##1° punto ###
library(MASS)
library(UsingR)
## Loading required package: HistData
## Loading required package: Hmisc
##
## Attaching package: 'Hmisc'
## The following objects are masked from 'package:base':
##
## format.pval, units
View(brightness)
library(ggplot2)
library(HistData)
## 2.2. a HIstograma y superimposed density plot##
data("brightness")
hist(brightness, freq = F)
density(brightness)
##
## Call:
## density.default(x = brightness)
##
## Data: brightness (966 obs.); Bandwidth 'bw' = 0.2425
##
## x y
## Min. : 1.342 Min. :0.0000156
## 1st Qu.: 4.296 1st Qu.:0.0025868
## Median : 7.250 Median :0.0226027
## Mean : 7.250 Mean :0.0845551
## 3rd Qu.:10.204 3rd Qu.:0.1267748
## Max. :13.158 Max. :0.3961890
plot(density(brightness))
hist(brightness, freq = F)
lines(density(brightness))
## 2.2. b Boxplot## LA data presenta Outliers.
#El segundo outlier más pequeño es 2.28
boxplot(brightness)
boxplot.stats(brightness)$out
## [1] 12.31 11.71 5.53 11.28 4.78 5.13 4.37 5.04 12.43 12.04 4.55 11.55
## [13] 12.14 11.63 4.99 11.67 4.61 11.99 12.04 5.55 12.17 11.55 11.79 12.19
## [25] 2.07 11.65 11.73 2.28 5.42 3.88 5.54 5.29 5.01 11.55 4.89 11.80
## [37] 5.41 5.24
sort(boxplot.stats(brightness)$out)
## [1] 2.07 2.28 3.88 4.37 4.55 4.61 4.78 4.89 4.99 5.01 5.04 5.13
## [13] 5.24 5.29 5.41 5.42 5.53 5.54 5.55 11.28 11.55 11.55 11.55 11.63
## [25] 11.65 11.67 11.71 11.73 11.79 11.80 11.99 12.04 12.04 12.14 12.17 12.19
## [37] 12.31 12.43
segundomenoroutlier = sort(boxplot.stats(brightness)$out)[2]
segundomenoroutlier
## [1] 2.28
## 2.2. c Crear variable sin outliers
obtener_outliers = which(brightness %in% c(boxplot.stats(brightness)$out))
obtener_outliers
## [1] 6 17 107 111 122 145 154 183 191 263 300 307 320 353 355 390 441 454 463
## [20] 475 522 548 560 569 676 730 736 744 759 763 811 812 839 896 908 909 928 948
#Defino la nueva variable brightness.without#
brightness.without = brightness[-c(obtener_outliers)]
brightness.without
## [1] 9.10 9.27 6.61 8.06 8.55 9.64 9.05 8.59 8.59 7.34 8.43 8.80
## [13] 7.25 8.60 8.15 11.03 6.53 8.51 7.55 8.69 7.57 9.05 6.28 9.13
## [25] 9.32 8.83 9.14 8.26 7.63 9.09 8.10 6.43 9.07 7.68 10.44 8.65
## [37] 7.46 8.70 10.61 8.20 6.18 7.91 9.59 8.57 10.78 7.31 9.53 6.49
## [49] 8.94 8.56 10.96 10.57 7.40 8.12 8.27 7.05 9.09 8.34 8.86 8.27
## [61] 6.36 8.08 11.00 8.55 7.83 8.79 8.33 10.42 8.26 8.97 6.90 9.93
## [73] 7.42 9.03 8.41 8.06 8.69 8.40 8.57 9.50 8.85 9.61 10.62 8.05
## [85] 7.80 5.71 7.87 7.64 7.66 8.68 8.12 10.10 8.67 10.46 9.87 9.48
## [97] 7.04 8.44 9.88 7.05 8.29 9.34 7.73 6.22 8.53 7.23 8.61 10.76
## [109] 8.93 7.95 7.46 8.60 8.55 9.20 6.82 8.29 6.83 7.21 5.58 8.70
## [121] 8.06 10.86 6.50 9.32 9.14 8.13 10.62 6.62 9.96 8.64 6.60 6.25
## [133] 7.83 10.03 9.04 8.47 7.33 8.66 10.35 8.96 8.49 11.26 8.15 7.04
## [145] 10.02 8.90 7.78 9.93 8.60 8.51 7.09 6.93 8.68 8.98 9.84 8.98
## [157] 7.98 10.16 8.86 8.58 9.56 9.24 9.63 5.80 9.05 8.45 8.86 7.84
## [169] 8.86 8.93 7.97 6.90 8.47 6.77 8.55 8.48 8.53 6.33 8.99 8.64
## [181] 9.55 8.74 8.16 9.46 5.70 7.62 8.95 8.97 8.94 7.24 10.32 8.24
## [193] 8.62 9.18 8.53 8.54 8.56 9.41 5.87 7.20 9.05 9.52 10.24 7.70
## [205] 8.17 7.29 9.26 7.94 8.42 8.56 7.52 7.74 8.85 9.01 7.17 9.04
## [217] 10.30 9.86 7.64 8.27 8.44 9.58 8.43 8.49 9.64 9.17 8.09 9.00
## [229] 6.25 8.56 10.81 8.76 7.76 7.82 7.90 8.52 9.73 9.19 8.10 8.75
## [241] 8.14 8.65 10.30 6.46 6.73 7.96 9.53 8.87 6.59 8.65 9.64 9.15
## [253] 9.04 8.42 8.09 9.06 8.09 8.18 8.77 7.36 9.16 8.82 11.14 6.24
## [265] 9.44 7.49 6.96 7.94 8.69 8.15 8.45 7.92 7.45 9.01 8.55 9.23
## [277] 9.16 7.90 8.68 7.78 8.21 8.11 8.29 7.89 9.67 8.24 6.80 8.18
## [289] 8.44 7.45 6.31 8.15 8.27 7.66 8.59 7.09 8.54 9.58 8.44 8.59
## [301] 8.01 8.29 9.62 7.26 7.91 9.45 8.19 8.93 7.65 8.53 7.38 8.56
## [313] 8.76 9.56 7.09 9.83 5.90 10.80 8.41 9.05 8.79 8.88 7.59 9.60
## [325] 10.66 8.55 8.11 9.44 9.60 5.78 10.66 6.38 8.80 7.79 8.60 7.77
## [337] 10.37 9.80 10.42 9.22 8.43 7.33 8.93 9.09 9.26 8.73 9.18 8.12
## [349] 9.26 8.94 6.11 9.13 7.90 9.34 7.13 10.82 7.46 8.72 7.02 9.08
## [361] 8.37 5.59 7.37 5.68 8.56 8.72 9.06 8.82 8.18 9.39 9.10 8.46
## [373] 9.15 8.28 8.18 7.93 9.21 6.09 8.31 7.83 8.72 6.61 6.25 7.82
## [385] 8.66 8.15 8.97 8.15 7.47 8.63 8.13 8.23 8.41 6.47 9.83 8.64
## [397] 7.73 8.64 8.94 8.84 6.32 5.80 8.97 7.53 7.41 7.80 8.14 6.71
## [409] 8.73 9.37 8.69 9.95 7.10 8.09 6.88 9.48 9.04 9.30 8.49 8.30
## [421] 7.95 7.08 6.93 8.38 8.56 8.78 7.42 8.26 7.71 6.91 9.16 8.99
## [433] 8.63 9.90 7.59 7.39 7.78 7.47 6.97 8.82 9.13 7.86 7.13 9.45
## [445] 8.78 7.23 9.73 7.36 7.36 8.47 9.37 6.99 8.20 8.36 8.22 9.91
## [457] 9.67 8.60 10.07 10.15 7.75 9.21 9.66 8.47 9.37 9.44 9.99 10.38
## [469] 7.51 8.91 7.45 9.57 8.99 8.58 6.90 7.55 7.93 9.71 9.57 8.55
## [481] 6.62 7.89 7.51 7.36 8.66 8.51 6.65 9.67 7.80 8.21 7.90 8.94
## [493] 9.82 8.69 8.57 8.89 5.98 7.92 7.60 8.22 5.70 8.75 6.93 7.97
## [505] 8.06 10.13 7.31 8.35 5.57 9.85 9.16 9.03 10.07 9.76 9.35 10.95
## [517] 8.87 6.68 9.69 8.05 10.30 6.07 8.51 7.71 8.56 8.26 8.62 10.92
## [529] 10.51 9.83 9.84 9.74 8.21 8.72 8.03 9.00 6.19 8.22 7.93 10.18
## [541] 8.98 9.13 6.91 8.79 8.23 10.24 8.83 7.62 8.96 10.41 8.97 9.61
## [553] 8.29 8.30 8.26 7.44 9.52 8.20 8.68 8.65 10.52 8.41 9.18 8.42
## [565] 8.86 7.92 10.97 8.85 9.31 10.28 7.56 7.88 7.99 8.23 8.52 9.14
## [577] 6.20 7.64 8.95 7.48 7.06 7.33 8.98 8.24 8.53 8.40 7.48 8.46
## [589] 9.29 8.57 8.70 8.50 8.37 6.87 7.50 7.39 8.19 7.56 8.37 7.39
## [601] 6.73 8.66 8.25 8.47 8.01 6.83 9.06 8.79 7.44 6.43 5.93 8.85
## [613] 9.86 8.55 7.66 7.82 9.08 10.10 8.21 8.85 7.79 7.58 7.85 7.18
## [625] 7.54 9.72 7.12 9.77 8.84 5.67 8.15 9.61 8.19 7.27 8.51 8.36
## [637] 10.00 8.74 6.18 10.26 10.16 8.31 8.58 7.04 8.81 5.99 8.22 9.86
## [649] 8.00 9.40 9.10 8.11 8.89 9.43 7.59 8.72 9.86 9.23 9.50 10.73
## [661] 7.59 7.41 9.26 7.78 7.76 8.94 8.95 6.41 6.11 7.76 7.38 6.21
## [673] 7.05 7.44 8.50 7.84 11.01 7.88 9.10 8.65 8.41 7.81 7.43 8.76
## [685] 7.58 9.55 6.82 10.24 6.24 7.31 10.52 9.27 7.13 9.14 8.48 8.57
## [697] 7.21 9.05 7.72 8.03 6.47 5.57 6.32 7.78 8.58 10.37 9.23 9.20
## [709] 6.93 9.32 7.11 9.79 8.21 8.42 7.05 9.26 8.77 9.25 9.30 10.63
## [721] 9.90 9.89 9.33 7.78 7.02 11.26 8.89 9.60 7.07 6.01 9.11 8.24
## [733] 8.97 8.59 7.17 7.94 7.27 9.59 7.94 8.52 7.59 9.17 8.08 9.80
## [745] 8.92 9.91 9.42 8.84 10.15 8.37 9.33 9.35 7.40 8.35 9.53 9.59
## [757] 10.05 8.57 8.48 8.43 8.45 8.84 11.18 8.64 8.42 6.34 7.93 8.36
## [769] 8.32 7.77 6.84 8.78 7.19 8.50 8.82 9.04 7.93 7.66 10.07 9.03
## [781] 8.13 7.51 9.08 7.10 7.88 9.40 9.06 8.38 10.65 7.77 8.50 8.61
## [793] 10.05 8.71 9.37 6.97 8.56 9.34 9.47 8.11 8.91 7.83 8.95 7.20
## [805] 9.37 5.84 9.81 9.27 9.50 9.32 8.92 8.38 7.74 8.60 9.49 8.35
## [817] 7.11 9.87 8.98 7.75 8.24 6.74 6.83 7.70 6.70 8.67 9.94 8.73
## [829] 9.63 6.66 8.29 8.47 8.16 8.97 7.51 8.97 8.55 5.84 7.85 8.68
## [841] 8.05 8.27 7.68 9.40 7.77 6.89 7.55 8.27 8.16 8.07 7.91 7.71
## [853] 10.16 8.41 8.88 9.64 7.93 7.78 8.90 8.55 9.15 10.86 9.08 7.44
## [865] 10.35 6.68 8.85 8.90 8.24 6.74 10.75 8.44 7.69 8.88 7.70 8.60
## [877] 8.44 9.50 9.03 7.15 7.95 8.23 9.81 8.48 9.33 8.97 8.08 7.47
## [889] 8.34 7.75 8.34 7.56 6.93 10.03 8.69 9.04 8.32 7.85 7.21 8.98
## [901] 7.09 8.85 9.21 8.61 7.91 7.47 8.65 8.53 9.92 8.09 7.06 8.45
## [913] 8.73 7.45 9.02 7.51 7.32 8.17 9.45 9.72 9.34 8.75 9.32 7.91
## [925] 7.49 6.53 6.18 8.69
boxplot(brightness.without)
##2.3. UScereal relationship between ...
# a) i manufacturer and shelf
data(UScereal)
barplot(table(UScereal$mfr, UScereal$shelf), beside = T, main = "Relationship manufacturer and shelf", xlab = "shelf", ylab = "# manufacturer")
# ii. fat and vitamins
barplot(table(UScereal$fat, UScereal$vitamins), beside = T, main = "Relationsship fat and vitamins",
xlab = "vitamins", ylab = "fat")
# iii. fat and shelf
barplot(table(UScereal$fat, UScereal$shelf), beside = T, main = "Relationship fat and shelf")
cor(UScereal$fat, UScereal$shelf)
## [1] 0.3256975
#baja correlación.
# iv. Carbohydrates and sugars
plot(UScereal$carbo, UScereal$sugars)
cor(UScereal$carbo, UScereal$sugars)
## [1] -0.04082599
# -0.0408 --> baja correlación negativa
# v. fiber and manufacturer (X = manufaturer , y = fiber)
plot(UScereal$mfr, UScereal$fibre)
# vi. sodium and sugars
cor(UScereal$sodium, UScereal$sugars) # 0.211, correlación baja positiva.
## [1] 0.2112437
plot(UScereal$sodium, UScereal$sugars)
## 2.4 mammals ---> relationship between body weight and brain weight of mammals.
# a) Linear correlation
data(mammals)
cor(mammals$body, mammals$brain) # 0.934, fuerte correlación positiva entre las variables.
## [1] 0.9341638
pairs(mammals$body ~ mammals$brain) # en este emparejamiento, se aprecia la correlación lineal existente #entre las variables.
# b) plot the data.
plot(mammals)
# c) Transform the data with log function.
pairs(log(mammals$body)~log(mammals$brain))
cor(log(mammals$body), log(mammals$brain)) # 0.959, mejora la realción lineal entre las variables
## [1] 0.9595748
#y así se puede apreciar en el plot.
## 2.5. emissions ---> CO2 and gross domestic products in 26 countries.
# a) Relationship GDP, perCapita and CO2 of each country.
data(emissions)
cor(emissions) # correlación altamente positiva de CO2 con GDP, media entre GDP y perCapita
## GDP perCapita CO2
## GDP 1.0000000 0.4325303 0.9501753
## perCapita 0.4325303 1.0000000 0.2757962
## CO2 0.9501753 0.2757962 1.0000000
# baja correlación positiva entre CO2 y perCapita y se puede apreciar en el gráfico.
pairs(emissions)
# b) Modelo de regresión lineal para predecir emisión de CO2 desde las variables.
reg_lineal= lm(emissions$CO2 ~ emissions$GDP + emissions$perCapita, data = emissions)
summary(reg_lineal)
##
## Call:
## lm(formula = emissions$CO2 ~ emissions$GDP + emissions$perCapita,
## data = emissions)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1037.3 -167.4 10.8 153.2 1052.0
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.100e+02 2.044e+02 2.495 0.0202 *
## emissions$GDP 8.406e-04 5.198e-05 16.172 4.68e-14 ***
## emissions$perCapita -3.039e-02 1.155e-02 -2.631 0.0149 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 382.8 on 23 degrees of freedom
## Multiple R-squared: 0.9253, Adjusted R-squared: 0.9188
## F-statistic: 142.5 on 2 and 23 DF, p-value: 1.102e-13
CO2_predict = predict(reg_lineal, emissions)
plot(emissions$GDP + emissions$perCapita, CO2_predict)
CO2_predict
## UnitedStates Japan Germany France UnitedKingdom
## 6403.720110 2357.274571 1328.457202 939.412260 915.510264
## Italy Russia Canada Spain Australia
## 888.117914 948.030553 418.174003 551.546727 203.695487
## Netherlands Poland Belgium Sweden Austria
## 137.905405 524.997147 3.295727 57.168681 6.260516
## Switzerland Portugal Greece Ukraine Denmark
## -65.251059 177.533136 235.468677 538.782254 -82.034073
## Norway Romania CzechRepublic Finland Hungary
## -213.820805 449.888660 273.235200 -5.729292 353.120804
## Ireland
## 59.239930
cor(emissions$CO2, CO2_predict) # 0.962 --> alta correlación positiva.
## [1] 0.9619321
# es un modelo ajustado con una buena confianza estadística, donde
#la variable GDP es la que mejor explica la predicción de emisión de CO2
# c) identificar outliers y se corre el modelo sin los outliers
boxplot(emissions)
boxplot.stats(emissions$CO2)$out
## [1] 6750 1320 1740 2000
outliers_emissions = which(emissions$CO2 %in% c(boxplot.stats(emissions$CO2)$out))
outliers_emissions
## [1] 1 2 3 7
CO2without.outliers = emissions$CO2[-c(outliers_emissions)]
CO2without.outliers
## [1] 550 675 540 700 370 480 240 400 145 75 80 54 75 125 420 75 56 160 150
## [20] 76 85 63
GDPwithout.outliers = emissions$GDP[-c(outliers_emissions)]
GDPwithout.outliers
## [1] 1320000 1242000 1240000 658000 642400 394000 343900 280700 236300
## [10] 176200 174100 172400 149500 137400 124900 122500 120500 114200
## [19] 111900 102100 73200 59900
perCapitawithout.outliers = emissions$perCapita[-c(outliers_emissions)]
perCapitawithout.outliers
## [1] 22381 21010 21856 21221 16401 20976 21755 7270 23208 19773 21390 23696
## [13] 15074 12833 2507 22868 27149 5136 10885 19793 7186 16488
reg_lineal2<- lm(CO2without.outliers ~ GDPwithout.outliers + perCapitawithout.outliers, data = emissions)
summary(reg_lineal2)
##
## Call:
## lm(formula = CO2without.outliers ~ GDPwithout.outliers + perCapitawithout.outliers,
## data = emissions)
##
## Residuals:
## Min 1Q Median 3Q Max
## -130.88 -63.84 -32.27 16.79 334.38
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.233e+02 7.202e+01 3.101 0.00588 **
## GDPwithout.outliers 4.912e-04 6.989e-05 7.028 1.09e-06 ***
## perCapitawithout.outliers -8.525e-03 4.114e-03 -2.072 0.05209 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 121.4 on 19 degrees of freedom
## Multiple R-squared: 0.7225, Adjusted R-squared: 0.6933
## F-statistic: 24.73 on 2 and 19 DF, p-value: 5.142e-06
plot(GDPwithout.outliers + perCapitawithout.outliers, CO2without.outliers )
CO2_predict_without_outliers = predict(reg_lineal2, emissions)
## Warning: 'newdata' had 26 rows but variables found have 22 rows
plot(GDPwithout.outliers + perCapitawithout.outliers, CO2_predict_without_outliers )
cor(CO2without.outliers, CO2_predict_without_outliers) # 0.85
## [1] 0.8499957
boxplot(emissions$CO2)
boxplot(CO2without.outliers)
## 2.6. Anorexia ---> cambio de peso en mujeres.
#a) cual tratamiento es más efectivo?
data(anorexia)
posicion_Wt = which(anorexia$Postwt>anorexia$Prewt)
exitosos = anorexia[c(posicion_Wt),]
mejor_tto = which.max(table(exitosos$Treat))
mejor_tto = c(paste(names(mejor_tto),": ",max(table(exitosos$Treat))," casos exitosos"))
table(exitosos$Treat)
##
## CBT Cont FT
## 18 11 13
mejor_tto # --> CBT con 18 casos exitosos-
## [1] "CBT : 18 casos exitosos"
# b) Cuántos pacientes ganaron y cuántos perdieron peso?
posicion_perdida_peso <- which(anorexia$Postwt<anorexia$Prewt)
casos_fracaso = anorexia[c(posicion_perdida_peso),]
ganaron_peso = length(posicion_Wt)
ganaron_peso # --> 42
## [1] 42
perdieron_peso = nrow(casos_fracaso)
perdieron_peso # -_> 29
## [1] 29
## 2.7. Generar 2 vectores de 50 valores desde una distribución normal con rnorm (50)
x = rnorm(50, mean = 0, sd = 1)
y = rnorm(50, mean = 0, sd = 1)
# a) Test Shapiro-Wilk (normalidad para cada vector)
vector1 = c(x)
shapiro.test(vector1) # pvalor > alfa --> no se rechaza Ho, lo que indica que es de distribución normal.
##
## Shapiro-Wilk normality test
##
## data: vector1
## W = 0.98636, p-value = 0.828
vector2= c(y)
shapiro.test(vector2) # pvalor > alfa --> No se rechaza Ho, lo que indica que la distribución es normal.
##
## Shapiro-Wilk normality test
##
## data: vector2
## W = 0.95103, p-value = 0.03761
# b) t-students de ambos vectores
TEST = t.test(vector1, vector2)
TEST # En este test t de student, tenemos un valor de p > 0.05 (alfa) lo que nos indica que ambas variables no difieren en su media ---> no hay diferencia significativa entre ambos vectores.
##
## Welch Two Sample t-test
##
## data: vector1 and vector2
## t = -0.082074, df = 97.575, p-value = 0.9348
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.3964094 0.3649236
## sample estimates:
## mean of x mean of y
## -0.09970833 -0.08396544
boxplot(vector1,vector2,names=c("X1","X2"))
medias <- c(mean(vector1),mean(vector2))
points(medias,pch=18,col="red") # se demuestra que no hay diferencia significativa en sus medias.
## 2.8. 50 valores con rnorm y 50 valores con rbinom
vectorr = rnorm(50, mean = 0, sd = 1)
vectorb = rbinom(50, size = 30, prob = 0.3)
# a) Shapiro para cada vector.
shapiro.test(vectorr) # pvalor > alfa, NO se rechaza Ho, hay normalidad.
##
## Shapiro-Wilk normality test
##
## data: vectorr
## W = 0.9412, p-value = 0.01503
shapiro.test(vectorb) # pvalor > alfa, NO se rechaza Ho, hay normalidad.
##
## Shapiro-Wilk normality test
##
## data: vectorb
## W = 0.97081, p-value = 0.2496
# b) t de student entre los dos vectores.
TEST2 = t.test(vectorr, vectorb)
TEST2 # p valor < alfa, lo que indica que hay diferencias significativas entre ambos vectores
##
## Welch Two Sample t-test
##
## data: vectorr and vectorb
## t = -19.209, df = 59.356, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -10.045922 -8.150601
## sample estimates:
## mean of x mean of y
## 0.1217386 9.2200000
## 2.9. 100 valores con rnor
vector12 = rnorm(100, mean = 0, sd = 1)
hist(vector12) # al repetir el ejercicio de creación del vector, cambia la distribución en el histograma de manera aleatoria
summary(vector12)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -2.3692 -0.8401 -0.2374 -0.1390 0.5221 2.8367
## 2.10. Función pwr.t.test. y dos ejemplos prácticos.
# la función pwr.t.test, nos proporciona el poder de la prueba t.
#ejemplo 1:
library(pwr)
pwr.t.test(n=110, d=0.65, sig.level = 0.05, type = "two.sample", alternative = "two.sided")
##
## Two-sample t test power calculation
##
## n = 110
## d = 0.65
## sig.level = 0.05
## power = 0.9977389
## alternative = two.sided
##
## NOTE: n is number in *each* group
#ejemplo 2:
pwr.t.test(n=35, d=0.8, sig.level = 0.01, type = "two.sample", alternative = "two.sided")
##
## Two-sample t test power calculation
##
## n = 35
## d = 0.8
## sig.level = 0.01
## power = 0.7545473
## alternative = two.sided
##
## NOTE: n is number in *each* group
# a medida que el nivel de significancia es menor, el poder de la prueba t disminuye