Qttte Essay

5833 words - 23 pages

CLASE 7 Intervalos de Confianza para Un Para´metro
El objetivo de la inferencia estad´ıstica es obtener conclusiones acerca de la poblacio´n usando una muestra aleatoria. Primariamente el esfuerzo esta´ en ajustar un modelo de pro- babilidad f(x; θ) a la caracter´ıstica aleatoria X que interesa investigar. Luego o simulta´nea- mente el intere´s se centra en el o los para´metros poblacionales del modelo θ. La Figura 1 da una visio´n general del proceso de inferencia estad´ıstica en el que se distinguen tres a´reas principales:
Estimacio´n Puntual
Estimacio´n por Intervalos
Pruebas de Hipo´tesis
En este cap´ıtulo se estudiara´ el proceso de estimacio´n por intervalos para un para´metro. Ahora se dara´n ciertas definiciones y nomenclaturas necesarias para construir la metodolog´ıa de la inferencia estad´ıstica.
1. Definiciones Previas
Definicio´n 1.1. Un para´metro poblacional, anotado θ, es un nu´mero que representa algu- na caracter´ıstica de la poblacio´n o del modelo f(x; θ). Esos para´metros pueden ser la media µ, la varianza σ2, una proporcio´n p, un total τ , un ma´ximo M , etc.
Definicio´n 1.2. El espacio de para´metros o espacio parame´trico , anotado Θ, es el conjunto de todos los valores posibles que puede asumir un para´metro o conjunto de para´metros.
Ejemplo 1. Para el modelo de probabilidad Normal identifique los para´metros y el espacio parame´trico. Solu´cio´n Si X tiene distribucio´n normal con media µ y varianza σ2 entonces la densidad de probabi- lidad es
f(x, µ, σ2) = 1√ 2piσ
e −1 2 (x−µ σ
)2 , con −∞ < x <∞, −∞ < µ <∞ y σ > 0
El vector de para´metros es θ = (µ;σ2) y el espacio parame´trico es Θ = R× R+ Ejemplo 2. (Servicio al Cliente) La Gerencia de Atencio´n al Cliente de un Banco co- mercial ha implementado un plan para que los ejecutivos de cuentas corrientes atiendan de una manera ma´s personal a su cartera de clientes. Despue´s de un mes, la Gerencia evalua la percepcio´n de sus clientes respecto a este nuevo plan implementando una encuesta en l´ınea. Un item de la encuesta solicita al cliente que evalu´e a su ejecutivo de cuentas ponie´ndole una nota entre 1 y 7 (nu´mero entero). ¿Cua´l es la variable aleatoria asociada a este problema? ¿Que´ tipo de modelo de probabilidad podr´ıa funcionar bie´n en este caso? ¿Cua´les ser´ıan los para´metros poblacionales? Solucio´n La variable aleatoria X es la nota que cada cliente...

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