REGRESIN LINEAL SIMPLE

Email:cgonzales1@usmp.edu.pe

Objetivos:Al finalizar este capitulo el alumno ser capaz de: Representar la recta que define la relacin lineal entre dos variables Estimar la recta de regresin por el mtodo de mnimos cuadrados e interpretar su ajuste. Realizar inferencia sobre los parmetros de la recta de regresin Construir e interpretar intervalos de confianza e intervalos de prediccin para la variable dependiente Realizar una prueba de hiptesis para determinar si el coeficiente de correlacin es distinto de cero

USOS DEL ANLISIS DE REGRESIN:

Prediccin: razn principal para usar regresin. Descripcin: La idea es establecer una ecuacin que describa la relacin entre la variable dependiente y las variables predictoras.Control: Controlar el comportamiento o variacin de la variable de respuesta. Seleccin de variables

REPRESENTACION GRAFICARelacin entre las variablesSugerir modelos posiblesExistencia de valores atpicos

GRAFICO DE DISPERSION

EL MODELO DE REGRESIN LINEAL SIMPLE

La variable X es no aleatoria y observada con la mejor precisin posible.Los errores ei son variables aleatorias con media 0 y varianza 2 constantes .Los errores ei y ej (i,j=1,n) son independientes entre si. Es decir, Cov(eiej)=0.

Suposiciones del modelo:

ESTIMACION DE LOS PARAMETROS

OBJETIVO:Hallar los estimadores bo y b1 de los parmetros desconocidos o, 1 respectivamente, y obtener la ecuacin de prediccin

ANALISIS DE VARIANZADescomposicin de la variacin total

CUADRO ANOVATest de F de la tabla del ANOVA

Hoja1

Fuentes de VariacinG.LSuma de Cuadrados(SC)Cuadrados Medios(CM)Test F

Regresin1SCRegCMReg=SCRegF=CMReg/CME

Errorn-2SCECME =SCE/n-2

Totaln-1SCY

Hoja2

Hoja3

El coeficiente de determinacin R2Corresponde a la porcin de la variacin total SCTo, de la variable dependiente que es explicada por el modelo de regresin.

Se han recogido datos de una localidad mediante sendas encuestas sobre el consumo (Y ) de productos de hogar y del ingreso (X) de los consumidores consultados, obtenindose los siguientes resultados:

Hoja1

XY

7.154.6

3.444.7

5.551

4.349.7

3.747.2

655

3.342.9

6.755.6

5.147.6

4.549.5

2.744.6

5.957.2

Hoja2

Hoja3

Cumple los supuestos de la regresin?Hallar la ecuacin de regresin estimadaHallar el ANOVADeterminar el coeficiente de determinacinProbar si existe relacin lineal entre X e Y. Usar un nivel de significacin del 5 %..

27.txtFitted Line Plot: Y2 versus X200Fitted Line Plot: Y2 versus X2;; HMF V1.24 TEXT;; (Microsoft Win32 Intel 386) HOOPS 4.50-11 I.M. 3.00-11(Selectability "windows=off,geometry=on")(Visibility "on")(Color_By_Index "Geometry,Face Contrast" 1)(Color_By_Index "Window" 0)(Window_Frame "off")(Window -1 1 -1 1)(Camera (0 0 -5) (0 0 0) (0 1 0) 2 2 "Stretched")

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28.txtNormal Prob Plot: RESI3

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The regression equation isY2 = 35.2 + 3.05 X2Predictor Coef SE Coef T PConstant 35.173 2.258 15.58 0.000X2 3.0503 0.4482 6.81 0.000

S = 2.115 R-Sq = 82.2% R-Sq(adj) = 80.5%

Analysis of VarianceSource DF SS MS F PRegression 1 207.21 207.21 46.31 0.000Residual Error 10 44.74 4.47Total 11 251.95

INTERVALO DE CONFIANZA PARA EL COEFICIENTE DE REGRESINSupuesto:Luego, un intervalo de confianza de 100(1-) para 1 est dado por:Donde:

PRUEBA DE HIPOTESIS PARA 1

ESTIMACIN DE LA RECTA MEDIAInteresa estimar la respuesta media en funcin de un valor especifico de Xh

Luego, un intervalo de confianza de 100(1-) para Y/X est dado por:Donde:

PREDICCION DE UNA NUEVA OBSERVACIONInteresa predecir una observacin que se producir para un valor especifico Xp

Luego, un intervalo de confianza de 100(1-) para la nueva observacin Yp est dado por:Donde:

COEFICIENTE DE CORRELACION

El anlisis de CORRELACION intenta medir la fuerza de la relacin lineal entre dos variables.Estimado por:

PRUEBA DE HIPOTESIS DE LA CORRELACION

Un comerciante al menudeo lleva a cabo un estudio para determinar la relacin entre los gastos semanales de publicidad y las ventas. Se registran los siguientes datos:Resolver:1. Analizar el diagrama de dispersin2. Ajustar un modelo de regresin lineal simple3. Determinar si el efecto del monto de las ventas sobre el costo promedio de la publicidad es significativo.

Hoja1

conrespConc 2 resp 2

0.4240,4 24,0

0.4240,4 24,0

0.423.50,4 23,5

0.424.50,4 24,5Costos por Publicidad402025203050402050402550

0.4230,4 23,0Ventas($)385400395365475440490420560525480510

0.530.50,5 30,5

0.5300,5 30,0

0.5300,5 30,0

0.5300,5 30,0

0.5300,5 30,0

0.745.50,7 45,5

0.7450,7 45,0

0.7450,7 45,0

0.7440,7 44,0

0.7450,7 45,0

0.8510,8 51,0

0.8510,8 51,0

0.850.50,8 50,5

0.8520,8 52,0

0.850.50,8 50,5

1671,0 67,0

166.51,0 66,5

166.51,0 66,5

1661,0 66,0

166.51,0 66,5

Hoja2

Hoja3

4.Calcule el intervalo de confianza del 95% para el parmetro .5. Podra afirmarse que por cada $10 de aumento en el costo de publicidad, el monto promedio de las ventas aumenta en $35?.6.Estime el monto promedio de las ventas si en una semana en particular se invierte en publicidad $35. Calcule un intervalo de confianza del 95% para esta estimacin.7.Suponga que la semana entrante se van a invertir en publicidad un total de $45. Cul ser el monto de las ventas? Encuentre un intervalo de confianza del 95 % para esta prediccin.8. Anlisis de los supuestos sobre el trmino de error.

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167.bin

168.bin