.69 '9=69.'9
689:.' '(:<5.; 69:<.8 '(698.' '(8('.( '5;=9.( '8669.( '::((.( '88'(.< '88'(.< '<:<<.: '<;95.' '<:<<.: ''559.< ':''8.9 '<:<<.: ':''8.9 8'<'9.( ':''8.9 '5858.< '5;;<.6 '5<=9.( '558'.< '=';9.8 '=';9.8 '=59=.9 '=59=.9 '=';9.8 8(::6.( '6;;'.( ';55(.: 8('9:.( ';6<5.; 88658.< 8<:(6.( 8''=(.<
5<'<.(( 5=8(.=: 55:;.<( 9'(:.(' 5696.'( =6<;.(( =';8.(( ;'9(.(( =;8<.:( =;8<.:( ;=8=.:9 ;6('.6( ;;(;.56 ;859.(( 6<;:.68 6'9=.;6 69:9.8; '8'';.(( '(858.': '(699.;( ''<:8.;: '':(;.(( '':=:.8< '85;5.88 '8=6:.6= '86;6.8' '<(55.:6 '8689.(= ':98;.6( ':99:.(( ':8(5.99 ':;8:.;( ':::;.=6 '=9<:.9( ';6'(.;( ';;59.;(
'(8.:< '(<.(' '(<.<9 '(:.95 '(5.88 '(;.(( '(;.(( '''.5= ''<.5= ''<.5= ''9.=; ''9.66 ''=.8; ''=.;5 ''6.:8 ''6.:6 '8(.66 '88.8= '8:.9< '89.:' '8=.8= '8;.96 '8;.=9 '<'.55 '<8.96 '<8.6= '<<.<< '<<.:( '<9.(: '<=.85 '<=.:= '<=.5: '<=.6= ':9.'( '5'.8: '55.58
4$ % &'as
'<:.: '<; 3#" &'a X
'5=.=5 ';(9<.<9 8:;;5.' 8'8('.9( '9: '6(:: 89;69.( 889<8.(( $%34% 331"1#1 "1"$4!4 4#224""%3 &'a Y &'a X^2 s'a Y^2 &'a XY
'5;.65 '9'.5'
180 160 140 120 100 80 60 40 20 0 20
40
n a b Sxx Syy Sxy
60
80
5( 9<.(:;85<8 (.='<5888; <=;:<.: 8(6(6.; 8=((8.'
100
S0? S0 S0 S@8 A
120
140
160
8(6(6.; '9:<.8 '6899.9
*ab+a ,-O., /'ente
&'a de c'adrados
e#resión rror
'6899.9 '9:<.8
?otal
8(6(6.;
0oeficiente de correlación
Grados de 0'adrados +ibertad edios ' :;
'6899.9 <:.8
fca+
fcrit
598.=;=6::; :.(:8958
0ov%,C) D%) D%C) r
5:(.( =59.6 :';.'695<5 (.6566(856
r r@8
(.6566(856< (.68':'86;=
-O,I, HoG los residuos o errores se austan a una distribución normal con media µ> ( H'G los residuos o errores no se austan a una distribución normal con media µ> n 5( E de clase = e eordenados e" :.6 F'.9( F'(.< F'.=9 2 (.' 8.5( F'(.( F'.=' '.;( F=.; F'.<< s@8 <:.8 '.9( F9.8 F'.(9 0lase *$mites F'.(( F5.; F(.66 ' F'(.< F5.5 F8.'( F5.8 F(.;6 8 F5.: F(.9 F=.;( F:.= F(.; < F(.5 :.< F(.6( F:.8 F(.=8 : :.: 6.8 F'.9( F:.8 F(.=8 5 6.< ':.' '.'( F<.9 F(.98 9 ':.8 '6.( '.'( F<.' F(.5< = '6.' <.6( F<.' F(.5< F9.8( F8.6 F(.5 F(.;( F8.< F(.<6 oi F5.8( F8.8 F(.<; 5 (.'( F8.' F(.<9 8: F:.8( F'.9 F(.8= ': ';.(( F'.9 F(.8= = 9.(( F'.9 F(.8= ;.:( F'.: F(.8: (.(' α F<.'( F'.< F(.88 Dalor 2 (.(';=9 F<.'( F'.8 F(.8' 2to cr$tico ''.<::;= F(.6( F'.8 F(.8' ecisiónG 0omo valor 2 es
(.;(
F'.(
F(.'=
s posible considerar que
F'.:(
F(.6
F(.'5
F'(.<(
F(.6
F(.'5
bservación G
F(.9( F<.9(
F(.; F(.;
F(.': F(.':
S@8 calculada de los resi
F8.8( 8<.=( F5.;( F8.6( F'.8( F:.=( F:.8( F(.5( F'.9( F(.:( F(.;( F8.<( =.(( <.=( F'.<( :.5( F(.:( 5.:( '.;( F'(.(( F'.8( 8.5(
F(.9 F(.5 F(.: F(.: (.' (.; '.' '.' '.9 '.; '.; 8.5 8.5 <.= <.6 :.5 5.: 9.( =.( ;.: ';.( 8<.=
F(.' F(.(6 F(.(= F(.(= (.(8 (.': (.'6 (.'6 (.8= (.<' (.<' (.:< (.:< (.9< (.9= (.== (.68 '.(< '.8 '.:: <.(; :.(5
S@8 estimada de los resi es sobreestimada
I-565-5-0I, HoG los residuos son independientes H'G los residuos son dependientes e F'.9:
%eiFeiF')@8
Errores estandarizados
ei@8 8.9=
eF(.8;
8.55 '.=9 '.55 F(.66 F8.(9 F=.;( F(.6< F'.5= '.'( '.'( <.6< F9.'; F(.;5 F5.'9 (.(5 F:.88 ';.(( 9.(( ;.:< F<.(= F<.(= F(.;= (.=9 F'.<= F'(.<5 F(.98 F<.56 F8.'6 8<.=< F5.;< F8.6' F'.8< F:.96 F:.'; F(.:5 F'.56
'=.:;<:<5< (.9'('96=; (.(:5'<9:= 9.:9('<;8 '.':55(95; <8.688685: :=.'96:8: (.:'8<;8<= =.''=68'5: ( ;.(:85=:5 '(8.<558:9 8;.:9:'9;6 ';.5666(;9 8=.8(<'(;8 ';.8<6;;:6 :6<.9(9=;' ':: 5.6'95'::' '<8.8((;<6 ( :.;(<<8;9= 8.99;'<=(= :.5:6<((:8 ;(.58(:<8= 6:.9(69;:6 ;.=6<<'<;< '.6:==(;<5 9='.;(8;5 ;=<.9;8995 ;.58:9':;= 8.;(95:(5< ''.69'5<(: (.89<<8(9: '<.6(<9=:: '.<(<<<'6:
9.:; <.'8 8.:' (.6; :.8: 9(.;( (.;9 8.:= '.8( '.8( '5.:9 <;.85 (.=8 89.95 (.(( '=.=6 <8<.66 <9.(( ='.'( 6.:( 6.:( (.=9 (.5; '.;6 '(=.(9 (.<; '8.;5 :.;( 59<.(; <<.6= ;.:9 '.58 88.(8 '=.:= (.8( 8.5<
(.:: (.<( (.8= F(.'= F(.<5 F'.<< F(.'9 F(.8= (.'6 (.'6 (.9= F'.(9 F(.'5 F(.;; (.(' F(.=8 <.(; '.(< '.:: F(.58 F(.58 F(.'5 (.'< F(.8< F'.== F(.'' F(.9' F(.<= :.(9 F'.(( F(.5( F(.8' F(.;( F(.=' F(.(; F(.8=
F(.:< F(.=6 F8.<' 9.69 <.=5 F'.8= :.:9 F(.<9 5.:( '.=9 F'(.(8 F'.8( 8.:6
'.<:686<;9 (.'6 (.'8=8=;5' (.98 8.<(5<6<:6 5.<8 ;5.;8'(65< :;.:8 '(.<8<86' ':.(< 85.':(=9:: '.9' <8.;'=:(5< '6.;6 8<.8<;<'<8 (.'< <<.'5;5:6< 86.': '<.8<(<6(9 <.'( '<;.=65'' '((.:8 ==.;;889=; '.:< '<.558''55 9.'; 3331#""# 1!4324 7
2%2"!2!
F(.(= F(.'< F(.<6 '.'6 (.9: F(.88 (.=9 F(.(9 (.68 (.<( F'.=' F(.8( (.:8 (.(( 0omo A se concluy
8OO05,&*I0I, HoG los residuos tienen varian-a constante H'G los residuos no tienen varian-a constante e F'.9: 8.55 '.=9 '.55 F(.66 F8.(9 F=.;( F(.6< F'.5= '.'( '.'( <.6<
eF(.8;; (.::; (.<'' (.8=< F(.'=: F(.<9< F'.<=< F(.'9: F(.8== (.'6< (.'6< (.96<
9 gorro ;'.;;58:' ;:.:5<688 ;;.=<5(55 6(.::=5(6 6<.('9';6 6:.(;9:=8 6=.86=<88 6=.86=<88 6=.6<6:6< 6;.'5<5:6 6;.'5<5:6 6;.<9=9(9
5.000 4.000 3.000
s o d a 2.000 z i r a d 1.000 n a t s e 0.000 s 70 o u -1.000 d i s e R -2.000
F9.'; F(.;5 F5.'9 (.(5 F:.88 ';.(( 9.(( ;.:< F<.(= F<.(= F(.;= (.=9 F'.<= F'(.<5 F(.98 F<.56 F8.'6 8<.=< F5.;< F8.6' F'.8< F:.96 F:.'; F(.:5 F'.56 F(.:< F(.=6 F8.<' 9.69 <.=5 F'.8= :.:9 F(.<9 5.:( '.=9 F'(.(8
F'.(;6 F(.'5( F(.6(6 (.((6 F(.=:< <.'=( '.(5= '.:;5 F(.5:( F(.5:( F(.'5: (.'<: F(.8:8 F'.;88 F(.'(6 F(.9<8 F(.<;9 :.';( F'.(8= F(.5'8 F(.8'= F(.;89 F(.=<9 F(.(=6 F(.8;( F(.(=9 F(.'<6 F(.:(9 '.889 (.99( F(.88< (.=;9 F(.(9: (.65' (.<'( F'.=95
'(8.:<:9; '(<.((55 '(<.<9889 '(:.9:99 '(5.8'=:8 '(;.((('9 '(;.((('9 '''.59=== ''<.5959< ''<.5959< ''9.==9:; ''9.66(5: ''=.8=565 ''=.;:9=9 ''6.:'95' ''6.:;=;= '8(.6;989 '88.8=(9 '8:.9858< '89.:(6(< '8=.89589 '8;.968< '8;.=9<95 '<'.5:9<6 '<8.9;;(< '<8.6=<:: '<<.<<(8 '<<.:('55 '<9.(:'5; '<=.85:5= '<=.:9;9< '<=.5<66; '<=.69;(6 ':9.'(885 '5'.8<69' '55.58(=:
-3.000
F'.8( 8.:6
F(.8'' (.:<;
'5;.6:595 '9'.5':<<
8a++e interva+os de confian=a de+ $< :ara α y β
s@8 n x barra Sxx D%") desves%") D%B) desves%B) t:;,(,(5 a b
<:.8 5( ==.'': <=;:<.: 9.(9:''6=' 8.:985:<<: (.(((6(:9< (.(<((==(: '.9==88:8 9<.(:;85<8 (.='<5888;
/ntervalo de confian-a del 657 para 5;.6'; 9=.'=; /ntervalo de confian-a del 657 para (.99<(=9 (.=9<69;
e+ ingreso :roedio; e+ interva+o de :redicción de+ $$< :ara +os ingresos 9
xo D%C) D%C#orro) t:;,(,(5
5 <6.98<<8<6 5.<;6':('= 8.('(9<:=9
/n#reso promedio yoI#orro 99.9'5;9:98 /ntervalo de confian-a del 657 para 9'.6:;8=8 ='.8;<:5=(: /ntervalo de predicción del 657 para 5<.6565' =6.8=88'6<:
a clientes minoristas en cierta ciudad. precios y f!ciles condiciones de crédito. unidades de '((((() y las cantidades
idad para un determinado n+mero de meses. endiente. uno de ellos. ba de hipótesis) do de cada uno de ellos. ableciendo en todos los casos las hipótesis
promedio, el intervalo de predicción del edio. para dar a conocer su producto4 xplique ndo este tipo de publicad, o hay que tener
e F'.9: 8.55 '.=9 '.55 F(.66 F8.(9 F=.;( F(.6< F'.5= '.'( '.'( <.6<
F9.'; F(.;5 F5.'9 (.(5 F:.88 ';.(( 9.(( ;.:< F<.(= F<.(= F(.;= (.=9 F'.<= F'(.<5 F(.98 F<.56 F8.'6 8<.=< F5.;< F8.6' F'.8< F:.96 F:.'; F(.:5 F'.56 F(.:< F(.=6 F8.<' 9.69 <.=5 F'.8= :.:9 F(.<9 5.:( '.=9 F'(.(8
F'.8( 8.:6
Inter:retación de +a :endiente> ";13 2or cada peso adicional que se invierte en publicidad los in#resos promedio por venta aumentar!n en =,'< pesos. Inter:retación de+ interce:to> !?3%4#2 0uando no se invierte en publicidad los in#resos promedio semanales son 9J<(:.;85
va+or 6 <.9=F(8;
varian-a σ^2 estimada con s^2 = 342 ( y varian-a σ^2 estimada con s^2 = 342
Kronteras F5.:5 F5.:5 F(.55 F(.55 :.<5 :.<5 6.85 6.85 ':.'5 ':.'5 '6.(5 '6.(5
ei
;.=; ':.<: '5.:5 ''.:8
Kronteras Li F(.6<8 F(.6<8 F(.(6: F(.(6: (.=:: (.=:: '.5;8 '.5;8 8.:8( 8.:8( <.85= <.85=
0.00
oi 5 8: ': 5 ( ' '
1
50
prob
0.176
0.287
0.309
0.172
0.049
0.007
ei
8.78
14.34
15.45
8.58
2.45
0.36
0.03 50
%oiFei)@8Mei '.9869;=' 9.:6656:9 (.'<9':;: '.=''6''= 6.6==<:';
menor que alfa recha-ar Ho, o el valor de la estad$stica es mayor que el punto cr$tic
los residuos se austan a una distribución normal con los par!metros dados.
uos
uos
<<.5(5<(9 <:.8<:
5.00 4.00 s o d a z i r a d n a t s e s e r o r r E
3.00 2.00 1.00 0.00 0
10
20
30
-1.00 -2.00 -3.00
Orden
40
50
!0
est! entre du>',56 y :Fdu > 8,:' entonces no se recha-a Ho, que los residuos son independientes.
90
110
130
150
170
e+ interva+o de confian=a de+ $$< :ara +os ingresos :roedio
gorro
C
o, recha-ar Ho
