.

Objetivo

Realizar cálculos de probabilida bajo la distribución Normal

Librerias

library(dplyr)

## Warning: package 'dplyr' was built under R version 3.6.3

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(mosaic)

## Warning: package 'mosaic' was built under R version 3.6.3

## Loading required package: lattice

## Loading required package: ggformula

## Warning: package 'ggformula' was built under R version 3.6.3

## Loading required package: ggplot2

## Warning: package 'ggplot2' was built under R version 3.6.3

## Loading required package: ggstance

## Warning: package 'ggstance' was built under R version 3.6.3

## 
## Attaching package: 'ggstance'

## The following objects are masked from 'package:ggplot2':
## 
##     geom_errorbarh, GeomErrorbarh

## 
## New to ggformula?  Try the tutorials: 
##  learnr::run_tutorial("introduction", package = "ggformula")
##  learnr::run_tutorial("refining", package = "ggformula")

## Loading required package: mosaicData

## Warning: package 'mosaicData' was built under R version 3.6.3

## Loading required package: Matrix

## Registered S3 method overwritten by 'mosaic':
##   method                           from   
##   fortify.SpatialPolygonsDataFrame ggplot2

## 
## The 'mosaic' package masks several functions from core packages in order to add 
## additional features.  The original behavior of these functions should not be affected by this.
## 
## Note: If you use the Matrix package, be sure to load it BEFORE loading mosaic.
## 
## Have you tried the ggformula package for your plots?

## 
## Attaching package: 'mosaic'

## The following object is masked from 'package:Matrix':
## 
##     mean

## The following object is masked from 'package:ggplot2':
## 
##     stat

## The following objects are masked from 'package:dplyr':
## 
##     count, do, tally

## The following objects are masked from 'package:stats':
## 
##     binom.test, cor, cor.test, cov, fivenum, IQR, median, prop.test,
##     quantile, sd, t.test, var

## The following objects are masked from 'package:base':
## 
##     max, mean, min, prod, range, sample, sum

library(readr)
library(ggplot2)
library(knitr)

Los datos

datos <- read.table("https://raw.githubusercontent.com/rpizarrog/probabilidad-y-estad-stica/master/datos/body.dat.txt", quote="\"", comment.char="")

Seleccionar y poner nombres de columnas a las columnas 23, 24 y 25 solamente

colnames(datos)[23:25] <- c("peso", "estatura", "genero")

kable(head(datos))

V1	V2	V3	V4	V5	V6	V7	V8	V9	V10	V11	V12	V13	V14	V15	V16	V17	V18	V19	V20	V21	V22	peso	estatura	genero
42.9	26.0	31.5	17.7	28.0	13.1	10.4	18.8	14.1	106.2	89.5	71.5	74.5	93.5	51.5	32.5	26.0	34.5	36.5	23.5	16.5	21	65.6	174.0	1
43.7	28.5	33.5	16.9	30.8	14.0	11.8	20.6	15.1	110.5	97.0	79.0	86.5	94.8	51.5	34.4	28.0	36.5	37.5	24.5	17.0	23	71.8	175.3	1
40.1	28.2	33.3	20.9	31.7	13.9	10.9	19.7	14.1	115.1	97.5	83.2	82.9	95.0	57.3	33.4	28.8	37.0	37.3	21.9	16.9	28	80.7	193.5	1
44.3	29.9	34.0	18.4	28.2	13.9	11.2	20.9	15.0	104.5	97.0	77.8	78.8	94.0	53.0	31.0	26.2	37.0	34.8	23.0	16.6	23	72.6	186.5	1
42.5	29.9	34.0	21.5	29.4	15.2	11.6	20.7	14.9	107.5	97.5	80.0	82.5	98.5	55.4	32.0	28.4	37.7	38.6	24.4	18.0	22	78.8	187.2	1
43.3	27.0	31.5	19.6	31.3	14.0	11.5	18.8	13.9	119.8	99.9	82.5	80.1	95.3	57.5	33.0	28.0	36.6	36.1	23.5	16.9	21	74.8	181.5	1

Poner la columna genero como factor 1, 0

datos$genero <- as.factor(datos$genero)

Ver estadísticos

summary(datos)

##        V1              V2              V3              V4       
##  Min.   :32.40   Min.   :18.70   Min.   :24.70   Min.   :14.30  
##  1st Qu.:36.20   1st Qu.:26.50   1st Qu.:30.60   1st Qu.:17.30  
##  Median :38.70   Median :28.00   Median :32.00   Median :19.00  
##  Mean   :38.81   Mean   :27.83   Mean   :31.98   Mean   :19.23  
##  3rd Qu.:41.15   3rd Qu.:29.25   3rd Qu.:33.35   3rd Qu.:20.90  
##  Max.   :47.40   Max.   :34.70   Max.   :38.00   Max.   :27.50  
##        V5              V6              V7              V8       
##  Min.   :22.20   Min.   : 9.90   Min.   : 8.10   Min.   :15.70  
##  1st Qu.:25.65   1st Qu.:12.40   1st Qu.: 9.80   1st Qu.:17.90  
##  Median :27.80   Median :13.30   Median :10.50   Median :18.70  
##  Mean   :27.97   Mean   :13.39   Mean   :10.54   Mean   :18.81  
##  3rd Qu.:29.95   3rd Qu.:14.40   3rd Qu.:11.20   3rd Qu.:19.60  
##  Max.   :35.60   Max.   :16.70   Max.   :13.30   Max.   :24.30  
##        V9             V10              V11              V12        
##  Min.   : 9.90   Min.   : 85.90   Min.   : 72.60   Min.   : 57.90  
##  1st Qu.:13.00   1st Qu.: 99.45   1st Qu.: 85.30   1st Qu.: 68.00  
##  Median :13.80   Median :108.20   Median : 91.60   Median : 75.80  
##  Mean   :13.86   Mean   :108.20   Mean   : 93.33   Mean   : 76.98  
##  3rd Qu.:14.80   3rd Qu.:116.55   3rd Qu.:101.15   3rd Qu.: 84.50  
##  Max.   :17.20   Max.   :134.80   Max.   :118.70   Max.   :113.20  
##       V13              V14              V15             V16       
##  Min.   : 64.00   Min.   : 78.80   Min.   :46.30   Min.   :22.40  
##  1st Qu.: 78.85   1st Qu.: 92.00   1st Qu.:53.70   1st Qu.:27.60  
##  Median : 84.60   Median : 96.00   Median :56.30   Median :31.00  
##  Mean   : 85.65   Mean   : 96.68   Mean   :56.86   Mean   :31.17  
##  3rd Qu.: 91.60   3rd Qu.:101.00   3rd Qu.:59.50   3rd Qu.:34.45  
##  Max.   :121.10   Max.   :128.30   Max.   :75.70   Max.   :42.40  
##       V17             V18             V19             V20             V21      
##  Min.   :19.60   Min.   :29.00   Min.   :28.40   Min.   :16.40   Min.   :13.0  
##  1st Qu.:23.60   1st Qu.:34.40   1st Qu.:34.10   1st Qu.:21.00   1st Qu.:15.0  
##  Median :25.80   Median :36.00   Median :36.00   Median :22.00   Median :16.1  
##  Mean   :25.94   Mean   :36.20   Mean   :36.08   Mean   :22.16   Mean   :16.1  
##  3rd Qu.:28.40   3rd Qu.:37.95   3rd Qu.:38.00   3rd Qu.:23.30   3rd Qu.:17.1  
##  Max.   :32.50   Max.   :49.00   Max.   :47.70   Max.   :29.30   Max.   :19.6  
##       V22             peso           estatura     genero 
##  Min.   :18.00   Min.   : 42.00   Min.   :147.2   0:260  
##  1st Qu.:23.00   1st Qu.: 58.40   1st Qu.:163.8   1:247  
##  Median :27.00   Median : 68.20   Median :170.3          
##  Mean   :30.18   Mean   : 69.15   Mean   :171.1          
##  3rd Qu.:36.00   3rd Qu.: 78.85   3rd Qu.:177.8          
##  Max.   :67.00   Max.   :116.40   Max.   :198.1

Visulizar datos con diagrama de cajas. Peso

ggplot(datos, aes(genero, peso)) +
  geom_boxplot(colour = c("deeppink", "dodgerblue4"))

Visulizar datos con diagrama de cajas. Estatura

ggplot(datos, aes(genero, estatura)) +
  geom_boxplot(colour = c("deeppink", "dodgerblue4"))

Filtrar masculinos y femeninos

masculinos <-  filter(datos, genero == 1)

summary(masculinos)

##        V1              V2              V3              V4       
##  Min.   :34.10   Min.   :19.40   Min.   :27.50   Min.   :14.40  
##  1st Qu.:40.00   1st Qu.:26.80   1st Qu.:31.40   1st Qu.:19.35  
##  Median :41.20   Median :28.00   Median :32.40   Median :20.60  
##  Mean   :41.24   Mean   :28.09   Mean   :32.53   Mean   :20.81  
##  3rd Qu.:42.60   3rd Qu.:29.45   3rd Qu.:33.80   3rd Qu.:22.10  
##  Max.   :47.40   Max.   :34.70   Max.   :38.00   Max.   :27.50  
##        V5              V6              V7              V8       
##  Min.   :24.70   Min.   :12.40   Min.   : 9.80   Min.   :16.90  
##  1st Qu.:28.65   1st Qu.:13.80   1st Qu.:10.80   1st Qu.:18.80  
##  Median :29.90   Median :14.40   Median :11.20   Median :19.50  
##  Mean   :29.95   Mean   :14.46   Mean   :11.25   Mean   :19.56  
##  3rd Qu.:31.40   3rd Qu.:15.10   3rd Qu.:11.65   3rd Qu.:20.40  
##  Max.   :35.60   Max.   :16.70   Max.   :13.30   Max.   :23.00  
##        V9             V10             V11              V12        
##  Min.   :12.30   Min.   :100.2   Min.   : 79.30   Min.   : 67.10  
##  1st Qu.:14.20   1st Qu.:112.2   1st Qu.: 95.95   1st Qu.: 77.90  
##  Median :14.80   Median :116.5   Median :101.00   Median : 83.40  
##  Mean   :14.74   Mean   :116.5   Mean   :100.99   Mean   : 84.53  
##  3rd Qu.:15.30   3rd Qu.:121.0   3rd Qu.:106.05   3rd Qu.: 90.00  
##  Max.   :17.20   Max.   :134.8   Max.   :118.70   Max.   :113.20  
##       V13              V14              V15             V16      
##  Min.   : 67.00   Min.   : 81.50   Min.   :46.80   Min.   :25.6  
##  1st Qu.: 81.55   1st Qu.: 93.25   1st Qu.:53.70   1st Qu.:32.5  
##  Median : 87.30   Median : 97.40   Median :56.00   Median :34.4  
##  Mean   : 87.66   Mean   : 97.76   Mean   :56.50   Mean   :34.4  
##  3rd Qu.: 92.70   3rd Qu.:101.55   3rd Qu.:59.15   3rd Qu.:36.4  
##  Max.   :111.80   Max.   :118.70   Max.   :70.00   Max.   :42.4  
##       V17             V18             V19             V20       
##  Min.   :22.70   Min.   :31.10   Min.   :28.90   Min.   :16.40  
##  1st Qu.:27.10   1st Qu.:35.75   1st Qu.:35.50   1st Qu.:22.00  
##  Median :28.40   Median :37.00   Median :37.30   Median :23.00  
##  Mean   :28.24   Mean   :37.20   Mean   :37.21   Mean   :23.16  
##  3rd Qu.:29.40   3rd Qu.:38.45   3rd Qu.:38.95   3rd Qu.:24.30  
##  Max.   :32.50   Max.   :45.70   Max.   :47.70   Max.   :29.30  
##       V21             V22             peso           estatura     genero 
##  Min.   :14.60   Min.   :18.00   Min.   : 53.90   Min.   :157.2   0:  0  
##  1st Qu.:16.50   1st Qu.:24.00   1st Qu.: 70.95   1st Qu.:172.9   1:247  
##  Median :17.10   Median :29.00   Median : 77.30   Median :177.8          
##  Mean   :17.19   Mean   :31.67   Mean   : 78.14   Mean   :177.7          
##  3rd Qu.:17.85   3rd Qu.:37.00   3rd Qu.: 85.50   3rd Qu.:182.7          
##  Max.   :19.60   Max.   :65.00   Max.   :116.40   Max.   :198.1

femeninos <- filter(datos, genero == 0)

summary(femeninos)

##        V1              V2              V3              V4              V5      
##  Min.   :32.40   Min.   :18.70   Min.   :24.70   Min.   :14.30   Min.   :22.2  
##  1st Qu.:35.17   1st Qu.:26.20   1st Qu.:30.00   1st Qu.:16.50   1st Qu.:24.9  
##  Median :36.40   Median :27.80   Median :31.50   Median :17.50   Median :25.9  
##  Mean   :36.50   Mean   :27.58   Mean   :31.46   Mean   :17.72   Mean   :26.1  
##  3rd Qu.:37.80   3rd Qu.:29.20   3rd Qu.:32.90   3rd Qu.:18.70   3rd Qu.:27.1  
##  Max.   :42.60   Max.   :33.30   Max.   :37.80   Max.   :26.80   Max.   :33.2  
##        V6              V7               V8             V9       
##  Min.   : 9.90   Min.   : 8.100   Min.   :15.7   Min.   : 9.90  
##  1st Qu.:11.80   1st Qu.: 9.400   1st Qu.:17.3   1st Qu.:12.40  
##  Median :12.40   Median : 9.800   Median :18.0   Median :13.00  
##  Mean   :12.37   Mean   : 9.874   Mean   :18.1   Mean   :13.03  
##  3rd Qu.:12.90   3rd Qu.:10.400   3rd Qu.:18.7   3rd Qu.:13.72  
##  Max.   :15.00   Max.   :12.200   Max.   :24.3   Max.   :15.50  
##       V10             V11              V12              V13        
##  Min.   : 85.9   Min.   : 72.60   Min.   : 57.90   Min.   : 64.00  
##  1st Qu.: 96.1   1st Qu.: 81.97   1st Qu.: 64.75   1st Qu.: 76.65  
##  Median : 99.5   Median : 85.50   Median : 68.30   Median : 82.35  
##  Mean   :100.3   Mean   : 86.06   Mean   : 69.80   Mean   : 83.75  
##  3rd Qu.:103.9   3rd Qu.: 89.50   3rd Qu.: 72.75   3rd Qu.: 90.03  
##  Max.   :129.5   Max.   :109.00   Max.   :101.50   Max.   :121.10  
##       V14              V15             V16            V17       
##  Min.   : 78.80   Min.   :46.30   Min.   :22.4   Min.   :19.60  
##  1st Qu.: 90.75   1st Qu.:53.77   1st Qu.:26.4   1st Qu.:22.68  
##  Median : 94.95   Median :56.40   Median :27.8   Median :23.60  
##  Mean   : 95.65   Mean   :57.20   Mean   :28.1   Mean   :23.76  
##  3rd Qu.: 99.50   3rd Qu.:59.80   3rd Qu.:29.8   3rd Qu.:24.70  
##  Max.   :128.30   Max.   :75.70   Max.   :40.3   Max.   :30.80  
##       V18             V19             V20             V21       
##  Min.   :29.00   Min.   :28.40   Min.   :17.40   Min.   :13.00  
##  1st Qu.:33.50   1st Qu.:33.10   1st Qu.:20.30   1st Qu.:14.50  
##  Median :35.00   Median :34.85   Median :21.10   Median :15.00  
##  Mean   :35.26   Mean   :35.01   Mean   :21.21   Mean   :15.06  
##  3rd Qu.:36.80   3rd Qu.:36.60   3rd Qu.:22.00   3rd Qu.:15.60  
##  Max.   :49.00   Max.   :45.40   Max.   :26.00   Max.   :18.20  
##       V22             peso          estatura     genero 
##  Min.   :18.00   Min.   : 42.0   Min.   :147.2   0:260  
##  1st Qu.:22.00   1st Qu.: 54.5   1st Qu.:160.0   1:  0  
##  Median :26.00   Median : 59.0   Median :164.5          
##  Mean   :28.77   Mean   : 60.6   Mean   :164.9          
##  3rd Qu.:34.00   3rd Qu.: 65.6   3rd Qu.:169.5          
##  Max.   :67.00   Max.   :105.2   Max.   :182.9

Calculando estadísticos de masculinos y femeninos peso

media.peso.m <- mean(masculinos$peso)
desv.std.peso.m <- sd(masculinos$peso)


media.peso.f <- mean(femeninos$peso)
desv.std.peso.f <- sd(femeninos$peso)


media.peso.m

## [1] 78.14453

desv.std.peso.m

## [1] 10.51289

media.peso.f

## [1] 60.60038

desv.std.peso.f

## [1] 9.615699

Calculando estadísticos de masculinos y femeninos estatura

media.estatura.m <- mean(masculinos$estatura)
desv.std.estatura.m <- sd(masculinos$estatura)


media.estatura.f <- mean(femeninos$estatura)
desv.std.estatura.f <- sd(femeninos$estatura)


media.estatura.m

## [1] 177.7453

desv.std.estatura.m

## [1] 7.183629

media.estatura.f

## [1] 164.8723

desv.std.estatura.f

## [1] 6.544602

Calcular probabilidades PESO

1.- ¿Cual es la probabilidad de elegir a una persona al azar que varón “masculinos” y que pese mas de 60 kilos

Se requiere los datos de masculinos
Visulizando la curva de campana

plotDist("norm", mean = media.peso.m, sd = desv.std.peso.m, groups = x > 60, type = "h", xlab = "Peso Hombres", ylab = "Densidad" )

Calculando la probabilidad

pnorm(q = 60, mean = media.peso.m, sd = desv.std.peso.m, lower.tail = FALSE)

## [1] 0.9578202

1 - pnorm(q = 60, mean = media.peso.m, sd = desv.std.peso.m)

## [1] 0.9578202

2.- ¿Cual es la probalidad de elegir a una persona al azar que sea varón “masculinos” y que pese entre 70 y 80 kilos?

Se requiere los datos de masculinos
Visualizando la curva de campana

plotDist("norm", mean = media.peso.m, sd = desv.std.peso.m, groups = x > 70, type = "h", xlab = "Peso Hombres", ylab = "Densidad" )

plotDist("norm", mean = media.peso.m, sd = desv.std.peso.m, groups = x < 80, type = "h", xlab = "Peso Hombres", ylab = "Densidad" )

Calculando la probabilidad.

pnorm(q = 70, mean = media.peso.m, sd = desv.std.peso.m, lower.tail = FALSE) - pnorm(q = 80, mean = media.peso.m, sd = desv.std.peso.m, lower.tail = FALSE)

## [1] 0.3507943

3. ¿Cual es la probabilidad de elegir a una persona al azar que sea varón “masculinos”y que pese menos de 75 kilos

Se requiere los datos de masculinos
Visualizando la curva de campana

plotDist("norm", mean = media.peso.m, sd = desv.std.peso.m, groups = x < 75, type = "h", xlab = "Peso Hombres", ylab = "Densidad" )

Calculando la probabilidad.

pnorm(q = 75, mean = media.peso.f, sd = desv.std.peso.f, lower.tail = FALSE)

## [1] 0.06713016

4. ¿Cual es la probabilidad de elegir a una persona al azar que sea mujer “femeninos” y que pese mas de 55 kilos?

-Se requieren los datos de femeninos -Visulizando la curva de campan

plotDist("norm", mean = media.peso.f, sd = desv.std.peso.f, groups = x > 55, type = "h", xlab = "Peso Mujeres", ylab = "Densidad" )

Calculando la probabilidad

pnorm(q = 55, mean = media.peso.f, sd = desv.std.peso.f, lower.tail = FALSE)

## [1] 0.7198584

5.- ¿Cual es la probabilidad de elegir a una persona al azar que sea mujer “femeninos” y que pese entre 50 y 60 kilos?

-Se requieren los datos de femeninos -Visulizando la curva de campana

plotDist("norm", mean = media.peso.m, sd = desv.std.peso.m, groups = x > 50, type = "h", xlab = "Peso Hombres", ylab = "Densidad" )

plotDist("norm", mean = media.peso.m, sd = desv.std.peso.m, groups = x < 60, type = "h", xlab = "Peso Hombres", ylab = "Densidad" )

Calculando la probabilidad.

pnorm(q = 50, mean = media.peso.m, sd = desv.std.peso.m, lower.tail = FALSE) - pnorm(q = 60, mean = media.peso.m, sd = desv.std.peso.m, lower.tail = FALSE)

## [1] 0.03846719

6. ¿Cual es la probabilidad de elegir a una persona al azar que sea mujer “femeninos” y que pese menos de 45 kilos?

Se requieren los datos de femeninos
Visulizando la curva de campana

plotDist("norm", mean = media.peso.m, sd = desv.std.peso.m, groups = x < 45, type = "h", xlab = "Peso Hombres", ylab = "Densidad" )

Calculando la probabilidad.

pnorm(q = 45, mean = media.peso.f, sd = desv.std.peso.f, lower.tail = FALSE)

## [1] 0.9476397

Calcular probabilidades ESTATURA

7. ¿Cual es la probabilidad de elegir a una persona al azar que sea varón masculinosy que mida mas de 165 cm de altura

Se requieren los datos de masculinos
Visulizando la curva de campana

plotDist("norm", mean = media.estatura.m, sd = desv.std.estatura.m, groups = x > 165, type = "h", xlab = "Estatura Hombres", ylab = "Densidad")

Calculando la probabilidad

pnorm(q = 165, mean = media.estatura.m, sd = desv.std.estatura.m, lower.tail = FALSE)

## [1] 0.9619867

8. ¿Cual es la probabilidad de elegir a una persona al azar que sea varón masculinos y que mida entre 175 y 185 de estatura?

-Se requieren los datos de masculinos -Visulizando la curva de campana

plotDist("norm", mean = media.estatura.m, sd = desv.std.estatura.m, groups = x > 175, type = "h", xlab = "Estatura Hombres", ylab = "Densidad")

plotDist("norm", mean = media.estatura.m, sd = desv.std.estatura.m, groups = x < 185, type = "h", xlab = "Estatura Hombres", ylab = "Densidad")

Calculando la probabilidad

pnorm(q = 175, mean = media.estatura.m, sd = desv.std.estatura.m, lower.tail = FALSE)- pnorm(q = 185, mean = media.estatura.m, sd = desv.std.estatura.m, lower.tail = FALSE)

## [1] 0.4925565

9. ¿Cual es la probabilidad de elegir a una persona al azar que sea varón masculinos y que mida menos de 170 de altura?

Se requieren los datos de masculinos
Visulizando la curva de campana

plotDist("norm", mean = media.estatura.m, sd = desv.std.estatura.m, groups = x > 170, type = "h", xlab = "Estatura Hombres", ylab = "Densidad")

Calculando la probabilidad

pnorm(q = 170, mean = media.estatura.m, sd = desv.std.estatura.m, lower.tail = FALSE)

## [1] 0.8595264

10. ¿Cual es la probabilidad de elegir a una persona al azar que sea mujer femeninosy que mida mas de 155 de altura?

Se requieren los datos de femeninos
Visulizando la curva de campana

plotDist("norm", mean = media.estatura.f, sd = desv.std.estatura.f, groups = x > 155, type = "h", xlab = "Estatura Mujeres", ylab = "Densidad")

Calculando la probabilidad

pnorm(q = 155, mean =media.estatura.f, sd = desv.std.estatura.f, lower.tail = FALSE)

## [1] 0.9342823

11. ¿Cual es la probabilidad de elegir a una persona al azar que sea mujer femeninos y que mida entre 165 y 170 de estatura?

Se requieren los datos de femeninos
Visulizando la curva de campana

plotDist("norm", mean = media.estatura.m, sd = desv.std.estatura.m, groups = x > 165, type = "h", xlab = "Estatura Hombres", ylab = "Densidad")

plotDist("norm", mean = media.estatura.m, sd = desv.std.estatura.m, groups = x < 170, type = "h", xlab = "Estatura Hombres", ylab = "Densidad")

Calculando la probabilidad

pnorm(q = 155, mean = media.estatura.m, sd = desv.std.estatura.m, lower.tail = FALSE)- pnorm(q = 170, mean = media.estatura.m, sd = desv.std.estatura.m, lower.tail = FALSE)

## [1] 0.1397016

12. ¿Cual es la probabilidad de elegir a una persona al azar que sea mujer femeninos y mida que menos de 155 de altura?

Se requieren los datos de femeninos
Visulizando la curva de campana

plotDist("norm", mean = media.estatura.m, sd = desv.std.estatura.m, groups = x > 155, type = "h", xlab = "Estatura Hombres", ylab = "Densidad")

Calculando la probabilidad

pnorm(q = 155, mean = media.estatura.m, sd = desv.std.estatura.m, lower.tail = FALSE)

## [1] 0.999228

.

Ingrid Jesabel Rodriguez Nuñez

8/6/2020