REGRESION LINEAL Y MULTIPLE
JOSE LUIS DIAZ SOTO JORGE LORA ISAURA FERNANDEZ
Tutor (a) MARCOS CASTRO
UNIVERSIDAD DE CARTAGENA Programa: ADMINISTRACIÓN FINANCIERA Área: MODELO INFERENCIAL VI SEMESTRE CERETE 2014
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1. Sobre un conjunto conjunto de conductores se ha realizado una encuesta para analizar su edad (Y) y el número de accidentes accidentes que han sufrido sufrido (X). A partir de la misma, misma, se obtuvieron los siguientes resultados:
X/Y (20,30] (30,40] (40,50] (50,60] (60,70] 74 82 78 72 7 0 7 6 5 6 5 1 3 2 2 1 1 2 A partir de estos datos, datos, se le pide que determin determine e para esta esta distribuci distribución ón las curvas de regresión de Y sobre X y de X sobre Y. Sea y: la variable edad(x) X: la variable números de accidentes (y) PROMEDIO DE EDAD 25 35 45 55 65 25 35 45 55 65 25 35 45 55 65
NUMERO DE ACCIDENTES 0 0 0 0 0 1 1 1 1 1 2 2 2 2 2
La curva de y sobre x es
b=
∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑
, a=
∑ ∑ ∑
NUMERO DE PERSONAS 74 82 76 72 7 7 6 5 6 5 3 2 2 1 1
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ahora de x/y
∑ ∑ ,
,
∑ ∑ ∑
b=
= =
= 0.000000001
a=
=
0.933
=455625
x/y=0.000000001x+0.933
2. Para la economía española, española, disponemos de los datos anuales redondeados redondeados sobre sobre consumo final de los hogares a precios corrientes (Y) y renta nacional disponible neta (X), tomados de la Contabilidad Nacional de España base 1995 del INE , para el período 1995-2002, ambos expresados en miles de millones de euros:
AÑO 1995 Yt 258’6 Xt 381’7
1996
1997
1998
1999
2000
2001
2002
273’6
289’7
308’9
331’0
355’0
377’1
400’4
402’2
426’5
454’3
86’5
520’2
553’3
590’0
Considerando que el consumo se puede expresar como función lineal de la renta (Yt=a + b ·Xt), determine:
a) Los parámetros a y b de la recta de regresión. Sea y’: la variable consumo X: la variable renta Los parametros son
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b=
b= 0.6834
a= a=
∑ ∑ ∑ =
a= 160138 b) La varianza de la variable consumo y su descomposición en varianza varianza explicada y no explicada por el modelo.
∑ = =
= 2235.966094
La varianza explicada es:
∑
El modelo es:
y= 0.6834x-1.60
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=
= 9658,88
La varianza del no explicada del modelo m odelo es:
∑ = 0.44842 =
c) el coeficiente de determinacion
Donde Sxy= Sxx= Syy=
= =
∑ ∑ ∑∑ ∑ ∑ ∑ ∑
=0.999800
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e) r=
√
donde
Sxx=
∑ ∑
∑ ∑ ∑ ∑ ∑ ∑ Syy=
Sxy=
Sxx=
Sxx =38289.638 Syy=
Syy =17887.72 Sxy=
Sxy =26168.26375 r=
=
=3.16
3. Se supone que se puede establecer cierta relación lineal entre las exportaciones de un país y la producci producción ón interna de de dicho dicho país. En el caso de España, España, tenemos tenemos los datos anuales (expresados en miles de millones de pesetas) para tales variables variables correspondientes correspondientes al quinquenio quinquenio 1992-96 en la siguiente siguiente tabla: Años Producción Exportacione 52.654 10.420 1992 53.972 11.841 1993
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b) Si sabemos que las exportaciones para 1997 fueron de 69.045.704 millones de pesetas, ¿cuál sería la producción interna aproximada para ese año? ¿Qué grado de confianza daría usted a esta predicción? c) ¿Qué tanto por ciento de la varianza de las exportaciones no vienen explicadas por la Producción interna, interna, y se debe a otro tipo tipo de variables? d) e) El coeficiente de correlación sea y: variable produccion X: variable exportacion
Y=bx + a #=bx + n a) Para saber la produccion de las exportaciones calculamos el modelo de regresion de la forma:
Y: bx + a donde b= a=
∑ ∑ ∑
n=5
∑ ∑ ∑∑ ∑ ∑ ∑ ∑
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b=1.55070 a=
= 179.2643
luego el modulo de regresion es: y^= 1.55070x +179.2643 1. Como la produccion para el año 1997 fue de 2210.6100 millones de pesetas. 2210.6100= 1.55070x+179.2643 2210.6100-179.2643=1.55070x 2031.345604=1.55070x
X=
X=1309.953959 millones de exportaciones
2. Para el grado de precision hacemos
Y-t
<
t
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∑ √ ∑ ∑ ----t
t0.05/2= 0,025, n=5
.025, v=n-1 ------t0,025,
=
4=2,776
=
Sxx=
=1089.28=46.830
2210.60-(2.776
+
<
+
a) Como las exportaciones para el año 1997 son de 69045704 millones, entonces la produccion interna seria para este año de:
Y=1.55070(69045709)+ 179.2643 Y =1070693525 millones de pesetas, comparando este valor y el del punto anterior el grado se confianza sea de 95%.
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2
S xy = 5273.78 Sxx=
∑ ∑ ∑ ∑
Sxx= 1089.28
= 46.830870
Syy= =
= 114.0283
2
r= 2
r= 2
r =0,9876 2
r =98.76% 100-98.76%=1.24% 100-98.76%=1.24% de varianza que no es explicada por las variables. Coeficiente de correlacion
√ √
r=
=
=
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como rxy=
=
mxy= rxsxsy
mxy=(0.9)(1.20)(2.10) mxy =2,26 bxy=
= bxy=
x=bxy(y-y^)+x x =0,51(y-10)+5 x =0,51(y-10)+5 x =0,51y-5,1+5 X= 0,51y-0,1
Y para y sobre x y= bxy(x-x)+y y=0.51(x-5)+10 y =0,51x – 2,25 + 10
=0.51
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Hallar los parametros parametros a y b de la ecuacion ecuacion y= bx + a
a= y – bx de la ecuacion ecuacion bxy 0,45, pero x=bxy(y-y) x=bxy(y-y) + x sobre y. como bxy= 0,45 axy= 10 dyx= y – bxy X r2= byxbxy entonces 0.9= byxbxy byx=
byx=2 entonces ayx= 10 – 0,45(20) ayx =1
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Sea y: variable aceleración X: variable motor a) Para determinar la función función de ajuste y ECM mediante mediante el modelo Ȳ = bx + a
b=
a=
∑ ∑ ∑∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ =
=
=
= -0.06796
= 20.85132, 20.85132, luego la función
de ajuste es:
Ȳ = -0.06796x + 20.85132
El error cuadratico medio es: SSE =
∑
∑ =
∑ ∑ ∑
=
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∑
Syy =
∑
√
= 724.25 =
= 30.875
= -0.9530
8. En un nuevo nuevo proceso artesanal de fabricación fabricación de cierto artículo artículo que está implantado, se ha considerado que era interesante ir anotando periódicamente el tiempo medio (medido en minutos) que se utiliza para realizar una pieza (variable Y) y el número de días desde que empezó dicho proceso de fabricación fabricación (variable X). Con ello, se pretende analizar cómo los operarios van adaptándose al nuevo proceso, mejorando paulatinamente su ritmo de producción conforme van adquiriendo más experiencia experiencia en él. A partir de las cifras cifras recogidas, que aparecen en la tabla adjunta, se decide ajustar una función exponencial que explique el tiempo de fabricación en función del número de días que se lleva trabajando con ese método.
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∑ ∑ ∑ ∑
donde
b = a=
y
∑ ∑ =
= -0.34642 = 35.5714
Ȳ = 35.5714- 0.34642(X)
Ahora para x = 100 días, el tiempo medio de fabricación de la pieza es de: Ȳ = 35.5714- 0.34642 (100) = 0.9294 b) Cuando y = 10 minutos, el tiempo transcurrido de fabricación de la pieza es:
Ȳ = 35.5714- 0.34642(X)
10 - 35.5714 = -0.34642(X) - 25.5714 = 0.34642(X)
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10. Un estudiante de la Escuela Universitaria de Estudios Empresariales de la 10. Universidad de Sevilla, para poder pagarse sus estudios, debe trabajar como camarero camarero en un bar de copas de su su localidad. localidad. A este este establecimiento, establecimiento, suelen acudir todos los jóvenes jóvenes de la zona. Este año, con los conocimientos aprendidos, decide por fin estudiar la relación existente entre la cantidad de sal de las galletas saladas y el consumo de bebidas, ya que es costumbre dar al cliente este aperitivo cuando pide una consumición. Se sabe que las galletas no pueden tener una concentración concentración de sal superior superior a 3'5 gramos gramos por cada 1000 galletas y, y, por ello, decide ir variando a partir de 1 gramo la concentración de 0'5 en 0'5 gramos cada semana e ir anotando el incremento en caja semanalmente, obteniendo la siguiente tabla: tab la: Gramos de sal por 1000 1 1,5 2 2,5 3
Ingresos 140300 150000 165000 175000 200000
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∑ ∑ ∑ ∑∑ ∑
∑ ∑ ∑ ∑ = 1732800
= 10
= 830300 = 22.5
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∑ ∑ ∑ ∑ ∑ ∑ =
=
= 2.5
= 2154472000
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