Facultad de Ciencias Veterinarias U.B.A.
Estadística Analítica
Área Bioestadística Bioestadística 2013 – 1er. Cuatrimestre
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Cronograma 2013 – 1er. Cuatrimestre Sem
1
2
lunes
4/3 4/3
11/3
Jueves: Revisión de conceptos relativos a inferencia. Intervalos de confianza y pruebas de hipótesis para la diferencia de medias de dos poblaciones.(hasta test t con varianzas desconocidas y distintas inclusive). Martes: Revisión de conceptos relativos a inferencia. Intervalos de confianza y pruebas de hipótesis para la diferencia de medias de dos poblaciones.(hasta test t con varianzas desconocidas y distintas inclusive). Jueves: Continuación con medias de dos poblaciones. Intervalos de confianza y pruebas de hipótesis para la media de las diferencias.
18/3
Martes: Continuación con medias de dos poblaciones. Intervalos de confianza y pruebas de hipótesis para la media de las diferencias. Jueves: Intervalos de confianza y pruebas de hipótesis para la diferencia de pro porciones de dos poblaciones. Distribución F. F. Intervalos de confianza y pruebas de hipótesis para el cociente de varianzas de dos poblaciones.
4
25/3
Martes: Intervalos de confianza y pruebas de hipótesis para la diferencia de pro porciones de dos poblaciones. Distribución F. F. Intervalos de confianza y pruebas de hipótesis para el cociente de varianzas de dos poblaciones.
5
1/4
6
8/4 8/4
3
7
15/4
Diseño de Experimentos. Diseño Completamente Aleatorizado. Modelo paramétrico. Diseño Completamente Aleatorizado modelo paramétrico (continuación) (continuación) y DiDi seño Completamente Aleatorizado no paramétrico. paramé trico. Sábado 20/4 Ejercitación e integración 1er. Parcial Sábado
Estadístico de Chi cuadrado para pruebas de bondad de ajuste. Pruebas de Prue bas de Independencia.
8
22/9
9
29/4
Estadístico de Chi cuadrado para Pruebas Pruebas de Homogeneidad Ejercitación.
10
6/5 6/5
Regresión Lineal Simple. Supuestos Supuestos del Modelo Modelo y Estimadores. Dócima de de hipótesis e intervalo de confianza utilizando la t de Student.
11
13/5
Regresión Lineal Simple. Intervalos Intervalos de predicción. predicción. Coeficiente de determinación. ANOVA en la regresión.
12
20/5
Regresión Lineal Múltiple. Correlación Lineal Simple paramétrica.
13
27/5
Correlación Simple no paramétrica. Ejercitación.
14
3/6 3/6
Integración. Revisión y consulta. 2do. Parcial Sábado 8/6 a las 11 hs
15
10/6
16
17/6
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Bibliografía Cantatore de Frank, Norma M.: Manual de Estadística Aplicada. Ed. Hemis-
ferio Sur. 1ra. Edición. Buenos Aires. Capítulos: 4, 5, 6, 7, 8, 12 y 13. Cappelletti, Carlos A.: Elementos de estadística. Cesarini Hnos. Editores.
2da. Edición. Bs. As. Capítulos 8, 9, 10, 11, 13 y 14. Daniel, Wayne W.: Bioestadística. Base para el análisis de las ciencias de la
salud. 3ra. Edición. Uteha, Noriega Editores. México. Capítulos: 5, 6, 8, y 10.
NOTA IMPORTANTE:
L a cáte cáted dra publ publiica sol solam amente la GUIA UI A DE DE TRAB T RABAJ AJ OS PRACTICO PRACTI COS S y la GUI A DE DE F ORMULAS RMUL AS Y TABLAS TABL AS para la cursada de esta materia. Cualquier otra publicación NO CUENTA CON CON LA L A APROBACI APROBACIO ON DE DE L A CATED CATEDRA.
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Sistema de Evaluación de Estadística Analítica Se tomarán dos parciales, que serán calificados en una escala de 0 a 10, en forma global. Las condiciones de LIBRE, ASISTENCIA CUMPLIDA, REGULAR Y PROMOcumplen las situaciones situaciones con respecto a calificación y CIÓN se obtienen si se cumplen asistencia que abajo se detallan. Concurrencia a las clases c lases teórico-prácticas en un porcentaje: ASISTENCIA: Concurrencia LIBRE: inferior al 75% ASISTENCIA CUMPLIDA y REGULAR: mayor o igual al 75% PROMOCIÓN: mayor o igual al 80%
CALIFICACIÓN: SEGUNDO PARCIAL 1
L A I C R A P R E M I R P
2
3
1 LIBRE
LIBRE
LIBRE
AC
AC
2 LIBRE
LIBRE
LIBRE
AC
AC
3 LIBRE
LIBRE
LIBRE
AC
AC
4 AC
AC
AC
AC
AC
5 AC
AC
AC
AC
AC
REC
REC
REC
REC
REC
2P
2P
2P
2P
2P
REC
REC
REC
REC
REC
2P
2P
2P
2P
2P
REC
REC
REC
REC
REC
2P
2P
2P
2P
2P
REC
REC
REC
REC
2P
2P
2P
2P
REC
REC
REC
2P
2P
2P
6 7 8 9 10
4
COL
5
6
7
8
9
10
REC
REC
REC
REC
REC
1P
1P
1P
1P
1P
REC
REC
REC
REC
REC
1P
1P
1P
1P
1P
REC
REC
REC
REC
REC
1P
1P
1P
1P
1P
REC
REC
REC
REC
1P
1P
1P
1P
REC
REC
REC
1P
1P
1P
REG
REG
REG
COL
COL
COL
PROM
PROM
PROM
PROM
PROM
PROM
PROM
PROM
PROM
PROM
PROM
PROM
COL
PROM
PROM
PROM
PROM
PROM
COL
PROM
PROM
PROM
PROM
PROM
Siendo: AC: asistencia cumplida COL: coloquio PROM: promoción REC 2P: recupera 2do. Parcial REC 1P: recupera 1er. Parcial REG: regular
NOTA 1. Los alumnos que estén ausentes a un parcial y presenten certificado oportunamente en la cátedra lo rendirán en la fecha de recuperatorio y si posteriormente quedan en situación de recuperar un parcial se les asignará una fecha. 2. Los alumnos que recuperan algún parcial consiguen como máximo la condición
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Unidad 1: INFERENCIA I NFERENCIA para DOS POBLACIONES Objetivos específicos:
Comprender la importancia de diseñar experimentos. Analizar Analizar la adecuación adecuación de cada cada diseño diseño en en función función del del contexto contexto de la investigac investigación. ión. Aplicar Aplicar los conceptos conceptos de inferencia estadístic estadística a a la comparación comparación de dos poblaciones, poblaciones, utilizando utilizando como procedimientos la estimación y la prueba de hipótesis. Seleccionar el procedimiento de inferencia adecuado en función del objetivo y del cumplimiento de los supuestos. Resolver problemas e interpretar conclusiones aplicando los métodos de análisis sobre dos poblaciones. Contenidos temáticos:
Diseño de experimentos: necesidad, ventajas, propósitos, definiciones previas. Tipos de diseños y alcances. Revisión de conceptos relativos a la estimación puntual y por intervalos. Intervalos de confianza para la diferencia de medias y para la media de las diferencias. Estimaciones para la diferencia de dos proporciones, para el cociente de varianzas, y para el cociente de desvíos estándar. Revisión de conceptos relacionados con las pruebas de hipótesis. Prueba de hipótesis para: diferencia de medias en base a dos muestras independientes: diferencia de medias, cociente de varianzas, diferencias de proporciones. Muestras apareadas: media de las diferencias. Relación entre intervalo de confianza y prueba de hipótesis bilateral. Aplicaciones.
Glosario: Diseño de experimentos : experimento, unidad experimental, tratamiento, factor, niveles de un factor,
observación, efecto. Repetición, aleatorización, control local. Estudios observacionales, preexperimentales, cuasiexperimentales y experimentales. experimentales. Inferencia para dos poblaciones : Población, muestra. Parámetro. Estimador. Estimación. Estimador puntual. Intervalo. Intervalo de confianza. Nivel de confianza. Hipótesis de trabajo. Hipótesis estadística. Hipótesis nula y alternativa. Error tipo I y tipo II. Nivel de significación. Región crítica. Regla de decisión. Distribución F de Snedecor. Diferencia de medias y de proporciones, cociente de varianzas para muestras independientes. Muestras apareadas: media de las diferencias. El diseño de experimentos
La ciencia, tiene entre sus objetivos la explicación y comprensión de los acontencimientos. acontencimientos. Un requisito fundamental en toda ciencia fáctica es el contraste de las hipótesis planteadas, poniendo a prueba las mismas mediante una confrontación con la experiencia. El diseño experimental crea las condiciones para el contraste de la hipótesis y brinda la metodología estadística correspondiente para el análisis de los datos. Es el proceso de planear un experimento para obtener datos apropiados que puedan ser analizados mediante métodos estadísticos, con objeto de producir conclusiones válidas y objetivas. La metodología estadística es el único enfoque objetivo para analizar un problema que involucre datos sujetos a errores experimentales. Así es que hay dos aspectos en cualquier problema experimental: el diseño del experimento y el análisis estadístico de los datos. El propósito del diseño experimental es controlar la máxima cantidad de información pertinente al problema bajo investigación. Sin embargo también es importante que el diseño o plan sea tan simple como sea posible, a fin de ahorrar tiempo, dinero, personal y material experimental. Para que la metodología de diseño de experimentos sea eficaz es fundamental que el diseño sea el adecuado. Un experimento puede realizarse por alguno de los siguientes motivos: Determinar los factores principales que influyen sobre la variable respuesta.
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Para poder realizar un buen diseño experimental, es necesario previamente comprender el problema que se desea estudiar, planteándose un conjunto de preguntas clásicas: 1- ¿Cuáles son las características de interés? 2- ¿Qué variables afectan a las características que se van a analizar? 3- ¿Cuántas veces debería repetirse el experimento? 4- ¿A partir de qué valor se considerará que el efecto efecto es significativo? Lo cual conduce a elegir las variables más apropiadas y sus niveles de medición, elegir la o las respuestas a evaluar y el modelo de diseño. Para responder estas preguntas es necesario definir claramente claramente algunos términos términos fundamentales: contr o Experimento : es un ensayo o una observación realizado bajo condiciones establecidas y controladas por el experimentador, susceptible de repetirse bajo las mismas condiciones. Variable de interés o respuesta : es la variable que se desea estudiar. Unidad experimental : es la parte más pequeña de material experimental, entidad física o sujeto, en la que se aplica un tratamiento una sola vez. También puede entenderse como cada una de las reproducciones del experimento. Tamaño del Experimento : es el número total de observaciones recogidas en la ejecución del experimento. Ejemplo: si se asignan 10 gallinas a cada una de tres dietas el tamaño del experimento es 30.
Factor : es una variable que se sospecha que puede ejercer influencia sobre la variable respuesta
Factor controlado : se denomina así a una variable manipulada por el investigador o variable in-
Tratamiento : conjunto de condiciones experimentales o procedimientos creados para el experi-
de interés.
dependiente, a fin de estudiar su influencia sobre la variable de interés o dependiente. Algunos autores la denominan variable de entrada al proceso. Ejemplo: si pensamos que la temperatura o la humedad pueden afectar a la conservación de cierta propiedad de un alimento o medicamento, se puede controlar manteniendo dicho producto con tres valores distintos de temperatura. Niveles del factor : son cada una de las categorías, o valores, o formas específicas que adopta la variable independiente o controlada. Ejemplo: en el caso de las tres dietas, el factor dieta tiene tres niveles; en el caso del rodeo, el factor tiene dos niveles. Tipos de factores : existen factores cuantitativos, cuyos niveles son cantidades numéricas, y cualitativas, cuyos niveles son procedimientos o cualidades. Ejemplo de factor cuantitativo puede ser la cantidad de fertilizante adicionado a las parcelas de cultivo por hectárea con niveles: 10kg/ha – 20 kg/ha -30 kg/ha de fertilizante. Ejemplo de factor cualitativo puede ser el tipo de nutriente adicionado a una dieta con niveles: potasio, magnesio y calcio. mento en función de la hipótesis de investigación a las que se someterá a las unidades experimentales en un diseño elegido. Con varios factores es una de las combinaciones específicas de los niveles de los factores de estudio, y en un diseño unifactorial es uno de los distintos niveles del factor en el caso. Por ejemplo: si se asignan tres dietas distintas a las gallinas de un criadero, cada una de las dietas es un tratamiento. Si en un tambo se combinan tres raciones de alimentación dos rodeos con vacas en ordeñe (uno con vacas de alta producción y el otro con las de baja producción). Cada combinación de rodeo y ración constituye un tratamiento (6 tratamientos). Observación : valor que asume una variable, también denominada variable respuesta, en una determinada realización del experimento, es decir cada registro realizado en el contexto del experimento de la variable respuesta. Efecto : diferencia entre los valores medios de la variable respuesta en presencia y ausencia de un nivel del factor. Si la variable respuesta de interés es el engorde semanal medido en gramos de una gallina con cierta dieta enriquecida, el efecto es la diferencia entre el engorde medio con la dieta enriquecida y el engorde medio con la dieta tradicional, ambos medidos en gramos. Diseño equilibrado o balanceado : es el diseño en el que todos los tratamientos son asignados a un número igual de unidades experimentales, en el cual se obtiene la misma cantidad de repeticiones
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Repetición: cuando un tratamiento es aplicado a más de una unidad experimental. Las observacio-
nes repetidas con las mismas condiciones experimentales en el contexto de un experimento no coinciden necesariamente, y por lo tanto una de las cuestiones fundamentales a la hora de diseñar un experimento es la selección del tamaño de muestra o número de repeticiones adecuado en cada contexto. contexto . Las razones por las cuales es deseable realizar repeticiones del experimento son: a- Proporcionar una estimación estimación del error experimental experimental (error generado por causas no controladas por el experimentador), que actúa como unidad básica de medida para indicar el significado de las diferencias. b- Obtener mayor mayor precisión en la estimación. c- Permitirnos extender el alcance alcance de la inferencia inferencia relativa al experimento. El error experimental según el contexto puede reflejar: errores de experimentación errores de observación errores de medición variación del material experimental El error experimental puede reducirse generalmente adoptando una o más de las técnicas siguientes: usando material experimental tan homogéneo como sea posible. utilizando información proporcionada por otras variables aleatorias teniendo cuidado al dirigir el experimento usando un diseño experimental más eficiente. Aleatorización: Todo procedimiento de prueba se basa en un conjunto de supuestos que deben
satisfacerse para que la prueba resulte válida. Una de las suposiciones más frecuentes es que las observaciones, o los errores en ellas, son independientes. Dicho en otras palabras la aleatorización hace válida la prueba. Control local: Se denomina de esta manera al conjunto de acciones que implementa el investigador con el fin de reducir al máximo posible el error experimental manteniéndolo en un rango de variación manejable. Por ejemplo: selección de unidades experimentales homogéneas, división en bloques, calibración de instrumentos, etc. Tipos de estudios de investigación
Los estudios observacionales son un conjunto de estudios en los que no hay intervención por parte del investigador y este se limita a medir las variables que define en el estudio. Por ejemplo, los estudios epidemiológicos. Ventajas de los estudios observacionales
1. Son más prácticos y factibles de realizar, ya que la cooperación de los sujetos es menos necesaria. 2. Sus resultados son más más generalizables a poblaciones, geográfica geográfica o demográficamente demográficamente defini das.
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Un estudio de intervención , también llamado estudio experimental , es un estudio caracterizado por la manipulación artificial del factor de estudio por el investigador y por la aleatorización de los casos o sujetos en dos grupos, llamados control y tratado. Cuando la característica de la aleatorización en el estudio no se cumple, se dice que el estudio es cuasiexperimental . La falta de aleatorización de los estudios cuasiexperimentales indica que no existe
manera de asegurar la equivalencia inicial de los grupos denominados experimental y de control. También es usual que, en un experimento, se utilicen controles históricos. El problema que presenta este tipo de diseño es que el grupo actualmente en tratamiento puede presentar importantes diferencias relativas al tratamiento respecto al grupo de control histórico. Los trabajos con controles históricos están generalmente sesgados a favor del tratamiento, mientras que los experimentos aleatorios evitan este tipo de sesgo. PROBLEMAS RESUELTOS
1) Mediante los estudios ecográficos, los bebés pueden actualmente ser observados mientras están en
el seno materno. Sin embargo, gran cantidad de experimentos desarrollados en animales de laboratorio dieron como resultado que la aplicación de ultrasonidos podía ser la causa de que el peso al nacer fuese inferior al normal. Ante el temor de que esta e sta conclusión concl usión fuese aplicable a los humanos, un grupo de especialistas esp ecialistas del Hospital John Hopkins de Baltimore puso en marcha un estudio para investigar el tema. En el mismo se observó el peso al nacimiento de los bebés que estuvieron expuestos a controles ecográficos (ultrasonido) y de los que no estuvieron expuestos. También en este caso los bebés expuestos al ultrasonido durante el embarazo pesaban en su mayoría al nacer menos que aquellos que no lo habían estado, pero un dato a tener tener en cuenta es que los obstetras recomendaban el ultrasonido cuando sospechaban que el embarazo no se desarrollaba con normalidad. a) ¿Se trata de un estudio observacional o experimental? ¿Por qué? b) ¿Puede concluirse que el ultrasonido influye sobre el peso del nacimiento? Solución:
a) Se trata de un estudio observacional, porque no hay intervención del investigador. b) Los bebés expuestos al ultrasonido y los no expuestos presentaban diferencias que no tenían nada
que ver con el hecho de ser tratados o no. De modo tal que los investigadores tuvieron un conjunto de factores de confusión con el cual enfrentarse. La conclusión del estudio fue, por lo tanto, que las ecografías y el menor peso de los bebés tenían una causa común: problemas durante el embarazo. 2) Mediante la siguiente experiencia se quiere determinar si una droga reduce el nivel promedio de gluco-
sa en sangre (glucemia) en una línea de ratas diabéticas. Se tomaron al azar 40 ratas de esta línea y se les suministró la droga (grupo tratado). Al mismo tiempo se tomaron otras 30 ratas de la misma línea y se les suministró un placebo (grupo control). Los niveles sanguíneos de glucosa (mg/ml) en las ratas fueron:
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2,17
) a g o r D ( s o d a v r e s b o s e l i t n a u C
n= 40 r= 0,994 (Droga)
1,95
Shapiro-Wilks (modificado) Variable n Droga 40
Media D.E. W* p(una cola) 1,73 0,20 0,97 0,7640
1,72
1,50
1,27 1,27
1,50
1,72
1,95
2,17
Cuantiles de una Normal(1,7328,0,04161)
2,48 ) o b e c a l P ( s o d a v r e s b o s e l i t n a u C
n= 30 r= 0,989 (Placebo)
2,27
Shapiro-Wilks (modificado) Variable n Media D.E. W* p(una cola) Placebo 30 2,02 0,20 0,97 0,7499
2,05
1,84
1,62 1,62
1,84
2,05
2,27
2,48
Cuantiles de una Normal(2,022,0,038086)
a) ¿Es la droga efectiva para reducir el nivel promedio de glucosa en sangre, al 5%? Asuma que la
droga no modifica la varianza poblacional del nivel de glucosa en sangre, y que ésta es conocida, simbólicamente 2droga= 2placebo =0,04 mg2/ml2
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Solución: a)
La hipótesis de trabajo que se desea poner a prueba es: “El empleo de la droga disminuye el nivel medio de glucosa en sangre de ratas diabéticas de la línea”
Verificación de supuestos: Para Para poder plantear las hipótesis hipótesis estadísticas y llevar a cabo la prueba, hay que verificar los supuestos teóricos necesarios. En este caso, los supuestos son que ambas variables (X1 y X2) sean independientes y se distribuyan normalmente. El supuesto de independencia se cumple por la forma en que se realizó el experimento: a un grupo de ratas se le suministró la droga y a otro grupo, también tomado al azar, se lo trató con placebo. Para X1: se realizó un gráfico de cuantil-cuantil (qqplot) para estudiar la normalidad de la variable. En este
gráfico se comparan dos distribuciones, la de los datos muestrales y la de una normal. Cuando los puntos están perfectamente alineados, se infiere que la distribución es exactamente normal, si los puntos estan muy cercanos a la línea, la distribución es aproximadamente normal, grandes apartamientos de esta estructura indican falta de normalidad. Esto sin embargo no tiene la fuerza de un test estadístico es una técnica exploratoria.
Qqplot
Observando el gráfico se puede ver que los puntos no se alejan mucho de la recta, sin embargo, por ser un gráfico, no se puede hacer inferencia sobre el comportamiento distribucional de la variable a nivel poblacional. Para poder concluir a nivel poblacional es necesario un test de normalidad. En Elementos de Estadística se estudió la prueba Shapiro-Wilks, para verificar normalidad, y cuyas hipótesis son:
2.17
) 1 X ( s o 1.95 d a v r e s 1.72 b o s e l i 1.50 t n a u C
1.27 1.27
1.50
1.72
1.95
2.17
Cuantiles de una Normal(1.7328,0.04161)
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Análogamente se estudia la normalidad de la variable X2: H 0 : X 2 ~N
Qqplot
2
;
2 2
H1 : X 2 no se distribuye normalmente N
2.48
) 2 X ( s o 2.27 d a v r e s 2.05 b o s e l i t 1.84 n a u C
2
2 2
;
Shapiro-Wilks (modificado)
Variable X2
n Media D.E. 30 2.02 0.20
W* p (una cola) 0.97 0,7499
1.62 1.62
1.84
2.05
2.27
2.48
Cuantiles de una Norm al(2.022,0.038086) al(2.022,0.038086)
Como p-valor= 0,7739 y es mayor que =0,10, no se rechaza la hipótesis nula, por lo tanto, con un nivel de significación del 10% se puede decir que la variable X 2 (nivel de glucosa en sangre de una rata tratada con placebo, en mg/ml) se distribuye normalmente. Una vez verificado el supuesto teórico se puede seguir adelante con la prueba. Nota:este test no será necesario si la información asegura distribución normal de la variable.
Hipótesis estadísticas. El interés del investigador es probar si la droga disminuye el nivel medio de glucosa en sangre, por lo tanto quiere saber si la media del nivel de glucosa en sangre de ratas tratadas con droga es menor que la media del nivel de glucosa en sangre de las ratas tratadas con placebo. Simbólicamente: 1 , esta expresión no lleva el signo igual, por lo tanto debe corresponder a 2 la hipótesis alternativa. Es decir que las hipótesis estadísticas son: :
1
2
H 1 :
1
2
H 0
Equivalentemente podría escribirse
H 0
:
1
2
0
H 1
:
1
2
0
o también
H 0
:
2
1
0
H 1
:
2
1
0
Cualquiera de estas formas expresan las mismas hipótesis estadísticas . Sin embargo hay que
elegir una expresión para poder continuar con la prueba manteniendo la elección a lo largo de todo el análisis y por sobre todo concluir para las hipótesis elegidas. Si esto no se mantiene deja de tener validez la prueba o peor aún, se podría estar estar concluyendo erróneamente. En este caso se va a trabajar trabajar H 0
:
1
2
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Región crítica: Observando la hipótesis alternativa (del par de hipótesis elegidas), se ve que la región crítica es unilateral izquierda. Por lo tanto el valor crítico es: Z 0,05 1,64 y la región crítica es: Z H 0 1, 64 Regla de decisión: Rechazo H0 si Z H 0 No rechazo H0 si Z H 0
1,64 1,64
Cálculo de ZHo: Hasta este momento no utilizamos los valores muestrales, excepto en la verificación de supuestos, sin embargo se podría haber hecho con muestras piloto y recién en esta instancia extraer las muestras para el análisis. Antes de calcular el valor del estadístico de prueba hay que calcular las medias muestrales utilizando las fórmulas dadas en la unidad de estadística descriptiva de Elementos de Estadística: X1 1, 73; X 2 2, 02 . Hay que tener en cuenta que la prueba se está realizando bajo la hipótesis nula que contiene el caso en que las medias poblacionales son iguales, por lo tanto la diferencia de las medias poblacionales es cero, es decir que 1 0 . Reemplazando estos valores y 2 el resto de la información en la fórmula nos queda:
Decisión: Se rechaza la hipótesis nula porque ZHo ZCALCULADO < ZCRITICO.
= -6, 01 017 ,
es menor que –1,64, o sea que
Conclusión: Con un nivel de significación de 5% tengo evidencia suficiente para rechazar la hipótesis nula ( H 0 : 1 ), por lo tanto la media poblacional del nivel de glucosa en sangre de ratas 2 diabéticas tratadas con droga es menor que la media poblacional del nivel de glucosa en sangre de ratas diabéticas tratadas con placebo, en estas poblaciones de ratas diabéticas en estudio. Por lo tanto puedo decir que la droga es efectiva.
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3) Se tomó una muestra aleatoria de 21 cerdos Yorkshire del norte de la provincia de Buenos Aires. Los
mismos tenían 3 meses de edad y pesos homogéneos, y se los separó, aleatoriamente, en dos lotes. Al lote 1 se le asignó una ración estándar (A) y al lote 2 otra con distinta formulación (B). La siguiente tabla contiene las ganancias de peso de cada animal, luego de 30 días de experiencia, expresadas en kg. Lote 1(A) Lote 2(B)
24 26
26 32
25 28
23 25
28 29
27 27
28 28
24 27
29 27
29 28
30
Por estudios anteriores se sabe que ambas variables se distribuyen normalmente con varianzas iguales, pero desconocidas. a) ¿Se puede suponer, al 5%, que la ganancia media de peso de los animales alimentados con la ración B supera significativamente la ganancia media de peso de los animales alimentados con ración A? b) Construir un intervalo para la diferencia de medias al 95%. ¿Qué puede concluir? Datos del problema:
Variables en estudio: X A: ganancia de peso de un cerdo Yorkshire de 3 meses de edad del norte de la provincia de Bs. As. alimentado con la ración estándar A XB: ganancia de peso de un cerdo Yorkshire de 3 meses de edad del norte de la provincia de Bs. As. alimentado con la formulación distinta B Tamaños de las muestras: muestras: n A=10 y nB=11 Varianzas Poblacionales:
2 A =
Nivel de significación: =0,05 Nivel de confianza: 1 Solución a)
= 0,95
2 B
=
2
(desconocidas)
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Nivel de significación: =0,05 Variable pivotal: En este caso, como en el ejercicio anterior, se está realizando un test para la diferencia de medias poblacionales, por lo tanto hay dos opciones para la variable pivotal (Z o t-Student). Como las varianzas poblacionales son desconocidas no se puede utilizar la variable Z, por lo tanto se utilizará la variable pivotal t de Student, cuya fórmula es:
. Donde Sa es la raíz cuadrada positiva de la varianza amalgamada, es decir que es un promedio ponderado entre la varianza muestral de la variable X A y la varianza muestral de la variable XB y estima a la única varianza poblacional que se desconoce, 2. Región crítica: Observando la hipótesis alternativa planteada se deduce que la región crítica es unilateral derecha (es decir que se rechaza la hipótesis nula a valores grandes de la variable pivotal). El valor crítico que se utiliza es tn A nB 2;1 t10 11 2;0,95 t 19;0,95 1,729 , por lo tanto la región crítica es: t 1,729 . Gráficamente:
Regla de decisión: Rechazo H0 si t H 0
1,729 y no rechazo H0 si t H 0
1,729
Cálculo: Para obtener el valor calculado del estadístico de prueba, hay que realizar ciertos cálculos auxiliares ( X A ; XB y S2a ) utilizando las fórmulas habituales para las medias y las varianzas muestrales, y la siguiente fórmula para la varianza amalgamada:
(
1) S 2
(
1)S 2
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Respuesta: Se puede suponer, al 5%, que la ganancia media poblacional del peso de los cerdos alimentados con la ración B supera significativamente a la media poblacional del peso de los cerdos alimentados con la ración A. Para este problema, la salida de InfoStat correspondiente es: Prueba T para muestras Independientes
Gr(1) {A}
Gr(2) {B}
n(1) 10
n(2) 11
media(1) media(1) media(2) 26,30 27,91
p(Var.Hom.) 0,6623
T -1,78
p prueba 0,0452 UnilatI
Nota: InfoStat compara grupos en orden alfabético, por lo cual la prueba es unilateral izquierda, o sea
que utiliza H1: A- B<0. Para la comparación es indistinta la forma en que se plantea la diferencia, siempre que se respete el sentido de la misma. El valor de t observado es el mismo que obtuvimos al aplicar la fórmula, pero de signo opuesto, por haber invertido el orden de la diferencia. Como puede verse, al realizar la Prueba T para muestras independientes, también se realiza una prueba para evaluar la Homogeneidad de Varianzas, el p-valor es 0,6623, por lo que se cumple este supuesto. En este caso, en que la región crítica es unilateral izquierda, el cálculo del p valor es:, p valor= P(t V.Calc.) = P(t19 -1,78)
Ahora, si consideramos la región crítica derecha que planteamos al principio, el p valor se grafica y se calcula de la siguiente forma, dado que la región crítica es unilateral derecha: p valor= P(t V.Calc) = P(t19≥ 1,78) b) La fórmula del intervalo de 95% de confianza para la diferencia de medias se deduce de la distribución
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Paciente Hemoglobina (mg%)
1
2
3
4
5
6
7
8
9
10
Inicial (I) (I)
12,2 11,3 14,7 11,4 11,5 12,7 12,3 13,0 12,7 13,0
Después de 3 meses (F, o final)
13,0 13,4 16,0 13,6 14,0 13,0 14,2 15,1 15,9 14,5
¿Hay aumento significativo de hemoglobina después del tratamiento al nivel del 5%? Qqplot -0,30
Shapiro-Wilks (modificado)
Variable n Media D.E. W* D 10 -1,79 0,84 0,97
p (una cola) 0,9425
) D ( s o d a v r e s b o s e l i t n a u C
-1,02
-1,75
-2,48
-3,20 -3,20
- 2,48
- 1,75
-1,02
- 0,3 0
Cuantiles de una Norm al(-1,79,0, al(-1,79,0,70989) 70989)
Datos del problema:
Variable en estudio: concentración de hemoglobina hemoglobina en sangre (en mg%) mg%) al inicio del tratamiento D: diferencia entre la concentración con vitamina B12 y la concentración de hemoglobina en sangre (en mg%) después de tres meses del
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Qqplot -0,30
H 0 : D se distribuye normal (
D
H1 : D no se distribuye normal (
2 D
; D
;
) 2 D
)
Shapiro-Wilks (modificado)
Variable n Media D.E. W* D 10 -1,79 0,84 0,97
p (una cola) 0,9425
) D ( s o d a v r e s b o s e l i t n a u C
-1,02
-1,75
-2,48
-3,20 - 3,20
- 2,48
- 1,75
- 1,02
- 0,30
Cuantiles de una Norm al(-1,79,0, al(-1,79,0,70989) 70989)
Como 0,9425 es mayor que 0,10, no se rechaza la hipótesis nula. Entonces, con un nivel de significación del 10% se puede concluir que la diferencia entre la concentración de hemoglobina en sangre (mg%) al inicio del tratamiento con vitamina B12 y la concentración de hemoglobina en sangre (mg%) luego de 3 meses de tratamiento con vitamina B12 en pacientes con anemia perniciosa se distribuye normalmente. 2 Simbólicamente D N ( )
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Cálculo: Para obtener el valor calculado del estadístico de prueba hay que realizar ciertos cálculos auxiliares ( d y s D2 ), utilizando las fórmulas habituales para la media muestral y la varianza muestral, sobre las 10 diferencias. Utilizando los valores calculados para d i (ver la tabla correspondiente al plantear la forma de realizar la misma), se obtuvo d 1, 79 y s D2 0, 71 , reemplazando en la fórmula de la variable pivotal:
t H 0
1, 79 0,84 10
1, 79 0,26
6,7 6, 7 .
Como el valor observado –6,7 es menor que –1,83, vale decir pertenece a la región crítica, se rechaza la hipótesis nula. Conclusión: Con un nivel de significación del 5% tengo evidencia suficiente para rechazar la hipótesis nula ( D 0 ), por lo tanto la media poblacional de las diferencias entre la concentración de hemoglobina en sangre (mg%) de pacientes con anemia perniciosa al inicio del tratamiento y la concentración de hemoglobina en sangre (mg%) de pacientes con anemia perniciosa después de tres meses de iniciado el tratamiento con vitamina vitamina B12 es menor que cero, en la población de pacientes con anemia perniciosa. Por lo cual la hemoglobina aumenta significativamente luego del tratamiento con vitamina B12. A continuación se da la salida del programa InfoStat para este problema Prueba T para un parámetro Valor del parámetro probado: 0
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cación de la normalidad de las variables, ya que solamente hay que verificar que las variables cumplan con las condiciones de una variable binomial, es decir: Que cada repetición del experimento tenga dos resultados posibles (éxito y fracaso). Si lo aplicamos al ejemplo veremos que las dos posibles respuestas que podemos obtener, al encuestar a una persona, son: “que le guste el programa del sábado a la noche” y “que no le guste el progr ama del sábado a la noche”.
Que los resultados (éxito y fracaso) sean mutuamente excluyentes en una misma repetición. Que los resultados (éxito y fracaso) sean independientes de repetición en repetición. Que el número de repeticiones esté prefijado de antemano y que la probabilidad de éxito sea constante a lo largo de todas las repeticiones del experimento aleatorio.
En este caso ambas variables cumplen con estas condiciones. Estimación puntual: Lo que se quiere estimar es: ”la diferencia entre las proporciones poblacionales poblacionales de adultos y adolescentes que ven el programa y les gus ta”, simbólicamente: p1-p2. Por lo tanto la estimación puntual de esta diferencia es la diferencia entre las proporciones estimadas, p1 p2 . ˆ
ˆ
p1
cantida cantidad d de adulto adultoss que ven ven el progr programa ama los los sábado sábadoss a la noche noche y les gusta gusta
100
cantidad total de adultos
400
ˆ
p2
0,25
cantidad cantidad de de adolesc adolescente entess que ven ven el programa programa los sábado sábadoss a la noche noche y les les gusta gusta
300
cantidad total de adolescentes
600
ˆ
0,5 0, 5
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La región crítica es bilateral, y está formada por los valores de Z mayores o iguales a 1,96, y los menores o iguales a -1,96. La regla de decisión es:
RECHAZO H0 si Z obs 1,96 o Zobs ≤ -1,96 NO RECHAZO H0 si -1,96 < Zobs < 1,96 p
x1
x2
100 300
n1
n2
400 600
ˆ
Como Zobs = -7,81 la decisión es
0,4 0, 4
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Utilizando la fórmula de la varianza muestral se obtiene:
2
2
Por lo tanto el estimador puntual del cociente entre S A y SB es:
S A2
4,9 4, 9
S B2
3,69
1,3279
(Nota: En este caso se estima el cociente entre la varianza poblacional de A sobre la varianza poblacional de B, pero también podríamos resolver este ejercicio haciendo el cociente inverso, dado que en el enunciado no hay ninguna orientación en especial para realizarlo.) Intervalo de confianza: La fórmula del intervalo de confianza para el cociente de varianzas se deduce de la variable pivotal que se utiliza para estudiar el cociente de varianzas, cuya fórmula es:
F
S A
2
S A
2
S B
A
2
S B
2
2
~F( n
A
1), ( nB 1)
o
F
2 A
~F ( n
A
1), ( nB 1)
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P
2 A 2 B 2 A 2 B
1
S
F
( n A 1),( nB 1);
P
1
1 F
( n A 1),( nB 1); 1
( n A 1),( nB 1); 1
S
2
2
1
F
2 A 2 B 2 A 2 B
S S
1
1
F
( n A 1),( nB 1);
2
2
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S A2 F
2 A 2 B 2 B
S
~F( n A
1),( nB 1)
o
F
S A2 S B2 2 A 2 B
~F ( nA
1), ( nB 1)
Región crítica: Observando la hipótesis alternativa, al ternativa, se ve que la región crítica es unilateral derecha. El valor crítico que la determina, debe buscarse en la tabla de la distribución de F de Snedecor y es: F n 1,n 1 ;1 F10 1,1,9 1 ;1 0, F 9,10 ;0,95 3,02 , por lo tanto la región crítica está determinada por 0 ,05 A
B
F 3,02
Regla de decisión: Rechazo H 0 si F H 0
3,02 y no rechazo H0 si F H 0
3,02
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Como podemos ver en el boxplot, las distribuciones de rendimiento académico de los individuos que provienen de establecimientos estatales y privados, son similares. Ambas son asimétricas positivas, ya que prácticamente coinciden el C 2 con el C1. La falta de normalidad ya la habíamos confirmado con la prueba de Shapiro Wilks.
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Al nivel del 5%, existen evidencias evidencias suficientes suficientes para rechazar H0, por lo que la mediana poblacional del rendimiento académico de los individuos que han realizado estudios secundarios en establecimientos estatales es distinta a la mediana poblacional de los que han realizado sus estudios secundarios en establecimientos privados. Como respuesta a la pregunta, podemos decir, que, al 5%, los rendimientos académicos difieren significativamente entre los individuos que han realizado sus estudios secundarios, al comparar establecimientos estatales y privados. Utilizando InfoStat, tenemos que aplicar el test de Wilcoxon para muestras independientes, y obtenemos la siguiente salida, en la que figuran medidas resumen, el estadístico correspondiente, y el p-valor de la prueba.
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Para X1: se realizó un test de Shapiro Wilks cuyas hipótesis son:
Al realizar el test, utilizando InfoStat, se obtuvieron los siguientes resultados: Shapiro-Wilks (modificado)
Variable X1
n 12
Media D.E. 119.58 1.049
W* p (una cola) 0.8715 0.1683
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rianzas poblacionales son desconocidas y desiguales, por lo tanto se utiliza una t, con la siguiente ex2
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reemplazando se obtiene que
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3) Las personas que tienen el síndrome de Raynaud sufren un súbito deterioro en la circulación sanguínea
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5) Un estudio llevado a cabo para probar si la aspirina afecta el tiempo de coagulación, se tomó una
muestra de 12 adultos varones. El tiempo de protrombina, que mide el tiempo en segundos entre el
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Cuadro de Análisis de F.V. SC Modelo 43,38 Aceite 43,38 Error 425,50 Total 468,88
la Varianza (SC tipo III) gl CM F p-valor 3 14,46 0,95 0,4292 3 14,46 0,95 0,4292 28 15,20 31
Como p-valor=0,4292, no rechazo H 0, y se puede considerar que las varianzas poblacionales de la concentración de esculeno en los aceites vegetales de maíz, maní, girasol y soja son iguales (supuesto de homoscedasticidad). d) Escriba las hipótesis de interés, y teniendo en cuenta la salida de computadora concluya al 5%, Por todo el análisis anterior no es válido utilizar un DCA Modelo 1 o fijo Paramétrico, Por otro lado, los cuatro vegetales son independientes y la variable concentración de esculeno en aceite es al menos de escala ordinal, por lo que selecciono la salida correspondiente a la Prueba de Kruskal Wallis. Prueba de Kruskal Wallis Variable Aceite Esculeno Girasol Esculeno Maíz Esculeno Maní Esculeno Soja Si simbolizamos con
N 8 8 8 8
Medias D.E. 10,00 3,38 21,00 6,48 21,50 7,98 11,25 5,06
Medianas 10,00 19,50 21,50 10,00
H 17,79
p 0,0005
a la mediana poblacional para la concentración de esculeno en aceite.
Estamos suponiendo en este caso el modelo: Yij = θi + ij Donde: i es el nivel del factor grupo y j es el número de observación dentro del grupo; θi es la mediana del i-simo grupo y la distribución de ij no se conoce. Los parámetros de este modelo son los θi. Hipótesis estadísticas H0 : θ1= θ2= θ3= θ4= θ H1 : algún θi≠ θ
i=1, 2, 3, 4
Es decir que se quiere probar si la variable concentración de esculeno en aceite tiene la misma posición para los cuatro vegetales: maíz, maní, girasol y soja. Conclusión: A un nivel de significación del 5%, hay evidencias suficientes para rechazar H 0 por lo que se puede suponer que al menos alguna de las medianas poblacionales de los cuatro vegetales (maíz, maní, girasol y soja), difiere de las restantes para la variable concentración de esculeno, en esta población de aceites vegetales en estudio. Nota: La potencia-eficiencia de la prueba de Kruskal Wallis comparada con el ANOVA de un factor modelo modelo 1 es de 3/ = 95,5%, Algunos Algunos experimentadores experimentadores para trabajar trabajar con el ANOVA paramétrico paramétrico ante ésta situación aplican alguna transformación a la variable respuesta (por ejemplo: ytranf = ln(y), ytranf = 0,5 y , ytranf = 1/y, etc,) para lograr que se cumplan los supuestos con respecto a al variable transformada. Nosotros no veremos transformaciones pues escapa de los alcances de esta materia.
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Tratamientos
Estadística descriptiva Tratamientos Variable 1 Ganancia 2 Ganancia 3 Ganancia 4 Ganancia 5 Ganancia
1
2
3
4
5
43
54
62
61
85
49
54
55
66
83
39
50
59
62
89
41
48
57
64
91
43
51
60
68
89
46
55
56
62
82
n 6 6 6 6 6
Media 43.50 52.00 58.17 63.83 86.50
D.E. 3.56 2.76 2.64 2,71 3.67
Mín 39.00 48.00 55.00 61.00 82.00
Máx 49.00 55.00 62.00 68.00 91.00
Mediana 43.00 52.50 58.00 63.00 87.00
A partir de los gráficos y las salidas correspondie correspondientes, ntes, responda responda los siguientes siguientes ítems: a) Verificar si se cumplen los supuestos para un DCA modelo fijo (concluya u opine para cada supuesto) Normalidad .................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... Homoscedasticidad ........................................................................................................................................ ......................................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... b) Teniendo en cuenta lo anterior y que el nivel es del 5% La salida elegida es ....................................................................................................... ................................ Las hipótesis plantea planteadas das son: ............................................................................................... ......................... Como p-valor es .......................... la decisión es ......................................, por lo que la conclusión es ......................................................................................................................................................................... ......................................................................................................................................................................... SALIDAS Y GRAFICOS OBTENIDOS POR COMPUTADORA: Tabla 1 Shapiro-Wilks (modificado) Variable n Media REGanancia 30 0.00
D.E. 2,88
W* 0.92
p (una cola) 0.1103
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Cuadro de Análisis de la Varianza (SC tipo III) F.V. SC gl CM F p-valor Modelo 4,03 4 1,01 0,41 0,8031 Tratamientos 4,03 4 1,01 0,41 0,8031 Error 62,21 25 2,49 Total 66,24 29 6,00
5,88 n= 30 r= 0,9 83 ( RDUO_C RDUO_Colum olumna2) na2)
) a i c n a n 2,94 a G _ O U D 0,00 R ( s b o s e l i -2,94 t n a u C
a 3,25 i c n a n a G 0,50 _ O U D R -2,25
-5,00 41,35
53 , 1 8
65 , 0 0
7 6, 8 3
-5,88 - 5,8 8
88,65
R² Aj 0,96
N 6 6 6 6 6
2,9 4
5 ,8 8
CV 5,10
Cuadro de Análisis de la Varianza (SC tipo III) F.V. SC gl CM Modelo 6320,13 4 1580,03 Tratamientos 6320,13 4 1580,03 Error 240,67 25 9,63 Total 6560,80 29 Tabla 4 Prueba de Kruskal Wallis Variable Tratamientos Ganancia 1 Ganancia 2 Ganancia 3 Ganancia 4 Ganancia 5
0 ,00
Cuantiles Normal(6,09E-015,8,29)
PRED_Ganancia
Análisis de la varianza Variable N R² Ganancia 30 0,96
- 2,9 4
Medianas 43,00 52,50 58,00 63,00 87,00
F 164,13 164,13
H 27,35
p-valor <0,0001 <0,0001
p <0,0001
4) Un fisiólogo estudió la función pituitaria de las gallinas ponedoras asociada a cada etapa del régimen estándar para muda forzada de plumas que usan los productores de huevos con el fin de mantener a las aves en producción. Las etapas de la dieta son cinco: (A) premuda, previa al inicio del régimen; (B) ayuno de 8 días; (C) 60 gr de salvado durante 10 días; (D) 80 gr de salvado durante 10 días; y (E) mezcla de malta durante 42 días. En el estudio se utilizaron 25 gallinas elegidas aleatoriamente de la población. Todas fueron puestas bajo la misma dieta en jaulas. Después de cada etapa, se seleccionaban aleatoriamente grupos de cinco y se las sacrificaba. Entre los compuestos medidos, el fisiólogo estaba interesado en saber si las distintas etapas afectaban la concentración de T3 en suero (medida en ng/dl). Se obtuvo como resultado el siguiente conjunto de observaciones:
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80 g de Salvado (D) Mezcla de malta (E) Estadística descriptiva Etapas Variable n a T3 5 b T3 5 c T3 5 d T3 5 e T3 5
112.47 83.14 Media 94,58 118,96 198,65 114,38 84,53
D.E. 4,03 5,80 6,11 4,15 4,15
117.51 89.59 Mín 90,45 115,00 192,50 110,00 79,21
Máx 99,38 129,06 207,31 119,92 89,59
119.92 87.76
112.01 82.94
Mediana Q1 94,09 91,00 117,61 115,23 197,18 194,00 112,47 112,01 83,14 82,94
110 79.21
Q3 98,00 117,90 202,25 117,51 87,76
A partir de los gráficos y las salidas salidas correspondientes, correspondientes, responda responda los siguientes ítems: ítems: a) Verificar si se cumplen los supuestos para un DCA modelo fijo (concluya u opine para cada supuesto) Normalidad .................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... Homoscedasticidad ........................................................................................................................................ ......................................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... b) Teniendo en cuenta lo anterior y que el nivel es del 5% La salida elegida es ........................................................................................................................................ Las hipótesis plantea planteadas das son: ............................................................................................... ......................... Como p-valor es .......................... la decisión es ......................................, por lo que la conclusión es ......................................................................................................................................................................... .........................................................................................................................................................................
SALIDAS Y GRAFICOS OBTENIDOS POR COMPUTADORA: Tabla 1 Shapiro-Wilks (modificado) Variable n Media RE_T3 25 -4,5E-03
D.E. 1,03
W* p (una cola) 0,90 0,0469
Tabla 2 Análisis de la varianza – Test de Levene Variable N R² R² Aj CV abs dif 25 0,04 0,00 97,95 Cuadro de Análisis de la Varianza (SC tipo III) F.V. SC gl CM F p-valor Modelo 8,66 4 2,17 0,19 0,9412 Etapas 8,66 4 2,17 0,19 0,9412 Error 228,87 20 11,44 Total 237,53 24
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Cuadro F,V, Modelo Etapas Error Total
de Análisis de la Varianza (SC tipo III) SC gl CM F 40488,68 4 10122,17 415,95 40488,68 4 10122,17 415,95 486,70 20 24,33 40975,38 24
Tabla 4 Prueba de Kruskal Wallis Variable Etapas N T3 a 5 T3 b 5 T3 c 5 T3 d 5 T3 e 5
Medias 94,58 118,96 198,65 114,38 84,53 84,53
D,E, 4,03 5,80 6,11 4,15 4,15
p-valor <0,0001 <0,0001
Medianas 94,09 117,61 197,18 112,47 83,14
H 22,24
p 0,0002
QQ-plot
Gráfico de densidad de puntos 2,29
2,47
) 3 T _ E 1,21 R ( s o d a v r 0,13 e s b o s e l i t n -0,95 a u C
1,45
3 T _ E R
0,42
-0,60
-2,02 - 2,02
-1,63 a
b
c
d
e
- 0,95
0,13
1,2 1
2 ,2 9
Cuantiles de una Normal(-0,0045184,1, Normal(-0,0045184,1,0556) 0556)
Etapas
5) (Un experimento ilegal). Es un hecho muy conocido que casi todos los caballos que corren carreras “cuadreras” o extraoficiales, reciben tratamientos medicamentosos que en las carreras oficiales no están permitidos. Todos los cuidadores afirman que de otro modo no es posible competir, pero la efectividad de esos tratamientos suele ser objeto de polémicas. A tal efecto se diseñó un experimento para comparar a tres de tales tratamientos, con 5 caballos cada uno y un grupo testigo, sin medicar. Se utilizaron, en consecuencia, 20 caballos de características lo más similares posibles (en velocidad, edad, sanidad). Se registraron en cada caso los tiempos (en segundos) empleados en una corrida a fondo sobre la distancia clásica cuadrera de 300 metros en pista normal, con los siguientes resultados: Trat 1 17,96 17,62 17,90 17,70 17,70 Estadística descriptiva Tratamiento Variable 1 Tiempo 2 Tiempo 3 Tiempo
Trat 2 17,80 17,90 17,68 17,72 18,00
n 5 5 5
Media 17.78 17.7 8 17.82 17.8 2 18.34 18.3 4
Trat 3 18,30 18,50 18,40 18,22 18,30
D.E. 0.15 0.13 0.11
Mín 17.62 17.68 18.22
Máx 17.96 18.00 18.50
Trat 4 (control) 18,60 18,80 18,60 18,90 18,80
Mediana 17.70 17.80 18.30
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Normalidad ................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... ........................................................................................................................................................................ Homoscedasticidad ........................................................................................................................................ ......................................................................................................................................................................... ......................................................................................................................................................................... ............................................................................................................................... b) Teniendo en cuenta lo anterior y que el nivel es del 5% La salida elegida es ............................................................................................................................. Las hipótesis plantea planteadas das son: ............................................................... ............................................... Como p-valor es .......................... la decisión es ......................................, por lo que la conclusión es ......................................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... SALIDAS Y GRAFICOS OBTENIDOS POR COMPUTADORA: Tabla 1 Shapiro-Wilks (modificado) Variable n Media D.E. RE_Tiempo 20 0.01 1.06 Tabla 2 Análisis de la varianza Variable N R² R² Aj abs dif 20 0,02 0,00
W* 0.87
p (una cola) 0.0201
CV 98,82
Cuadro de Análisis de la Varianza (SC tipo III) F.V. SC gl CM Modelo 2,9E-03 3 9,6E-04 Trat 2,9E-03 3 9,6E-04 Error 0,14 16 0,01 Total 0,15 19 Tabla 3 Análisis de la varianza Variable N R² Tiempo 20 0,92
R² Aj 0,91
CV 0,72
F 0,11 0,11
p-valor 0,9550 0,9550
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Gráfico de densidad de puntos
QQ-plot
19,00
2,00 ) o p m e i T 1,01 _ E R ( s o d 0,01 a v r e s b o s e -0,98 l i t n a u C
18,50 o p m 18,00 e i T
17,50
17,00
-1,97 - 1,97
1 -0,98
0, 0 1
1, 0 1
2,00
2
3
4
Tratamiento
Cuantiles de una Norm al(0,014126,1, al(0,014126,1,1307) 1307)
6) Interesa determinar si existen diferencias significativas entre las concentraciones medias de glucosa registradas después de aplicar diferentes drogas a conejos de cierta raza. Para ello se empleó un conjunto de 18 conejos de esa raza y se lo dividió aleatoriamente en tres grupos. Cada grupo recibió una droga diferente. Al cabo de cierto tiempo se midió la concentración de glucosa en plasma, en mg/100 ml, y se obtuvieron los siguientes datos: Droga A: 94 97 84 92 95 107 Droga B: 82 Droga C: 91 Estadística descriptiva Droga Variable A Concentración Concentración B Concentración Concentraci ón C Concentración Concentraci ón
n 6 6 6
Media 94.83 78.33 100.33
73
77
81
106 102 104 D.E. 7.47 4.72 7.06
Mín 84.00 73.00 91.00
84
73
107
92
Máx Mediana 107.00 94.50 84.00 79.00 107.00 103.00
A partir de los gráficos y las salidas correspondie correspondientes, ntes, responda responda los siguientes siguientes ítems: a) Verificar si se cumplen los supuestos para un DCA modelo fijo (concluya u opine para cada supuesto) Normalidad .................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... ......................................................................................................................................................................... Homoscedasticidad ........................................................................................................................................
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SALIDAS Y GRAFICOS OBTENIDOS POR COMPUTADORA: Tabla 1 Shapiro-Wilks (modificado) Variable n RE_Concentración RE_Concentra ción 18
Media D.E. 0,00 1,03
W* 0,96
p (una cola) 0,7830
Tabla 2 Análisis de la varianza – Test de Levene Variable N R² R² Aj CV abs dif 18 0,02 0,00 91,27 Cuadro de Análisis de la Varianza (SC tipo III) F.V. SC gl CM F Modelo 5,44 2 2,72 0,15 Droga 5,44 2 2,72 2, 72 0,15 Error 278,67 15 18,58 Total 284,11 17 Tabla 3 Análisis de la varianza Variable N Concentración Concentraci ón 18
R² 0,71
R² Aj 0,67
CV 7,16
Cuadro de Análisis de la Varianza (SC tipo III) F,V, SC gl CM Modelo 1573,00 2 786,50 Droga 1573,00 2 786,50 Error 639,50 15 42,63 Total 2212,50 17 Tabla 4 Prueba de Kruskal Wallis Variable Droga Concentración Concentraci ón A Concentración Concentraci ón B Concentración Concentración C
N 6 6 6
p-valor 0,8649 0,8649
F 18,45 18,45
Medianas 94,50 79,00 103,00
p-valor 0,0001 0,0001
H 11,38
Gráfico de densidad de puntos
QQ-plot ) n ó i c a r t n e c n o C _
p 0,0033
110,00
2,08
100,00
1,08 n ó i c
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CUESTIONARIO 1.- ¿Cuáles son los supuestos supuestos del D.C.A. paramétrico? paramétrico? ¿Cómo puede verificarlos y/o garantizarlos? 2.- ¿Cuáles son los supuestos del D.C.A. no paramétrico? ¿Cómo puede verificarlos y/o garantizarlos? 3.- En los problemas propuestos 3) y 4) indique: Para el problema propuesto 3:
a) Factor en estudio: .......................................................................................................................... b) Tratamientos: ................................................................................................................................ c) Unidad experimental: experimental: ................................................................................................................ ..... d) Observación: ................................................................................................................................. Para el problema propuesto 4:
a) Factor en estudio: .......................................................................................................................... b) Tratamientos: ................................................................................................................................ experimental: ........................................................................................................ ............. c) Unidad experimental:
d) Observación: ................................................................................................................................. 4.- En los problemas propuestos 5) y 6) indique los supuestos que tuvo en cuenta para elegir el tipo de análisis adecuado. 5.- En un Análisis de la Varianza paramétrico, ¿por qué la región crítica es unilateral derecha? 6.- Indique los modelos teóricos que se utilizan para el análisis de la varianza paramétrico y no paramétrico. Señale en cada caso los parámetros del modelo y sus estimadores.
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Unidad 5 – CORRELACIÓN CORRELACIÓN LINEAL (Pearson) Y NO PARAMÉTRICA (Spearman) (Spearman) Objetivos específicos Destacar la necesidad y utilidad del análisis de correlación. Diferenciar el análisis de correlación del análisis de regresión. Analizar Analizar resultad resultados os y casos casos espec específico íficos s y sus sus aplicac aplicacion iones es en en el el campo campo de de los los proce procesos sos biológ biológico icos. s.
Contenidos Contenidos temáticos Concepto de correlación. Diagrama de dispersión. Coeficiente de correlación: cálculo e interpretación. Prueba de hipótesis para = 0. Correlación no paramétrica. Actividad globalizante de los conocimientos adquiridos basándose en el estudio de casos que se presentan en la experimentación o investigación biológica.
Glosario Distribución bivariada (dos variables aleatorias conjuntamente distribuidas). Distribución normal bivariada. Parámetros de esta distribución. Elipse de concentración. Asociación entre variables aleatorias normales y no normales. Coeficiente de correlación lineal y no lineal. Aplicaciones a inferencia estadística: Estimación y contraste.
PROBLEMAS RESUELTOS 01.- Los siguientes datos corresponden al peso vivo (PV, en Kg) y al espesor de grasa dorsal (EGD, en mm) de 30 lechones elegidos al azar de una población de porcinos Duroc Jersey del Oeste de la provincia de Buenos Aires:
Obs
1
2
3
4
5
6
7
8
9
10
PV EGD
56,81 16,19
70,40 22,00
71,73 19,52
75,10 31,00
79,65 23,58
51,43 16,58
52,10 17,20
69,10 26,70
35,70 21,60
76,44 24,51
Obs
11
12
13
14
15
16
17
18
19
20
PV EGD
27,51 16,21
67,90 13,80
51,61 16,43
69,40 33,60
48,93 25,07
55,02 20,52
62,70 26,20
84,00 11,50
67,50 21,50
58,62 27,26
Obs
21
22
23
24
25
26
27
28
29
30
PV EGD
68,58 12,60
67,10 23,30
49,10 32,46
73,80 19,90
61,40 20,48
63,44 7,29
93,00 40,90
58,90 9,21
58,70 27,60
66,45 23,27
a.- Dibujar el diagrama de dispersión e interpretarlo. b.- Calcular el coeficiente de correlación muestral. suficiente evidencia evidencia para admitir admitir asociación entre entre el peso y el el espesor de de grasa? ( = 0,05) c.- ¿Hay suficiente Datos del problema: Variables en estudio: X1: peso vivo de un lechón, en kg; X2: espesor de grasa dorsal de un lechón, en mm; Tamaño de muestra: n = 30
Diagrama de dispersión
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variables aumentan los valores de la otra, y viceversa).
b) Existen dos coeficientes de correlación muestral: el de Pearson y el de Spearman. Las variables en estudio en este caso cumplen, en principio, con el supuesto teórico de Spearman, ya que ambas variables (Peso vivo y espesor de grasa dorsal) son al menos ordinales, es decir que tienen un orden natural porque son números. Sin embargo podríamos docimar la hipótesis de normalidad, para ver si se cumple el supuesto de Pearson Dócima de normalidad para X 1 al 10%
Salida del programa InfoStat para X 1
Shapiro-Wilks (modificado) Variable PV
n 30
Media D.E. 63.07 13.54
W* 0.98
p (una cola) 0.9463
Como el p-valor = 0,9463>0,10 no se rechaza la hipótesis nula. Por lo tanto al 10% se puede suponer que la variable el peso vivo de los porcinos Duroc Jersey del Oeste de la provincia de Buenos Aires se distribuye normalmente. Dócima de normalidad para X 2 al 10%
Salida del programa InfoStat para X 2 2
Shapiro-Wilks (modificado) Variable EGD
n 30
Media D.E. 21.59 7.44
W* 0.98
p (una cola) 0.9485
Como el p-valor = 0,2501>0,10 no se rechaza la hipótesis nula. Por lo tanto al 10% se puede suponer que el espesor de la grasa dorsal de los porcinos Duroc Jersey del Oeste de la provincia de Buenos Aires se
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c) Para responder a esta pregunta debemos realizar la dócima correspondiente cuyas hipótesis son: Hipótesis estadísticas:
H0 :
= 0 (no hay correlació correlación n lineal) lineal)
H1:
0 (hay correlació correlación n lineal) lineal)
Como no nos dice asociación positiva o asociación negativa, se plantea igual versus distinto. Nivel de significación: significación:
= 0,05
Estadístico de prueba: t =
r n-2
~ t n-2
1-r 2
Región crítica: Para este punto vamos a necesitar los valores críticos, son dos por que la región crítica es bilateral, y van a estar determinados por t n 2, / 2 y t n 2,1 / 2 . En este caso, t 28;0,025 y t 28;0,975 . Valores Críticos: t 28;0,025 = -2,048 y t 28;0,975 = 2,048. Región Crítica: t
Regla de decisión:
-2,048 conjuntamente t
2,048
RECHAZO HO si tH0 - 2,048 o tH0 2,048 NO RECHAZO HO si -2,048 < tH0 < 2,048
Cálculo: Reemplazando los datos muestrales en el estadístico
t
Ho
=
0, 25 2572 28 1 - (0,257 (0,2572 2)
2
=
1,3606 = 1, 408 0,9663
Decisión: Como 1,5746<2,048 y 1,5746>-2,048 no se rechaza la hipótesis nula al 5% Conclusión : Al nivel de significación del 5%, no hay evidencias suficientes para rechazar H 0 (H0: =0), =0), por lo tanto tanto no existirí existiría a asociació asociación n lineal poblacional entre el peso vivo y el espesor de grasa dorsal en la población de lechones Duroc Jersey del Oeste de la provincia de Buenos Aires.
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X1: concentración de colesterol en un hombre X2: concentración de ácido úrico en un hombre. Tamaño de muestra: n = 10
Solución: Para poder resolver esta situación hay dos posibilidades, utilizar el coeficiente de correlación de Pearson o el de Spearman. Las variable en estudio, en este caso, cumplen con el supuesto teórico de Spearman, ya que ambas variables (concentración de colesterol y concentración de ácido úrico) son al menos ordinales (es decir que tienen un orden natural porque son números). Sin embargo podríamos docimar la hipótesis de normalidad, para ver si se cumple el supuesto de Pearson Dócima de normalidad para X 1 al 10%
Salida del programa InfoStat para X 1 Shapiro-Wilks (modificado) Variable X1
n 10
Media 282,00
D.E. W* 29,44 0,82
p (una cola) 0,0341
Como el p-valor = 0,0341<0,10 se rechaza la hipótesis nula. Por lo tanto al 10% la concentración de colesterol en la población de hombres que habitan la ciudad de Córdoba no se puede suponer que se distribuye normalmente. Dócima de normalidad para X 2 al 10%
Salida del programa InfoStat para X 2 2 Shapiro-Wilks (modificado)
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RECHAZO H0 si r S -0,5515 NO RECHAZO H0 si r S > -0,5515
Regla de decisión:
Cálculo: Para calcular el valor del estadístico se utilizarán los órdenes o rangos de los datos muestrales sabiendo que: ui: rango correspondiente a la observación “i” de la variable X 1. vi: rango correspondiente a la observación “i” de la variable X 2. di: diferencia entre u i y vi; 2
di : cuadrado de la diferencia entre u i y vi. Se obtiene la tabla: 2
x1i
x2i
ui
vi
di
di
269
43
5
1
4
16
279
65
6
4
2
4
248
78
1
9
-8
64
318
73
8,5
8
0,5
0,25
318
71
8,5
7
1,5
2,25
254
69
3
6
-3
9
263
67
4
5
-1
1
320
45
10
2
8
64
251
81
2
10
-8
64
300
62
7
3
4
16 240,5
La estimación del coeficiente de correlación de Spearman es: r S
1-
6*240,5
1-
1443
1-
1443
1 - 1 4575
- 0 4575
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Shapiro-Wilks (modificado) Variable T P
n 10 10
Media D.E. 38.36 1.07 71.20 14.12
W* 0.93 0.80
p (una cola) 0.5915 0.0190
c.- Comprobar estadísticamente si existe asociación positiva entre la temperatura y el pulso, usando un nivel de significación del 5%.
04.- Considerando que la estimación del coeficiente de correlación entre el peso al nacer y al destete en una especie fue igual a 0,40 (r=0,40) a partir de una muestra de 20 individuos. ¿Podría afirmar que el peso al nacer y el peso al destete están significativamente asociados? ( =0,05). Suponer que ambas variables tienen distribución normal.
05.- Sabiendo que el contenido de celulosa, hemicelulosa, y lignina se estudian a través de la fibra detergente neutro (FDN) de la dieta, se quiere analizar su relación con el pH ruminal generado. Se tomaron 11 bovinos y se midieron ambas variables:
FDN (D) 37,9 39,2 41,2 43,1 44,6 45,8 46,2 55,8 57,0 59,0 47,1 pH (P) 6,35 5,92 6,01 6,10 5,92 6,56 5,93 6,20 6,30 6,57 6,90 Suponiendo que ambas variables se distribuyen normalmente, y que
= 0,05, responder:
Hipótesis estadísticas:................................................. estadísticas:................................................................................................................. .................................................................................. .................. Región crítica:............................................................................ crítica:.............................................................................................................................................. .................................................................. Valor de la variable pivotal:.................................................................................... pivotal:.......................................................................................................................... ...................................... Población:............................................ Población:............................................................................................................................ ........................................................................................................ ........................
Datos:
dipi = 3239 3239,, 38; 38;
2
di = 2481 24811,39 1,39;;
2
pi = 430,85 430,85 ;
di = 516,9; 516,9;
pi = 68,76 68,76
06.- El director de un centro de internación veterinaria les pide a dos enfermeros que califiquen de 1 a 100 los pacientes internados de acuerdo a la dificultad de cuidados médicos que los mismos requieren. La siguiente tabla tiene la respuesta de los enfermeros.
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X1 X2
1 13005
8 11096
Responder, sabiendo que
3 8720
2 9166
6 10111
2 12937
0 13734
4 8861
5 9747
6 9568
= 0,05.
Hipótesis estadísticas:........................................................................................................................... Regla de decisión................................................................................................................................. Decisión:.............................................................................................................................................. Conclusión: ......................................................................................................................................... ........................................................................................................................................................... ...........................................................................................................................................................
CUESTIONARIO 1.- Suponer que las variables en estudio son A: peso y B: longitud. a.- Explicar cómo seleccionaría las unidades experimentales y qué observaría en cada una para que se pueda realizar un análisis de correlación entre ambas variables. b.- Modificar el enunciado para que corresponda a un estudio de regresión.
2.-Indicar verdadero o falso según corresponda.
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b.- ¿Cuántos y cuáles son los parámetros de la distribución? c.- Proponga un ejemplo donde identifique los parámetros e indique su significado biológico. 6.-Elegir la opción correcta: a.- Si (X1,X2) se distribuyen normal bivariada y son independientes entre si, entonces: 1) 1 2)
0
3)
0
4)
1
b.- Si dos variables no están correlacionadas linealmente, para estimar la correlación se emplea: 1) El coeficiente de Spearman 2) El coeficiente de Pearson 3) No se puede estimar la correlación 4) No tienen sentido estimar la correlación. c.- El supuesto teórico para emplear el coeficiente de correlación de Spearman es 1) Ambas variables no se distribuyan normalmente 2) Ambas variable se distribuyan normalmente 3) Ambas variable al menos ordinales 4) Una de las variable no se distribuye normalmente
7.- ¿Cuándo se utiliza el análisis de correlación no paramétrico? Proponer un ejemplo donde se
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PROBLEMAS PROBLE MAS COMPL COMPLEMEN EMENTARIO TARIOS S UNIDADES UNIDADES TEMÁTICA S 1 y 2 01.- Dos analistas, X e Y, midieron la dureza del agua en una ciudad. Se desea comparar si los resultados de X e Y son similares, por lo cual se toma una muestra, de tamaño 10, de agua de distintas regiones de la Ciudad de Buenos Aires. Cada analista midió las 10 muestras. Estime un intervalo de confianza del 95% para la media de las diferencias entre estos dos analistas. ¿Qué supuesto debe hacer para poder realizar el intervalo?
xi
0,46
0,62
0,37
0,40
0,44
0,58
0,48
0,53
0,59
0,68
y i
0,72
0,61
0,73
0,51
0,33
0,48
0,43
0,35
0,67
0,78
), se estimó 02.- En un estudio sobre llamadas de apareamiento, realizado en el sapo arbóreo ( Hyla ewingi ), en Tasmania, en una muestra de 29 observaciones, que la duración de las llamadas tenía una media de 189 ms (milisegundos) y un desvío estándar de 32 ms; y en Bristbane, en una muestra de 31 observaciones, una media de 216 ms (milisegundos) y un desvío estándar de 28 ms. a) Estime un intervalo de confianza del 95% para el cociente de varianzas. b) Asumiendo igualdad de varianzas, pruebe la hipótesis: “el tiempo de llamada del sapo arbóreo es mayor en Britsbane que en Tasmania”.
03.- A 300 pacientes que sufrían una enfermedad se los dividió al azar en tres grupos iguales de 100 pacientes cada uno. A cada grupo se le asignó, al azar, uno de los siguientes tratamientos: a) estándar, b) nuevo, y c) placebo. Durante la primera semana de tratamiento, en el primer grupo se recuperaron 78
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Tipo de tenencia de tierra
Forman parte de una cooperativa SI
NO
Propietarios
52
68
Aparceros
25
65
Arrendatarios
43
107
¿Cuál es la H0 a contrastar? Usando un estadístico Dar conclusiones.
2
y un nivel de significación del 5%, ¿se rechaza H 0?
08.- Se realizó un ensayo sobre el rendimiento de cultivares de papa utilizando diez parcelas con plantas sanas y nueve con plantas enfermas por el mosaico deformante. El rendimiento promedio en plantas sanas fue de 16.042 kg/ha con un desvío de 35,21 kg/ha y en plantas enfermas se obtuvo un promedio de 12.027 kg/ha y un desvío de 42,43 kg/ha. Estimar, con una confianza del 90%, la diferencia de los rendimientos medios poblacionales. ¿En qué supuestos teóricos se basa para realizar esta estimación?
09.- Con el objeto de orientar su propaganda a fin de neutralizar el efecto de la competencia, una empresa productora de artículos de consumo realizó una encuesta piloto sobre la preferencia del público a personas de cada una de las cuatro regiones del país. Los resultados son los siguientes:
Zona
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Método A
14
37
35
43
34
36
48
33
33
Método B
18
37
38
36
47
38
57
28
42
Determinar si hay diferencias entre los dos métodos al nivel del 5%
14.- Se ha demostrado que un alto contenido de nitrato en la composición de los alimentos da origen a numerosos efectos nocivos. En un experimento se tomaron 16 ratas al azar de una línea, las mismas fueron alimentadas con una dieta estándar. A 9 de ellas, elegidas al azar, se les dio de beber agua con 2000 ppm de nitrato. Luego se midió la ganancia de peso y se expresó en porcentaje:
A
12,7
19,3
20,5
10,5
14,0
10,8
16,6
B
18,2
22,9
10,1
14,3
10,2
17,1
15,7
14,0
17,2
¿Se puede concluir que la dosis de nitrato disminuye la ganancia de peso de las ratas? Justifique estadísticame estadísticamente nte su respuesta respuesta ( = 0,05). ¿En qué supuesto supuestos s se basa para realizar el test de hipótesis? hipótesis?
15.- Un Veterinario sabe, por archivo de datos históricos, que en su clínica durante el mes de diciembre los casos se distribuyen así: infecciones 50%, vacunaciones 35%, cirugías 10% y otros 5%. Durante diciembre de 2003 se encontró con las siguientes distribución de casos: infecciones 90, vacunaciones 80, 80, cirugías 30 y otros 20. Con un nivel de significación de 5%, 5%, ¿cree Ud. que que estos
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Control
610
405
805
560
570
620
730
620
En función del costo del fertilizante, los productores no estarían dispuestos a invertir en fertilizante si el aumento debido al uso es de a lo sumo 140 kg/ha. ¿Con un nivel de significación del 10%, cree que los productores productores invertirán en fertilizante?
19.- Los siguientes datos indican el aumento de peso, en g, de 20 ratas elegidas al azar de las cuales la mitad recibió proteína de maní crudo y la otra mitad de maní tostado. Probar si el tostado del maní ha tenido efecto sobre su valor proteico proteico ( =0,05). Suponga Suponga homog homogeneidad eneidad de varianzas, varianzas, y tenga presente presente que se sospecha que el tostado aumenta el valor proteico del maní, y que, obviamente, a mayor nivel proteico mayor aumento de peso.
Crudo
61
61
56
63
56
63
59
56
44
61
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24.- Dos establecimientos dedicados al cultivo de maíz híbrido siembran en quince parcelas diferentes, obteniendo los siguientes rendimientos (en Kg/parcela): ESTABLECIMIENTO 1: 114 - 86 - 93 - 75 - 102 - 89 - 83 - 89 - 92 - 96 – 100 - 98 - 87 - 80 - 86 ESTABLECIMIENTO 2: 107 - 94 - 86 - 70 - 78 - 90 - 82 - 77 - 95 - 84 - 100 - 89 - 92 - 99 - 85 Construir un intervalo del 99% de confianza para el cociente de las varianzas de los rendimientos de maíz híbrido entre los establecimientos. ¿En qué supuestos se basa para realizar el intervalo de confianza?
25.- Dos tratamientos A y B fueron asignados al azar a cada uno de dos lotes de animales tomados aleatoriamente de una población. La respuesta registrada fue el aumento de peso, en kg, durante el período experimental. Datos: X =1,57g ; X =1,89g ; n =11 ; n =10 ; S2 =0,15g2; S2 =0,12g2. Calcular un
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¿Considera que estos medicamento medicamentos s se comportan comportan de manera manera similar, use un =0,05?
31.- Se consideró una muestra de 97 niños escolarizados de 10 años que fueron clasificados según las siguientes variables: La Conducta en Clase, que fue clasificada por los maestros en Buena (B) y No Buena (NB) y la Adversidad Escolar en baja (b), media (m) o alta (a). Adversidad Escolar Conducta en clase Indicar:
b
m
a
B
20
49
8
NB
5
11
4
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.................................................................................................................................................................... .................................................................................................................................................................... ......
35.- Con el objeto de medir el efecto del ejercicio en enfermedades coronarias, un grupo de investigadores decidió comparar el índice de enfermedad en dos grandes grupos de personas que trabajan en los colectivos de Buenos Aires: choferes e inspectores. Los inspectores realizan más ejercicio, ya que su actividad requiere que estén caminando gran parte del día, mientras mientras que la l a tarea de los choferes es más sedentaria. Se consideraron aquellas personas que vinieran realizando el mismo trabajo durante los últimos 8 años y además la distribución de las l as edades en ambos grupos es similar. Se observó que el índice de enfermedades coronarias entre los conductores co nductores era sustancialmente mayor. mayor. experimento ¿es observacional o experimental? Justificar a) Este experimento
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El problema de detectar la aparición de flebitis es de particular importancia para los investigadores, ya que se pueden prevenir complicaciones cardíacas. Este estudio se diseñó con la finalidad primordial de buscar mecanismos para la detección temprana de la misma. Para ello se trabajó con conejos del bioterio y se eligió como droga a la amiodarona (antiarrítmico), para ser aplicada por vía endovenosa, ya que como efecto colateral se observa la aparición de flebitis en el lugar de la aplicación. Se sospecha que un aumento en la temperatura de los tejidos cercanos al lugar de la administración intravenosa serían señal de una inflamación i nflamación inminente. Se administraron tres tratamientos intravenosos en conejos. Estos fueron: amiodarona en una solución excipiente amiodarona sólo una solución excipiente una solución salina (control)
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artificial que el aire ambiental ambiental del empaque empaque comercial? b. ¿Son más efectivos los gases que el el vacío total?
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Shapiro-Wilks (modificado) Región
Variable
n
Media D.E.
W*
p (una cola)
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De acuerdo con los resultados anteriores, plantear al 5% y resolver el análisis de correlación que considere adecuado.
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Estimar puntualmente y con un nivel de confianza del 95%, el aumento de bacterias en función del tiempo. a) Estimación puntual e Interpretación
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11.- Para estudiar la relación entre la edad y presión sanguínea entre bovinos, se seleccionaron 40
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