1. Introducción.

Se desea comparar dos genotipos de papa con base al rendimiento (biomasa de tuberculos).

Un ensayo utilizó dos variedades (criolla y pastusa) involucrando 180 plantas de la primera variedad y 200 de la segunda. Los datos de rendimiento de la cossecha se presentan en los siguientes vectores.

2. Configuración.

criolla= rnorm(n=180,mean=2.8,sd=0.2)
pastusa=rnorm(n=200,mean=3.0,sd=0.21)
criolla
##   [1] 2.966113 3.103719 2.930128 2.894651 3.015479 2.520695 3.005356 2.566926
##   [9] 2.231120 2.612667 2.773101 2.781413 3.039396 3.014167 2.520091 2.542491
##  [17] 3.011926 2.740569 2.461011 2.773978 2.664966 3.020263 2.873455 2.834710
##  [25] 2.801759 2.693793 3.139308 2.872382 2.494922 2.937002 2.704264 2.657051
##  [33] 2.714802 2.710109 2.658831 2.745416 2.666917 2.741439 2.970199 2.816084
##  [41] 2.721792 2.787996 2.671350 2.778864 2.937291 2.567439 3.108516 3.323116
##  [49] 2.998199 2.799699 2.402791 2.635430 2.954392 2.912052 2.819698 2.793156
##  [57] 2.939067 2.950727 2.885299 3.022475 2.767766 2.849286 2.929312 2.832327
##  [65] 2.945751 2.754225 3.011872 2.874404 2.588560 2.855335 2.861501 2.693313
##  [73] 3.083735 3.073343 2.677244 2.581331 2.824609 3.102372 2.922093 2.670597
##  [81] 2.806016 3.052401 2.782242 3.256488 2.579484 2.963278 2.541120 2.479107
##  [89] 2.652826 2.659078 2.864642 2.791220 2.831981 2.862675 2.642415 2.553136
##  [97] 2.877453 2.757925 2.558972 2.826313 2.839034 2.691483 2.664359 2.645133
## [105] 2.359079 2.908138 2.516295 2.401445 2.741218 2.919267 2.915700 2.718726
## [113] 2.939715 3.134436 2.565613 3.250662 2.441295 2.822330 3.032733 2.739839
## [121] 2.866173 2.974672 2.832480 2.848755 2.675209 2.637583 3.184808 2.706803
## [129] 2.782831 2.526937 3.018651 2.462334 2.599289 2.413863 2.991041 2.648081
## [137] 2.748508 3.209423 3.073188 2.908412 2.785592 2.826106 2.712084 3.021889
## [145] 2.633200 3.032607 2.864902 2.823633 2.913835 2.864566 2.740889 2.661775
## [153] 2.561459 3.006243 2.916136 2.820064 2.767697 3.103166 2.758683 2.670985
## [161] 2.712479 3.327096 3.037094 2.778246 2.885370 2.616361 2.930000 2.906949
## [169] 2.932821 2.961629 2.614793 2.768474 2.269116 3.214116 2.589011 2.908274
## [177] 3.010421 2.664557 2.847186 2.809652
pastusa
##   [1] 3.484303 2.683919 2.933043 2.504802 2.693382 3.180744 2.683025 3.015655
##   [9] 2.894107 3.229029 2.715904 3.117870 2.939864 2.827469 2.855918 2.992641
##  [17] 2.646764 2.872841 2.630584 2.913561 2.862150 3.295817 3.140635 2.913948
##  [25] 2.826017 3.096989 2.867045 3.177345 3.037240 3.053008 2.778036 2.979093
##  [33] 3.150302 2.664696 2.720589 3.102863 3.219413 2.911087 2.958407 2.985596
##  [41] 3.061036 3.122990 3.122354 3.375718 2.695669 2.893782 3.127633 3.400415
##  [49] 2.872272 2.875886 3.346452 3.025261 3.093295 3.038598 2.213848 3.313646
##  [57] 2.840197 2.969676 2.687457 2.851497 2.973148 3.193485 3.153121 2.811915
##  [65] 3.066757 3.028615 3.033774 3.076977 2.840826 3.177540 3.315743 3.035322
##  [73] 3.004835 3.155152 3.308309 2.886127 2.792537 3.231395 2.965240 2.819118
##  [81] 3.507385 3.082909 3.104456 2.797337 3.009733 2.769801 2.818996 3.236042
##  [89] 3.042187 2.976448 2.894769 3.267491 2.704910 2.926635 2.774128 3.270265
##  [97] 3.095486 3.008228 3.224919 2.869791 3.311679 3.241083 2.636474 3.121520
## [105] 3.227618 2.964363 3.088502 2.818605 2.959151 2.943479 3.365831 2.782050
## [113] 2.981302 2.928604 2.844831 3.399319 3.087715 2.878443 2.860950 3.111008
## [121] 2.994251 2.696581 2.910874 2.714278 3.061788 2.927201 2.700490 2.486756
## [129] 2.846640 2.873971 3.038653 2.957759 3.182081 3.160423 3.080343 2.982864
## [137] 2.913733 3.444116 3.140191 2.888696 2.681459 2.907941 3.232051 3.401562
## [145] 2.732993 3.108951 3.135880 2.963576 2.947440 3.049815 3.380878 3.151738
## [153] 3.125055 2.948859 3.296462 3.534728 3.181730 2.890931 3.155398 2.556476
## [161] 3.032700 2.786744 3.044433 2.842082 2.832966 3.112076 2.900413 2.936914
## [169] 2.804688 3.069845 2.975226 2.829231 2.925815 3.234545 2.703375 3.113502
## [177] 2.931970 3.016960 3.060411 2.848790 3.168632 3.138853 2.999185 2.896186
## [185] 2.650398 3.072935 3.050344 3.125027 2.998042 2.881063 2.766305 3.139941
## [193] 3.305382 3.237064 2.907517 2.564400 3.040938 3.023527 2.892521 3.315107
criolla=round(criolla,2)
pastusa=round(pastusa,2)

par(mfrow=c(1,2))

hist(criolla)
abline(v=mean(criolla),col="red", lwd=3)

hist(pastusa)
abline(v=mean(pastusa),col="red", lwd=3)

par(mfrow=c(1,2))
boxplot(criolla, main ="criolla", ylab="RTO(Kg/planta)")
boxplot(pastusa, main ="pastusa", ylab="RTO(Kg/planta)")

summary(criolla)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.230   2.667   2.815   2.806   2.940   3.330
summary(pastusa)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.210   2.860   2.990   2.994   3.130   3.530
library(psych)
## Warning: package 'psych' was built under R version 4.2.2
psych::describe(criolla)
##    vars   n mean  sd median trimmed mad  min  max range  skew kurtosis   se
## X1    1 180 2.81 0.2   2.82    2.81 0.2 2.23 3.33   1.1 -0.05     0.04 0.01
psych::describe(pastusa)
##    vars   n mean   sd median trimmed mad  min  max range  skew kurtosis   se
## X1    1 200 2.99 0.21   2.99    2.99 0.2 2.21 3.53  1.32 -0.11     0.38 0.01

coeficiente de variación cv=100*sd/mean

cvcriolla=100*sd(criolla)/mean(criolla)
cvpastusa=100*sd(pastusa)/mean(pastusa)
cvcriolla
## [1] 7.135159
cvpastusa
## [1] 7.044854

3. Conclusión.

Desde el analisis descriptivo, ambos coeficientes de variación bajos se pueden omitir el problema de diferente variabilidad, se selecciona la variedad de mayor rendimiento promedio.

Analisis inferencia a traves de pruebas de hipotesis

\[ H_o : μ_{pastusa} = μ_{criolla} \\H_a : μ_{pastusa} ≠ μ_{criolla} \]

Prueba t-student para comparar dos muestras independientes.

Modalidad 1: Varianzas iguales. Modalidad 2: Varianzas desiguales.

Prueba para la comparacion de dos varianzas

\[ H_0 : σ^2_{pastusa} = σ^2_{criolla} \\H_a : σ^2_{pastusa} ≠ σ^2_{criolla} \]

var(criolla)
## [1] 0.04007713
var(pastusa)
## [1] 0.04449592
vt=var.test(pastusa,criolla)
vt$p.value
## [1] 0.4754762
ifelse(vt$p.value<0.025,"Rechazo Ho","No rechazo Ho")
## [1] "No rechazo Ho"

Prueba t-student para comparar las dos medias con varianzas iguales.

pt = t.test(pastusa, criolla,alternative ="t",var.equal=TRUE)
pt
## 
##  Two Sample t-test
## 
## data:  pastusa and criolla
## t = 8.9112, df = 378, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  0.1469289 0.2301267
## sample estimates:
## mean of x mean of y 
##  2.994250  2.805722

4. Conclusión final.

Los datos proporcionan evidencia a favor de la hipotesis nula, es decir que estadisticamente se consideran ambas variedades de igual rendimiento Cualquiera de las variedades es igual de buena.