EJERCICIO PROPUESTO

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Regresión y Correlación simple
		REGRESIÓN Y CORRELACIÓN SIMPLE
		Son dos herramientas para investigar la dependencia de una variable dependiente Y en función de
		una variable independiente X
		Y = f(X)
		Y = Variable dependiente que se desea explicar o predecir, también se llama regresor o respuesta
		X = Variable independiente, también se llama variable explicativa, regresor o predictor
		Regresión lineal - La relación entre X y Y se representa por medio de una línea recta
		Regresión curvilinea - La relación entre X y Y se representa por medio de una curva
		 Y			 * *
			 **	*	 *			* *	 
			 * *		 *			 *
			 *	 b1		 * *	 *	 *	 
			 *			* *	 *		 
		 * *				 * *		 * *	 
		 b0							 
								 X	 
			Correlación positiva		Correlación negativa		 Sin correlación
		La ecuación de la recta es la siguiente:
							e = error aleatorio
		El término de error es la diferencia entre los valores reales observados Yi y los valores estimados por 
		la ecaución de la recta. Se trata de que estos sean mínimos, para lo cual se utiliza el método de 
		mínimos cudrados
		 Y
				 *
			 *
						 X
		Se trata de minimizar la suma de todos los errores o residuos:
		Al aplicar esta regla a las ecuaciones correspondientes se obtienen las siguientes sumas de cuadrados:
						De esta forma los mínimos cuadrados
						se basan en las diferencias al cuadrado
						respecto a las medias tanto en X como
						en Y
		Los coeficientes de la recta son:
		Ejemplo:
		Observación	Publicidad X	Pasajeros Y	 XY	 X*X	 Y*Y
		1	10	15	150	100	225
		2	12	17	204	144	289
		3	8	13	104	64	169
		4	17	23	391	289	529
		5	10	16	160	100	256
		6	15	21	315	225	441
		7	10	14	140	100	196
		8	14	20	280	196	400
		9	19	24	456	361	576
		10	10	17	170	100	289
		11	11	16	176	121	256
		12	13	18	234	169	324
		13	16	23	368	256	529
		14	10	15	150	100	225
		15	12	16	192	144	256
	Suma	 	187	268	3490	2469	4960
	Media		12.4667	17.8667
		SCx =	137.7333
		SCy =	171.7333
		SCxy =	148.9333
		b1 =	1.0813
		b0 =	4.3865
		El modelo de regresión es:			Para pronosticar en X = 10 se tiene:
		*			*
		Y = 4.40 + 1.08 X			Y = 15.2
	SUPUESTOS DEL MODELO DE REGRESIÖN LINEAL
		1. Los errores o residuos se distribuyen normalmente alrededor de la recta de regresión poblacional
			 	*
				*
			 *	*
			 *	*
			 *	*
			 *
		2. Las varianzas de los errores son las mismas en todos los valores de X (Homoscedasticidad)
		 en caso contrario se tiene (Heteroscedasticidad)
		3. Los errores o residuos son independientes: No se muestra algun patrón definido+B170
		Los residuos o errores se muestran a continuación:
		Obs X Y Fit SE Fit Residual St Resid
		 1 10.0 15.000 15.199 0.302 -0.199 -0.23 
		 2 12.0 17.000 17.362 0.237 -0.362 -0.41 
		 3 8.0 13.000 13.037 0.417 -0.037 -0.05 
		 4 17.0 23.000 22.769 0.421 0.231 0.29 
		 5 10.0 16.000 15.199 0.302 0.801 0.94 
		 6 15.0 21.000 20.606 0.305 0.394 0.46 
		 7 10.0 14.000 15.199 0.302 -1.199 -1.40 
		 8 14.0 20.000 19.525 0.262 0.475 0.55 
		 9 19.0 24.000 24.931 0.556 -0.931 -1.30 
		 10 10.0 17.000 15.199 0.302 1.801 2.11R 
		 11 11.0 16.000 16.281 0.260 -0.281 -0.32 
		 12 13.0 18.000 18.443 0.238 -0.443 -0.51 
		 13 16.0 23.000 21.687 0.360 1.313 1.58 
		 14 10.0 15.000 15.199 0.302 -0.199 -0.23 
		 15 12.0 16.000 17.362 0.237 -1.362 -1.56
		Cuando los errores forman un patrón como los siguientes, indicaría que los errores tienen 
		autocorrelación (no son independientes)
	 +e				+e
		 		 *	 	**	*	*
	 '0	 *	 * 		0	 
		 *	 **			 
			 * 			 *	 *	 *
	 - e				 -e
		Autocorrelación Positiva de los residuos				Autocorrelación negativa de los residuos
		Para probar la autocorrelación se utiliza el estadístico de Durbin Watson:
						Haciendo los cálculos d = 2.48
						De MINITAB
						Durbin-Watson statistic = 2.48
		De la tabla III del Apéndice K para un alfa = 0.01, n = 15 y k=1 variables independientes, se tiene:
		dl = 0.81 du = 1.07
		 INDEFINIDO		SIN AUTOCORRELACIÓN		INDEFINIDO
		 dl = 0.81	 du =1.07		4 - du = 2.93	 4 - dl = 3.19
		AUTOCORRELACIÓN POSITIVA				AUTOCORRELACIÓN NEGATIVA
		Se rechaza Ho				Se rechaza Ho
		NOTA: Normalmente no habrá Autocorrelación si el estadístico Durbin Watson es cercano a 2.
	ERROR ESTÁNDAR 		Es una medida del grado de dispersión de los valores Yi alrededor de la recta de regresión
	DE ESTIMACIÓN Se		y proporciona una medida del error que se presentará en la estimación de Yi.
		En el caso del ejemplo:
		SCE = 10.6893
		CME = 0.8222
		Se = 0.907
	ANALISIS DE CORRELACIÓN
		El coeficiente de Correlación r desarrollado por Carl Pearson es un indicador de la fuerza de la relación
		entre las variables X y Y, puede asumir valores entre -1 y 1 para correlación negativa y positiva perfecta
		respectivamente.
		Se identifican tres medidas de desviación como sigue:
			Y
							Yest = 4.4 + 1.08 X
				 Yi = 23	 *		Desviación no explicada
							Error = (Yi - Yest) = 1.32
				Variación total
				(Yi-Ymedia)=5.13	 	Desviació explicada
						(Yest-Ymedia) = 3.81
		Ymedia =17.87
					X = 16			X
		Calculando el valor de r = 0.9683
		Y el coeficiente de determinación r2 = 0.94
	PRUEBAS PARA EL COEFICIENTE DE LA RECTA BETA 1
	 
		Ho: Beta1 = 0
		Ha: Beta1 <>0
		La prueba t es:
		Sb1 = 0.0772
		tm = 13.995	 t de excel para alfa=0.05 y gl=n-2=13 es 2.169 
			Por tanto se rechaza Ho y el coeficiente Beta 1 es signif.
		IC Beta 1 = 1.08 +- (2.16)(0.07726) = 0.913 <= Beta 1 <= 1.247
	PRUEBAS PARA EL COEFICIENTE DE CORRELACIÓN POBLACIONAL RO
		Ho: Ro = 0
		Ha: Ro<>0
		La prueba t es:
		Sr = 0.069
		t = 13.995
		La t de Excel es para un alfa del 5% y gl = n-2 = 12 es 				2.1603686565
		Por tanto se rechaza la hipótesis nula y el coeficiente de correlación Ro es significativo
	INTERVALO DE CONFIANZA PARA LA MEDIA DE LA Y ESTIMADA CONDICIONADA A Xo
		El intervalo de confianza para la media de la Yestimada dado un valor de Xo se calcula como sigue:
								Banda del
				Y				95% de
								Confianza
		En el ejemplo:
						Xo		 X
		Xo = 10	Yest = 15.2
		Sy = 0.303
		IC = 15.2 + - 2.160(0.303) 14.55 <= Mu y <= 15.85
		Considerando todos los valores se forma una banda de confianza alrededor de la recta de regresión
	INTERVALO DE PREDICCIÓN PARA UN VALOR ÚNICO DE Y ESTIMADA
		El intervalo de predicción para un valor único de Y estimada dado un valor de Xo se calcula como sigue:
								Banda de
								Predicción
								del 95%
		Xo = 10	Yest = 15.2
		Sy10 = 0.907
		IP = 15.2 + - 2.160(0.956) 13.14 <= Mu y <= 17.27
		Considerando todos los valores se forma una banda de predicción alrededor de la recta de regresión
		la cual es más amplia que la banda de confianza
	ANÁLISIS DE VARIANZA EN LA REGRESIÓN
		Fuente de	Suma de 	Grados de	Cuadrado 
		variación	cuadrados	libertad	medio	Fm
		Regresión	SCR	k	CMR = SCR/k	CMR/CME
		Error	SCE	n-k-1	CME=SCE / (n-k-1)
		Total	SCT
				NOTA: k variables predictoras 
		 
		Fuente de	Suma de 	Grados de	Cuadrado 
		variación	cuadrados	libertad	medio	Fm
		Regresión	161.04	1	161.04	196.39
		Error	10.69	13	0.82
		Total	SCT	14
		A un valor de alfa de 0.05 el estadístico F de Excel es F0.05,1,13 = 4.67
		Por tanto se comprueba que existe la regresión y es significativao sea Beta 1<>0
		NOTA: Aquí el valor F es el cuadrado del valor de t para comprobar la hipótesis de Beta 1<> 0
	SALIDA DE MINITAB
		Regression Analysis: Y versus X
		The regression equation is
		Y = 4.39 + 1.08 X
		Predictor Coef SE Coef T P
		Constant 4.3863 0.9913 4.42 0.001
		X 1.08132 0.07726 13.99 0.000
		S = 0.9068 R-Sq = 93.8% R-Sq(adj) = 93.3%
		Analysis of Variance
		Source DF SS MS F P
		Regression 1 161.04 161.04 195.86 0.000
		Residual Error 13 10.69 0.82
		Total 14 171.73
Datos de ejercicio
	Problema . Investigar si existe una relación entre los niveles de Consumo (Y) y el ingreso de los consumindores (X). 
		a) Establecer la ecuación de regresión
		b) Hacer un diagrama de Dispersión para los datos (Agregar línea de tendencia, ecuación y R2)
		c) ¿Qué consumo se pronostica para alguien que gane $27,000?
		Datos:
		X=Ingreso	Y=Consumo
		24.3	16.2
		12.5	8.5
		31.2	15
		28	17
		35.1	24.2
		10.5	11.2
		23.2	15
		10	7.1
		8.5	3.5
		15.9	11.5
		14.7	10.7
		15	9.2
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X1
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Y = 4.38625 + 1.08132 X1
S = 0.906780 R-Sq = 93.8 % R-Sq(adj) = 93.3 %
Regression
95% CI
95% PI
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