Universidad Nacional Mayor de San Marcos Estadstica

Prctica N5

1. La tabla siguiente presenta el mejoramiento (en la velocidad de lectura de ocho estudiantes que cursaron un programa de lectura de velocidad y el nmero de semanas que siguieron el programa.X352 88934

Y861184919316323273109

a) Ajustar una recta a estos datos por el mtodo de mnimos cuadrados.b) Por cada semana ms de permanencia en el programa en cuantas palabras incrementan la verdad de lecturac) Estimar la velocidad de lectura (palabras por minuto) para un estudiante que ha permanecido en el programa 7 semanas

Solucin:

a) Determinando los datos necesarios para encontrar la relacin de Y con X:

Diagrama de dispersin segn los datos de la tabla

Y = a + bx

Ecuaciones normales de la recta:

Remplazando los valores hallados en la ecuacin:

Realizando las respectivas operaciones se halla a y b entonces:

Remplazando en la ecuacin de regresin de la recta:

b) De la ecuacin de regresin, se tiene:

Donde la pendiente es decir 22.45 representa el aumento de velocidad por cada semana, entonces:

Por cada semana ms de permanencia la velocidad de lectura se incrementa en 22.45.

c) De la ecuacin de regresin, se tiene:

Un alumno que permanece en el programa por 7 semanas, se puede decir que su velocidad de lectura ser de 167.5048 (Palabras por minuto).

2. Con los datos de la tabla que se acompaa, relacionados con ingresos (X) y consumos (Y) de 5 personas.X200300400600900

Y180270320480700

Hallar:

a) La grfica de dispersin de los puntos (X, Y)b) El ndice de correlacinc) La recta de ajuste de mnimos cuadrados de Y en X Ajusta satisfactoriamente esta recta a los datos?d) Qu porcentaje de la varianza de los valores de Y explica la recta de regresin?Solucin: a) Determinamos los datos necesarios para encontrar la relacin de Y con X:

Diagrama de dispersin segn los datos de la tabla

Y = a + bx

Ecuaciones normales de la recta:

Remplazando los valores hallados en la ecuacin:

Realizando las respectivas operaciones se halla a y b entonces:

Remplazando en la ecuacin de regresin de la recta:

b) Coeficiente de Correlacin:

c) La recta si se ajusta satisfactoriamente a los datos.

d) El porcentaje de la varianza de los valores de Y es 80 %.

3. La siguiente tabla muestra las respectivas estaturas de una muestra de doce padres (X) y sus hijos mayores (Y):

X (en pulg)656367646862706668676971

Y (en pulg)686668656966686571676870

a) Construir el diagrama de dispersinb) Determinar la recta de regresin de Y dado Xc) Hallar el coeficiente de correlacin

Solucin:

a) Determinamos los datos necesarios para encontrar la relacin de Y con X:

Diagrama de dispersin segn los datos de la tabla

Ecuaciones normales de la recta:

Remplazando los valores hallados en la ecuacin:

Realizando las respectivas operaciones se halla a y b entonces:

Remplazando en la ecuacin de regresin de la recta:

b) Coeficiente de Correlacin:

Por lo tanto la ecuacin de la recta se ajusta a los datos.

4. La siguiente tabla consiga las calificaciones obtenidas por 12 estudiantes en las pruebas parcial y final del curso de estadstica.P. parcial(X)1410161581211101391210

P. final (Y)13121211814101112111411

a) Obtener las ecuaciones de regresin de Y sobre X y de X sobre Y.b) Con los resultados de:i. Estimar la calificacin de un estudiante en la prueba final si en la prueba parcial obtuvo 15; y la calificacin de un estudiante en la prueba parcial si en la prueba final obtuvo 09 c) Con los resultados de:I. Calcular e interpretar los coeficientes de correlacin.Solucin:

a) Determinamos los datos necesarios para obtener la relacin de Y con X:

Ecuacin de regresin de Y sobre X:

Diagrama de dispersin segn los datos de la tabla.

La ecuacin que utilizaremos ser la de la funcin potencial

Y=a.xb

Las ecuaciones de la funcin potencial son:

Reemplazando valores:

12.7124 = 12 (loga) +b (12.6974)13.4895 = 12.6974 (loga) + b (13.5275)

Los valores de a y b son: a= 4.1543 b= 0.4155

Finalmente la ecuacin de Y en X es: Y = 4.1543. X0.4155

Ecuacin de regresin de X sobre Y:

Diagrama de dispersin segn los datos de la tabla

La ecuacin que utilizaremos ser la de la funcin potencial

X=a.yb

Las ecuaciones de la funcin potencial son:

Reemplazando valores:

12.6974 = 12 (loga) +b (12.7124)13.4895 = 12.7124 (loga) + b (13.5156)

Los valores de a y b son:a= 1.6661b= 0.7895

Finalmente la ecuacin de X en Y es: X = 1.6661. y0.7895

b) Tenemos:

Y = 4.1543 (15 0.4155)Y = 12.7986

Es decir que si un alumno saco 15 en el examen parcial en el examen final probablemente su nota sea de 13.

X= 1.661 (9 0.7895)X= 9.4423

Es decir que si el alumno saca en su examen final 9 en su examen parcial probablemente haya tenido una nota de 9 tambin.

c) Coeficiente de correlacin:

Correlacin de Y sobre X:

Correlacin de X sobre Y:

5. Se han estudiado las calificaciones de 200 estudiantes en la asignaturas de matemtica y estadstica, obtenindose los siguientes resultados:

X: puntaje en matemticaY: puntaje en estadstica

X =13Y = 12.5Sx = 2Sy = 1.2r = 0.9

a) Estimar el puntaje de un estudiante en Estadstica si en matemtica obtuvo 14b) Para un estudiante que en Estadstica obtuvo 10, que puntaje se estima obtendra en matemtica.

Solucin:

A la vez X e Y son medias aritmticas de las calificaciones de los 200 estudiantes.Se sabe que:r = SXY__ SXSY

0.9 = SXY__ 2(1.2)

SXY = 2.16

Tambin se sabe que la pendiente de la recta es b:

b = SXYSX2 2.16 = 0.54 4

Y = a + bx

12.5 = a + 0.54 (13)

a = 5.48

La ecuacin de regresin es:Yc = 5.48 + 0.54X

6. Los directivos de una industria desean conocer si el ingreso personal disponible es un indicador confiable para predecir las ventas de la industria. Para tal efecto se desea ajustar una recta a los datos de la tabla (de Y en X). X representa ingreso personal disponible e Y representa ventas de la industria (X en Y se tomaron en miles de u. m).

X356781039

Y23556728

a) Realizar el ajuste por el mtodo de los mnimos cuadrados.b) Estimar e interpretar Y cuando:I. X = 9II. X = 12Solucin:a) Determinamos los datos necesarios para obtener la relacin de Y con X

Diagrama de dispersin segn los datos de la tabla

Como es una funcin rectilnea entonces se procede ajustar por el mtodo de mnimos cuadrado:

Ecuaciones normales de la recta:

Sustituyendo informacin de la tabla:

39 = 6a + 28b204 = 28a + 148bEntonces los valores de a y b son:

a = 0.5769; b = 1.2692

Sustituyendo en la ecuacin de la recta que es de la forma:

Y = a + bx

En conclusin:Y = 0.5769 + 1.2692x

b) Estimando el valor de Y cuando:i. X = 9Reemplazando el valor de X que nos pide en la ecuacin de la recta

Y = 0.5769 + 1.2692 (9)Y = 11.9997

Interpretacin.- Cuando el ingreso personal es de 9 u.m. entonces las ventas son de 11.9997 u.m.

ii. X = 12Reemplazando el valor de X que nos pide en la ecuacin de la recta

Y = 0.5769 + 1.2692 (12)Y = 15.8073

Interpretacin.- Cuando el ingreso personal es de 12 u.m. entonces las ventas son de 15.8073 u.m.

7. Se tiene los datos de la siguiente tabla, correspondiente la informacin sobre ventas (X) en u.m y horas hombre (Y) ocupadas en la diferentes secciones de 6 tiendas de un cooperativa. Este estudio servir de base para desarrollar guas para el nmero de distribucin de personal

X1357910

Y237111517

a) Hallar la ecuacin que relacione las variables.b) Hallar e interpretar el valor de Y correspondiente al incremento unitario de X.c) Hallar e interpretar el error estndar de estimacin.Solucin:a) Determinamos los datos necesarios para obtener la relacin de Y con Xb)

Diagrama de dispersin segn los datos de la tabla

La ecuacin que utilizaremos ser la de la funcin exponencial

Y=a.bx

Las ecuaciones de la funcin potencial son:

Reemplazando valores:5.0712 = 6 (loga) + 35 (logb)

36.1364 = 35 (loga) + 265 (logb)

Entonces los valores de a y b son:

a = 0.1078

b = 1.2817

Sustituyendo en la ecuacin que es de la forma:

Y = abx

En conclusin:Y = 0.1078 (1.2817)x

c) Si incrementa el X en 1 se necesitara menos de 1 hora para realizar una venta.

8. Ajustar una parbola de segundo grado por el mtodo de los mnimos cuadrados a los datos de la tabla adjunta que se refieren a 8 lotes de artculos (X) en miles y costo unitario promedio (Y) en u. m.X1.42.32.83.24.28.39.71.3

Y123467911

a) Estimar el valor de Y cuando X = 10. Interpretarb) Hallar el error estndar de estimacin

Solucin:a) Determinamos los datos necesarios para obtener la relacin de Y con X

Diagrama de dispersin segn los datos de la tabla

La ecuacin de la parbola:

Reemplazando los valores dados:

11 = 8a + 33.2b + 207.64c

212.1 = 33.2a + 207.64b + 1630.376c

1530.49 = 207.64a + 1630.37b + 14110.9348c

Entonces los valores de a, b y c son:

a = 6.6730b = -1.3597c = 0.1673

Reemplazando los valores de a y b en la ecuacin de la parbola:

Y = a + bx + cx

Y = 6.6730 1.3597 x + 0.1673 x

Para: x = 10Y = 6.6730 1.3597 (10) + 0.1673 (10)

Y = 9.806 u. m.

Interpretacin.- 8 lotes de 10 artculos cada uno tienen un costo promedio de 9.806 u. m.

b) El error estndar:

9. Para una regin de cierto pas se tiene los datos de la tabla:Aos199319941995199619971998

X200220245270300340

Y180210230250280320

Calcular la elasticidad ingreso del consumo mediante la funcin adecuada de Y (consumo por habitante) sobre X (ingreso por habitante en u. m).

Solucin:

Determinamos los datos necesarios para obtener la relacin de Y con X:

Diagrama de dispersin segn los datos de la tabla

La ecuacin que utilizaremos ser la de la funcin potencial

Y=a. xb

Las ecuaciones de la funcin potencial son:

Reemplazando valores:

14.2895 = 6 (loga) +b (14.4726)34.5055 = 14.4726 (loga) + b (34.9458)

Los valores de a y b son: a= 0.7581 b= 1.0374

Finalmente la ecuacin de Y en X es:

Y = 0.7581. X1.0374

10. a) Calcular el coeficiente de correlacin en los ejercicios: 1, 2, 3, 4, 6 y 7.b) Interprete en cada caso el coeficiente de determinacin

Solucin: a) Coeficiente de Correlacin:

En el ejercicio 1:Segn los datos obtenidos:

Interpretacin.- El coeficiente de correlacin es alto lo que amerita la buena relacin entre las variables. En el ejercicio 2:Segn los datos obtenidos:

Interpretacin.- El coeficiente de correlacin demuestra que la ecuacin de regresin se ajusta satisfactoriamente a los datos. En el ejercicio 3:Segn los datos obtenidos:

Por lo tanto la ecuacin de la recta se ajusta satisfactoriamente a los datos.

En el ejercicio 4:Segn los datos obtenidos:

Correlacin de Y sobre X:

Correlacin de X sobre Y:

Por lo tanto ambas regresiones se ajusta satisfactoriamente a los datos.

En el ejercicio 6:Segn los datos obtenidos:

Interpretacin.- El coeficiente de correlacin demuestra que la ecuacin de regresin se ajusta satisfactoriamente a los datos. En el ejercicio 7:Segn los datos obtenidos:

Interpretacin.- El coeficiente de correlacin demuestra que la ecuacin de regresin se ajusta satisfactoriamente a los datos.b) Coeficiente de Determinacin (r2):

En el ejercicio 1:

Interpretacin.-El 76.63% de la variabilidad total es explicada por la ecuacin de regresin.

En el ejercicio 2:

Interpretacin.- El 99.6% de la variabilidad total es explicada por la ecuacin de regresin.

En el ejercicio 3:

Interpretacin.- El 41.15% de la variabilidad total es explicada por la ecuacin de regresin.

En el ejercicio 4:Segn la regresin de X sobre Y:

Interpretacin.- El 32.11% de la variabilidad total es explicada por la ecuacin de regresin.

En el ejercicio 6:

Interpretacin.-El 73.12% de la variabilidad total es explicada por la ecuacin de regresin.

En el ejercicio 7:

Interpretacin.-El 95.45% de la variabilidad total es explicada por la ecuacin de regresin.

11. Si Sy = 0.2 Sy y se sabe qu r = 0.8. Hallar el coeficiente de regresin de la recta: y = a + bx

Solucin:

El coeficiente de regresin de la recta es nada ms que la pendiente. bEntonces:

0.8 =SXY__ 0.2 SX2

0.16 = SXY SX2

b = 0.16 Coef. de regresin

Por lo tanto el coeficiente de regresin es 0.16.

12. Demostrar que otra frmula para calcular el coeficiente de correlacin rectilneo es:

Solucin:

La demostracin resulta de las ecuaciones normales de la recta que son sacados por M.M.C. Mtodos de mnimos cuadrados.

r = SXY__ SXSY

SXY: Es la covarianza, la cual por frmula es:

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XYx2LOGY(LOGY)XLOGY21210.30100.30100.09063390.47711.43140.227657250.84514.22550.7142711491.04147.28971.0845915811.176110.58481.383210171001.230412.30451.514035552655.071236.13695.0141XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000

Hoja2

Hoja3

Grfico1

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XYx2LOGY(LOGY)XLOGY21210.30100.30100.09063390.47711.43140.227657250.84514.22550.7142711491.04147.28971.0845915811.176110.58481.383210171001.230412.30451.514035552655.071236.13695.0141XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000

Hoja2

Hoja3

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

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Hoja2

Hoja3

Grfico2

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Hoja2

Hoja3

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100XiYiXi(Yi)(Xi)2(Yi)2656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

XYx2LOGY(LOGY)XLOGY21210.30100.30100.09063390.47711.43140.227657250.84514.22550.7142711491.04147.28971.0845915811.176110.58481.383210171001.230412.30451.514035552655.071236.13695.0141XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000

Hoja2

Hoja3

Grfico3

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100XiYiXi(Yi)(Xi)2(Yi)2656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

XYx2LOGY(LOGY)XLOGY21210.30100.30100.09063390.47711.43140.227657250.84514.22550.7142711491.04147.28971.0845915811.176110.58481.383210171001.230412.30451.514035552655.071236.13695.0141XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000

Hoja2

Hoja3

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100XiYiXi(Yi)(Xi)2(Yi)2656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

XYx2LOGY(LOGY)XLOGY21210.30100.30100.09063390.47711.43140.227657250.84514.22550.7142711491.04147.28971.0845915811.176110.58481.383210171001.230412.30451.514035552655.071236.13695.0141XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000

Hoja2

Hoja3

Grfico4

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100XiYiXi(Yi)(Xi)2(Yi)2656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

XYx2LOGY(LOGY)XLOGY21210.30100.30100.09063390.47711.43140.227657250.84514.22550.7142711491.04147.28971.0845915811.176110.58481.383210171001.230412.30451.514035552655.071236.13695.0141XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000

Hoja2

Hoja3

Grfico5

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100XiYiXi(Yi)(Xi)2(Yi)2656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

XYx2LOGY(LOGY)XLOGY21210.30100.30100.09063390.47711.43140.227657250.84514.22550.7142711491.04147.28971.0845915811.176110.58481.383210171001.230412.30451.514035552655.071236.13695.0141XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000

Hoja2

Hoja3

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100XiYiXi(Yi)(Xi)2(Yi)2656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

XYx2LOGY(LOGY)XLOGY21210.30100.30100.09063390.47711.43140.227657250.84514.22550.7142711491.04147.28971.0845915811.176110.58481.383210171001.230412.30451.514035552655.071236.13695.0141XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000

Hoja2

Hoja3

Grfico6

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100XiYiXi(Yi)(Xi)2(Yi)2656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

XYx2LOGY(LOGY)XLOGY21210.30100.30100.09063390.47711.43140.227657250.84514.22550.7142711491.04147.28971.0845915811.176110.58481.383210171001.230412.30451.514035552655.071236.13695.0141XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000

Hoja2

Hoja3

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100XiYiXi(Yi)(Xi)2(Yi)2656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

XYx2LOGY(LOGY)XLOGY21210.30100.30100.09063390.47711.43140.227657250.84514.22550.7142711491.04147.28971.0845915811.176110.58481.383210171001.230412.30451.514035552655.071236.13695.0141XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000

Hoja2

Hoja3

Grfico8

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100XiYiXi(Yi)(Xi)2(Yi)2656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

XYx2LOGY(LOGY)XLOGY21210.30100.30100.09063390.47711.43140.227657250.84514.22550.7142711491.04147.28971.0845915811.176110.58481.383210171001.230412.30451.514035552655.071236.13695.0141XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000

Hoja2

Hoja3

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100XiYiXi(Yi)(Xi)2(Yi)2656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

XYx2LOGY(LOGY)XLOGY21210.30100.30100.09060.30103390.47711.43140.22764.294157250.84514.22550.714221.1275711491.04147.28971.084551.0282915811.176110.58481.383295.263410171001.230412.30451.5140123.044935552655.071236.13695.0141295.0591XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000

Hoja2

Hoja3

Grfico9

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100XiYiXi(Yi)(Xi)2(Yi)2656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

XYx2LOGY(LOGY)XLOGY21210.30100.30100.09060.30103390.47711.43140.22764.294157250.84514.22550.714221.1275711491.04147.28971.084551.0282915811.176110.58481.383295.263410171001.230412.30451.5140123.044935552655.071236.13695.0141295.0591XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000

Hoja2

Hoja3

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100XiYiXi(Yi)(Xi)2(Yi)2656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

XYx2LOGY(LOGY)XLOGY21210.30100.30100.09060.30103390.47711.43140.22764.294157250.84514.22550.714221.1275711491.04147.28971.084551.0282915811.176110.58481.383295.263410171001.230412.30451.5140123.044935552655.071236.13695.0141295.0591XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000XYLOGXLOGYLOGY(LOGX)(LOGX)2(LOGY)2(LOGY)(LOGX)22001802.30102.25535.18945.29475.086311.94112202102.34242.32225.43965.48695.392712.74192452302.38922.36175.64265.70815.577813.48102702502.43142.39795.83035.91155.750114.17553002802.47712.44726.06196.13615.988615.01613403202.53152.50516.34176.40846.275816.05401575147014.472614.289534.505534.945834.071283.4095

Hoja2

Hoja3

Grfico1

Hoja1XYXYX2386258951185902524998481931544648163130464923220888137321994109436164210236537272XiYiXi(Yi)(Xi)2(Yi)22001803600040000324003002708100090000729004003201280001600001024006004802880003600002304009007006300008100004900002400195011630001460000928100XiYiXi(Yi)(Xi)2(Yi)2656844204225462463664158396943566768455644894624646541604096422568694692462447616266409238444356706847604900462466654290435642256871482846245041676744894489448969684692476146247170497050414900800811541075341854849XYLOGXLOGYLOGX(LOGY)LOGX2LOGY214131.14611.11391.27671.31361.240910121.00001.07921.07921.00001.164616121.20411.07921.29951.44991.164615111.17611.04141.22481.38321.0845880.90310.90310.81560.81560.815612141.07921.14611.23691.16461.313611101.04141.00001.04141.08451.000010111.00001.04141.04141.00001.084513121.11391.07921.20211.24091.16469110.95421.04140.99370.91061.084512141.07921.14611.23691.16461.313610111.00001.04141.04141.00001.084514013912.697412.712413.489513.527513.5156XYXYX2Y223649351592556302536573525496848366471070491002839204148283

XYx2LOGY(LOGY)XLOGY21210.30100.30100.09060.30103390.47711.43140.22764.294157250.84514.22550.714221.1275711491.04147.28971.084551.0282915811.176110.58481.383295.263410171001.230412.30451.5140123.044935552655.071236.13695.0141295.0591XYXYX2X2YX3X4Y21.411.40001.96001.96002.74403.841612.324.60005.290010.580012.167027.984142.838.40007.840023.520021.952061.465693.2412.800010.240040.960032.7680104.8576164.2625.200017.6400105.840074.0880311.1696368.3758.100068.8900482.2300571.78704745.8321499.7987.300094.0900846.8100912.67308852.9281811.31114.30001.690018.59002.19702.856112133.243212.1000207.64001530.49001630.376014110.9348317.0000XYLOGXLOGYLOGY(LOGX)(LOGX)2(LOGY)2(LOGY)(LOGX)22001802.30102.25535.18945.29475.086311.94112202102.34242.32225.43965.48695.392712.74192452302.38922.36175.64265.70815.577813.48102702502.43142.39795.83035.91155.750114.17553002802.47712.44726.06196.13615.988615.01613403202.53152.50516.34176.40846.275816.05401575147014.472614.289534.505534.945834.071283.4095

Hoja2

Hoja3