Sol4

EJERCICIOS 4
1) a) Diagrama de dispersión:
DIAGRAMA DE DISPERSION
INGRESOS
10
8
6
4
2
0
0
5
10
15
20
25
30
PRACTICA
c) Dada la recta de regresión, ninguna de las rectas estimadas en b) es adecuada.
d) Coeficiente de correlación: 0.8588.
Correlations
INGRESOS
PRACTICA
-------------------------------------------------------------INGRESOS
0.8588
(
10)
0.0015
PRACTICA
0.8588
(
10)
0.0015
-------------------------------------------------------------Correlation
(Sample Size)
P-Value
1
2) Diagrama de dispersión:
DIAGRAMA DE DISPERSION
DISTANCIA
40
30
20
10
0
3.1
5.1
7.1
9.1
11.1
CARGA
b) Recta de regresión:
β̂1 =
donde cov(x, y) =
cov(x, y)
= 3,5441,
Sx2
n
n
1X
1X
(xi − x̄)(yi − ȳ) y Sx2 =
(xi − x̄)2 .
n i=1
n i=1
β̂0 = ȳ − x̄β̂1 = −3,6878
Regression Analysis - Linear model: Y = a + b*X
----------------------------------------------------------------------------Dependent variable: DISTANCIA
Independent variable: CARGA
----------------------------------------------------------------------------Standard
Parameter
Estimate
Error
T
Statistic
P-Value
----------------------------------------------------------------------------Intercept
-3.6878
4.65993
-0.791386
0.4515
Slope
3.54412
0.623294
5.68611
0.0005
-----------------------------------------------------------------------------
2
Analysis of Variance
----------------------------------------------------------------------------Source
Sum of Squares
Df
Mean Square
F-Ratio
P-Value
----------------------------------------------------------------------------Model
706.782
1
706.782
Residual
174.882
8
21.8602
32.33
0.0005
----------------------------------------------------------------------------Total (Corr.)
881.664
9
Correlation Coefficient = 0.895347
R-squared = 80.1646 percent
R-squared (adjusted for d.f.) = 77.6852 percent
Standard Error of Est. = 4.67549
c) En la tabla siguiente tenemos Yi (CARGA), Ŷi (PREDICCION), Yi −Ŷi (RESIDUOS)
y (Yi − Ŷi )2 (RESIDUOS2 ). Los cálculos pueden comprobarse mediante
n
X
(Yi − Ŷi ) = 0.
i=1
--------------------------------------------------------DISTANCIA
CARGA
PREDICCION
RESIDUOS
RESIDUOS2
22.4
6.8
20.4122
1.9878
3.9513
36.8
10.5
33.5254
3.2746
10.7227
14.4
4.0
10.4887
3.9113
15.2985
27.2
7.9
24.3107
2.8893
8.3478
16.0
8.1
25.0196
-9.0196
81.3525
35.2
9.5
29.9813
5.2187
27.2345
8.0
3.1
7.2990
0.7010
0.4914
19.2
7.2
21.8299
-2.6298
6.9161
9.6
4.5
12.2607
-2.6607
7.0795
25.6
9.3
29.2725
-3.6725
13.4873
---------------------------------------------------------
RECTA DE REGRESION
DISTANCIA
40
30
20
10
0
3.1
5.1
7.1
CARGA
3
9.1
11.1
3) a) Recta de regresión: Y = 7,9398 + 0,4419 · X
Regression Analysis - Linear model: Y = a + b*X
----------------------------------------------------------------------------Dependent variable: TIEMPO
Independent variable: PASAJEROS
----------------------------------------------------------------------------Standard
Parameter
Estimate
T
Error
Statistic
P-Value
----------------------------------------------------------------------------Intercept
Slope
7.93978
6.70688
1.18383
0.2577
0.441876
0.0206224
21.427
0.0000
-----------------------------------------------------------------------------
Analysis of Variance
----------------------------------------------------------------------------Source
Sum of Squares
Df
Mean Square
F-Ratio
P-Value
----------------------------------------------------------------------------Model
71665.7
1
71665.7
Residual
2029.23
13
156.095
459.12
0.0000
----------------------------------------------------------------------------Total (Corr.)
73694.9
14
Correlation Coefficient = 0.986136
R-squared = 97.2464 percent
R-squared (adjusted for d.f.) = 97.0346 percent
Standard Error of Est. = 12.4938
b) Residuos: Yi − Ŷi , con Ŷi = 7,9398 + 0,4419 · Xi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
RESIDUOS
-10.3368
-19.7384
7.86798
16.2134
-10.7588
10.8244
0.357416
-19.2238
7.8442
8.54434
1.75704
8.5001
1.19826
12.4294
-15.4788
4
q
73694,933
14
= 72,5529;
√
Desviación tı́pica residual = 156,0946 = 12,4938
c) Desviación tı́pica de Y =
d) Coeficiente de correlación = 0,9861
4) a)
DIAGRAMA DE DISPERSION
66
TIEMPO
61
56
51
46
41
36
6
9
12
15
18
21
DEFECTOS
b) Recta de regresión: Y = 31 + 1,4167 · X
RECTA DE REGRESION
66
TIEMPO
61
56
51
46
41
36
6
9
12
15
DEFECTOS
5
18
21
Regression Analysis - Linear model: Y = a + b*X
----------------------------------------------------------------------------Dependent variable: TIEMPO
Independent variable: DEFECTOS
----------------------------------------------------------------------------Standard
Parameter
Estimate
T
Error
Statistic
P-Value
----------------------------------------------------------------------------Intercept
Slope
31,0
4,92293
6,29706
0,0002
1,41667
0,383948
3,68973
0,0061
-----------------------------------------------------------------------------
Analysis of Variance
----------------------------------------------------------------------------Source
Sum of Squares
Df
Mean Square
F-Ratio
P-Value
----------------------------------------------------------------------------Model
409,417
1
409,417
Residual
240,583
8
30,0729
13,61
0,0061
----------------------------------------------------------------------------Total (Corr.)
650,0
9
Correlation Coefficient = 0,793645
R-squared = 62,9872 percent
Standard Error of Est. = 5,48388
c) Residuos:
1
2
3
4
5
6
7
8
9
10
-7.08333
3.83333
6.25000
4.16667
-5.00000
6.50000
-4.91667
-2.91667
2.66667
-3.50000
r
650
= 8,4984
9
√
Desviación tı́pica residual = 30,07292 = 5,4839
d) Desviación tı́pica de Y =
6
5) Coeficiente de correlación = 0,7911
Correlations
FAMILIA
UNIDADES
-------------------------------------------------------FAMILIA
0,7911
(
10)
0,0064
UNIDADES
0,7911
(
10)
0,0064
--------------------------------------------------------
Correlation
(Sample Size)
P-Value
6) Coeficiente de correlación = 0,7608
Correlations
INVENTARIO
GANANCIAS
----------------------------------------------------------INVENTARIO
0,7608
(
5)
0,1353
GANANCIAS
0,7608
(
5)
0,1353
----------------------------------------------------------Correlation
(Sample Size)
P-Value
7
7) a)
Regression Analysis - Linear model: Y = a + b*X
----------------------------------------------------------------------------Dependent variable: GASOLINA
Independent variable: PETROLEO
----------------------------------------------------------------------------Standard
Parameter
Estimate
T
Error
Statistic
P-Value
----------------------------------------------------------------------------Intercept
35,5054
6,30403
5,63218
0,0001
Slope
2,90637
0,296128
9,81458
0,0000
-----------------------------------------------------------------------------
Analysis of Variance
----------------------------------------------------------------------------Source
Sum of Squares
Df
Mean Square
F-Ratio
P-Value
----------------------------------------------------------------------------Model
7375,54
1
7375,54
Residual
995,392
13
76,5686
96,33
0,0000
----------------------------------------------------------------------------Total (Corr.)
8370,93
14
Correlation Coefficient = 0,938664
R-squared = 88,109 percent
Standard Error of Est. = 8,75035
Recta de regresión: Y = 35,5054 + 2,9064 · X
b)
RECTA DE REGRESION
GASOLINA
137
117
97
77
57
10
15
20
25
30
35
40
PETROLEO
La recta de regresión estimada parece apropiada para explicar la relación entre las dos
variables.
8
c) Predicción: Ŷ0 = 35,5054 + 2,9064 · X0 = 79,1014, donde X0 = 15
8) a) Diagrama de dispersión y recta de regresión:
DIAGRAMA DE DISPERSION
VENTAS
72
62
52
42
32
1400
1700
2000
2300
2600
2900
INDICE D.J.
Regression Analysis - Linear model: Y = a + b*X
----------------------------------------------------------------------------Dependent variable: VENTAS
Independent variable: INDICE D.J.
----------------------------------------------------------------------------Standard
Parameter
Estimate
T
Error
Statistic
P-Value
----------------------------------------------------------------------------Intercept
Slope
4,71478
11,4398
0,412137
0,6889
0,0224277
0,00558354
4,01675
0,0025
-----------------------------------------------------------------------------
Analysis of Variance
----------------------------------------------------------------------------Source
Sum of Squares
Df
Mean Square
F-Ratio
P-Value
----------------------------------------------------------------------------Model
Residual
846,47
1
846,47
524,639
10
52,4639
16,13
0,0025
----------------------------------------------------------------------------Total (Corr.)
1371,11
11
Correlation Coefficient = 0,785723
R-squared = 61,7361 percent
Standard Error of Est. = 7,2432
b) Como vemos, la variabilidad explicada por la regresión es R2 = 61,74 %, con lo cual
podemos concluir que existe relación entre las ventas y el ı́ndice Dow Jones.
9