C23
JORGEPAREDES
17/1/2021
Objetivo Simular el teorema del limite central.
Descripcion Con un conjunto de datos y librerías adecuadas, simular el valor de la media muestral comparado con el valor de la media poblacional asociando con ello con el teorema del límite central.
library(dplyr)## Warning: package 'dplyr' was built under R version 4.0.3##
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## intersect, setdiff, setequal, unionlibrary(mosaic)## Warning: package 'mosaic' was built under R version 4.0.3## Registered S3 method overwritten by 'mosaic':
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## fortify.SpatialPolygonsDataFrame ggplot2##
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## max, mean, min, prod, range, sample, sumlibrary(readr)## Warning: package 'readr' was built under R version 4.0.3library(ggplot2) # Para gráficos
library(knitr) # Para formateo de datos## Warning: package 'knitr' was built under R version 4.0.3library(fdth) # Para tablas de frecuencias## Warning: package 'fdth' was built under R version 4.0.3##
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## sd, varlibrary(gtools) # Para combinaciones y permutaciones## Warning: package 'gtools' was built under R version 4.0.3##
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## logitlibrary(Rmpfr) # Para factoriales de números muy grandes## Warning: package 'Rmpfr' was built under R version 4.0.3## Loading required package: gmp## Warning: package 'gmp' was built under R version 4.0.3##
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## %*%, apply, crossprod, matrix, tcrossprod## C code of R package 'Rmpfr': GMP using 64 bits per limb##
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## cbind, pmax, pmin, rbindN <- 1000000;
edad.poblacion <- round(rnorm(N, mean = 35, sd = 5), 0)
summary(edad.poblacion)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 8 32 35 35 38 59paste("El valor de edad de una pobacion. Los primeros cincuenta valores ...")## [1] "El valor de edad de una pobacion. Los primeros cincuenta valores ..."head(edad.poblacion, 50) ## [1] 28 35 31 42 37 36 25 40 25 34 31 34 33 42 39 35 33 33 27 32 39 30 27 38 36
## [26] 40 31 28 38 35 25 34 33 34 30 28 42 40 33 30 36 34 29 32 32 42 36 42 32 39paste("El valor de edad de una pobacion. Los últimos cincuenta valores ...")## [1] "El valor de edad de una pobacion. Los últimos cincuenta valores ..."tail(edad.poblacion, 50)## [1] 35 32 30 25 38 42 40 38 30 36 40 37 24 28 33 38 38 41 28 31 30 39 47 41 43
## [26] 34 36 31 38 45 33 32 27 40 27 39 35 38 31 34 38 38 37 45 32 33 36 34 31 39media.pob <- mean(edad.poblacion)
desv.std <- sd(edad.poblacion)
paste("Los parámetros de la media y desviación estándard de la población")## [1] "Los parámetros de la media y desviación estándard de la población"media.pob; desv.std## [1] 35.00175## [1] 5.011395Medias y Desviaciones Muestrales -Determinar cinco muestras de n=500 casos por medio de la función sample(), se guardan en un data.frame llamado muestras. -Se visualizan los estadísticos principales por medio de la función summary() -Se utiliza un ciclo para determinar las medias de cada muestra. -Se construye un data frame con los valores de los errores estadísticos.
n <- 500
muestras <- data.frame(m1=sample(edad.poblacion, n),
m2=sample(edad.poblacion, n),
m3=sample(edad.poblacion, n),
m4=sample(edad.poblacion, n),
m5=sample(edad.poblacion, n))
summary(muestras)## m1 m2 m3 m4
## Min. :20.00 Min. :20.00 Min. :23.00 Min. :20.00
## 1st Qu.:31.00 1st Qu.:32.00 1st Qu.:32.00 1st Qu.:31.00
## Median :35.00 Median :35.00 Median :35.00 Median :35.00
## Mean :34.68 Mean :35.01 Mean :35.21 Mean :35.15
## 3rd Qu.:38.00 3rd Qu.:38.00 3rd Qu.:39.00 3rd Qu.:39.00
## Max. :48.00 Max. :48.00 Max. :49.00 Max. :49.00
## m5
## Min. :21.00
## 1st Qu.:31.00
## Median :35.00
## Mean :34.84
## 3rd Qu.:38.00
## Max. :52.00kable(head(muestras, 10), caption = "Muestras de la población. Los primeros diez de 500 registros")| m1 | m2 | m3 | m4 | m5 |
|---|---|---|---|---|
| 29 | 36 | 32 | 35 | 37 |
| 34 | 42 | 32 | 30 | 32 |
| 31 | 31 | 32 | 39 | 35 |
| 23 | 30 | 33 | 39 | 35 |
| 36 | 33 | 33 | 34 | 26 |
| 41 | 37 | 31 | 44 | 37 |
| 41 | 36 | 34 | 38 | 34 |
| 37 | 29 | 30 | 25 | 36 |
| 29 | 31 | 30 | 33 | 22 |
| 30 | 38 | 33 | 32 | 40 |
kable(head(muestras, 10), caption = "Muestras de la población. Los últimos diez de 500 registros")| m1 | m2 | m3 | m4 | m5 |
|---|---|---|---|---|
| 29 | 36 | 32 | 35 | 37 |
| 34 | 42 | 32 | 30 | 32 |
| 31 | 31 | 32 | 39 | 35 |
| 23 | 30 | 33 | 39 | 35 |
| 36 | 33 | 33 | 34 | 26 |
| 41 | 37 | 31 | 44 | 37 |
| 41 | 36 | 34 | 38 | 34 |
| 37 | 29 | 30 | 25 | 36 |
| 29 | 31 | 30 | 33 | 22 |
| 30 | 38 | 33 | 32 | 40 |
medias <- 0
error <- 0
for(i in 1:5) {
medias[i] <- mean(muestras[,i])
error[i] <- medias[i] - media.pob
}
error.muestreo <- data.frame(Media.Poblacion = media.pob, Media.Muestras = medias, Errores = error)
kable(error.muestreo, caption = "Error de media de edad de cada muestra con respecto a la media de la población")| Media.Poblacion | Media.Muestras | Errores |
|---|---|---|
| 35.00175 | 34.684 | -0.317754 |
| 35.00175 | 35.012 | 0.010246 |
| 35.00175 | 35.214 | 0.212246 |
| 35.00175 | 35.150 | 0.148246 |
| 35.00175 | 34.838 | -0.163754 |
hist(edad.poblacion, main = "Histrograma de la edad de la población")
hist(muestras$m1, main = "Histrograma de la edad de la muestra 1", ylab = "Edades", xlab="Observacaiones")
hist(muestras$m2, main = "Histrograma de la edad de la muestRa 2", ylab = "Edades", xlab="Observacaiones")
hist(muestras$m3, main = "Histrograma de la edad de la muestRa 3", ylab = "Edades", xlab="Observacaiones")
hist(muestras$m4, main = "Histrograma de la edad de la muestRa 4", ylab = "Edades", xlab="Observacaiones")
hist(muestras$m5, main = "Histrograma de la edad de la muestRa 5", ylab = "Edades", xlab="Observacaiones")
options(scipen = 999) # Para mostrar notación normal y no científica en el valor de N: 1e+06
N; n; ## [1] 1000000## [1] 500options(scipen = 0) # Regresa a notación numérica normal
summary(edad.poblacion)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 8 32 35 35 38 59media.pob; desv.std## [1] 35.00175## [1] 5.011395N <- 10; n <- 2
edad.poblacion <- round(rnorm(N, mean = 35, sd = 5), 0)
edad.poblacion## [1] 41 33 32 26 32 36 35 23 45 28media.pob <- mean(edad.poblacion)
desv.std <- sd(edad.poblacion)
media.pob; desv.std## [1] 33.1## [1] 6.640783n.combinaciones <- factorialMpfr(N) / (factorialMpfr(n) * (factorialMpfr(N-n)))
as.integer(n.combinaciones)## [1] 45muestras <- cbind(1:as.integer(n.combinaciones))
muestras <- cbind(muestras, combinations(N, n, 1:N))
muestras <- cbind(muestras, edad.poblacion[muestras[,2]], edad.poblacion[muestras[,3]])
medias <- 0
error <- 0
for(i in 1:as.integer(n.combinaciones)) {
medias[i] <- mean(muestras[i,c(4,5)])
error[i] <- medias[i] - media.pob
}
muestras <- cbind(muestras, medias)
muestras <- cbind(muestras, media.pob)
muestras <- cbind(muestras, error)
muestras <- data.frame(muestras)
colnames(muestras) <- c("Muestra", "Pos.1", "Pos.2", "Valor.1", "Valor.2", "Media muestra", "Media pob.", "Error")
kable(muestras, caption = "Las muestras")| Muestra | Pos.1 | Pos.2 | Valor.1 | Valor.2 | Media muestra | Media pob. | Error |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 41 | 33 | 37.0 | 33.1 | 3.9 |
| 2 | 1 | 3 | 41 | 32 | 36.5 | 33.1 | 3.4 |
| 3 | 1 | 4 | 41 | 26 | 33.5 | 33.1 | 0.4 |
| 4 | 1 | 5 | 41 | 32 | 36.5 | 33.1 | 3.4 |
| 5 | 1 | 6 | 41 | 36 | 38.5 | 33.1 | 5.4 |
| 6 | 1 | 7 | 41 | 35 | 38.0 | 33.1 | 4.9 |
| 7 | 1 | 8 | 41 | 23 | 32.0 | 33.1 | -1.1 |
| 8 | 1 | 9 | 41 | 45 | 43.0 | 33.1 | 9.9 |
| 9 | 1 | 10 | 41 | 28 | 34.5 | 33.1 | 1.4 |
| 10 | 2 | 3 | 33 | 32 | 32.5 | 33.1 | -0.6 |
| 11 | 2 | 4 | 33 | 26 | 29.5 | 33.1 | -3.6 |
| 12 | 2 | 5 | 33 | 32 | 32.5 | 33.1 | -0.6 |
| 13 | 2 | 6 | 33 | 36 | 34.5 | 33.1 | 1.4 |
| 14 | 2 | 7 | 33 | 35 | 34.0 | 33.1 | 0.9 |
| 15 | 2 | 8 | 33 | 23 | 28.0 | 33.1 | -5.1 |
| 16 | 2 | 9 | 33 | 45 | 39.0 | 33.1 | 5.9 |
| 17 | 2 | 10 | 33 | 28 | 30.5 | 33.1 | -2.6 |
| 18 | 3 | 4 | 32 | 26 | 29.0 | 33.1 | -4.1 |
| 19 | 3 | 5 | 32 | 32 | 32.0 | 33.1 | -1.1 |
| 20 | 3 | 6 | 32 | 36 | 34.0 | 33.1 | 0.9 |
| 21 | 3 | 7 | 32 | 35 | 33.5 | 33.1 | 0.4 |
| 22 | 3 | 8 | 32 | 23 | 27.5 | 33.1 | -5.6 |
| 23 | 3 | 9 | 32 | 45 | 38.5 | 33.1 | 5.4 |
| 24 | 3 | 10 | 32 | 28 | 30.0 | 33.1 | -3.1 |
| 25 | 4 | 5 | 26 | 32 | 29.0 | 33.1 | -4.1 |
| 26 | 4 | 6 | 26 | 36 | 31.0 | 33.1 | -2.1 |
| 27 | 4 | 7 | 26 | 35 | 30.5 | 33.1 | -2.6 |
| 28 | 4 | 8 | 26 | 23 | 24.5 | 33.1 | -8.6 |
| 29 | 4 | 9 | 26 | 45 | 35.5 | 33.1 | 2.4 |
| 30 | 4 | 10 | 26 | 28 | 27.0 | 33.1 | -6.1 |
| 31 | 5 | 6 | 32 | 36 | 34.0 | 33.1 | 0.9 |
| 32 | 5 | 7 | 32 | 35 | 33.5 | 33.1 | 0.4 |
| 33 | 5 | 8 | 32 | 23 | 27.5 | 33.1 | -5.6 |
| 34 | 5 | 9 | 32 | 45 | 38.5 | 33.1 | 5.4 |
| 35 | 5 | 10 | 32 | 28 | 30.0 | 33.1 | -3.1 |
| 36 | 6 | 7 | 36 | 35 | 35.5 | 33.1 | 2.4 |
| 37 | 6 | 8 | 36 | 23 | 29.5 | 33.1 | -3.6 |
| 38 | 6 | 9 | 36 | 45 | 40.5 | 33.1 | 7.4 |
| 39 | 6 | 10 | 36 | 28 | 32.0 | 33.1 | -1.1 |
| 40 | 7 | 8 | 35 | 23 | 29.0 | 33.1 | -4.1 |
| 41 | 7 | 9 | 35 | 45 | 40.0 | 33.1 | 6.9 |
| 42 | 7 | 10 | 35 | 28 | 31.5 | 33.1 | -1.6 |
| 43 | 8 | 9 | 23 | 45 | 34.0 | 33.1 | 0.9 |
| 44 | 8 | 10 | 23 | 28 | 25.5 | 33.1 | -7.6 |
| 45 | 9 | 10 | 45 | 28 | 36.5 | 33.1 | 3.4 |
paste("La media poblacional es: ", media.pob, " y la media de la edad de la distribución muestral es: ", mean(muestras$`Media muestra`))## [1] "La media poblacional es: 33.1 y la media de la edad de la distribución muestral es: 33.1"