Pruebas de hipotesis

Problema 1:

Se desea comparar dos genotipos de papa con base al rendimiento (biomasa de tuberculos). Un ensayo utilizo dos variedades (Criolla y pastusa) involucrando 180 plantas de la primera variedda y 200 de la segunda. Los datos de rendimiento en la cosecha se presentan en los siguientes vectores

criolla = rnorm(n = 180, mean = 2.8, sd = 0.2)
pastusa = rnorm(n = 200, mean = 3.0, sd =0.21)
criolla
##   [1] 2.792753 2.868761 2.719878 2.666651 2.298494 2.570088 2.898854 2.954451
##   [9] 2.684129 3.028810 2.918895 2.955362 2.738748 2.918407 2.520066 2.995732
##  [17] 2.726035 3.052121 2.693141 2.954735 3.110292 2.613743 2.533385 3.137412
##  [25] 2.723156 2.939152 2.705454 2.934443 3.066505 3.049951 2.493375 2.780009
##  [33] 2.864221 2.683656 2.958226 2.983761 2.911409 2.548196 2.876387 3.028146
##  [41] 2.632319 3.030976 2.711356 2.782764 2.636094 2.756331 2.599838 2.892644
##  [49] 2.829805 2.770474 2.668992 3.025978 2.597059 2.637827 2.737686 2.786508
##  [57] 3.061385 2.426104 2.811881 2.865964 2.697372 2.927608 2.917723 2.962369
##  [65] 2.825674 2.750494 2.779137 2.921300 3.125617 2.766495 2.741157 3.109010
##  [73] 2.566970 2.887718 3.039528 2.767257 2.873484 2.971344 2.668733 2.473050
##  [81] 2.964861 2.936889 2.382100 2.609811 2.635296 3.012684 2.868917 2.891189
##  [89] 2.857643 2.941129 2.889006 2.904382 2.803749 3.027368 2.752771 3.123342
##  [97] 2.642240 3.161537 2.620079 2.743383 2.936850 2.505061 2.683717 2.684950
## [105] 2.535246 2.852434 2.655612 3.075579 2.604307 3.101002 2.448770 2.668043
## [113] 2.434855 2.678888 2.570080 2.822971 2.868786 2.684068 2.873373 2.849364
## [121] 2.838262 2.625865 2.864565 3.167501 2.559780 2.788568 2.815953 2.830768
## [129] 2.779835 2.372772 2.704138 2.819432 2.373170 2.279779 2.979701 2.766197
## [137] 2.582818 2.826580 2.764209 2.476769 2.662668 2.744635 2.750284 2.740046
## [145] 2.806101 2.646211 2.906456 3.025345 3.041186 2.836844 2.627060 3.047017
## [153] 3.074512 3.067591 2.959258 2.694489 2.942127 3.038852 2.486697 2.744078
## [161] 2.967716 2.827714 2.967130 3.171051 2.753061 2.830828 2.840165 2.977654
## [169] 2.663844 2.933403 2.549820 2.619864 2.519973 2.928281 2.956975 2.738179
## [177] 2.662801 2.517071 2.740010 3.124481
pastusa
##   [1] 2.859504 2.838597 2.970585 3.312228 3.004486 3.269155 3.041759 3.302456
##   [9] 3.491993 2.676623 3.140786 3.050510 2.885801 3.340257 2.674593 2.728243
##  [17] 2.994855 2.775080 2.587473 2.813871 3.248211 2.940531 3.020637 3.146594
##  [25] 3.226868 3.051559 2.931308 3.191571 3.214265 2.811423 3.122156 2.945297
##  [33] 2.957322 2.900621 2.913203 3.253388 2.906448 3.180656 3.235074 3.373372
##  [41] 3.054485 2.749928 3.133769 3.411905 2.814795 3.231525 3.064199 2.595735
##  [49] 2.940088 2.991673 3.106892 2.795561 3.038495 2.949534 2.900761 3.109375
##  [57] 2.915953 2.976912 2.899542 3.500798 3.507903 3.157522 3.106284 3.169773
##  [65] 2.969968 3.099668 2.928148 3.129759 3.428498 2.806495 2.961971 3.146072
##  [73] 2.775133 2.971406 3.023890 2.907363 3.151073 2.827487 3.001509 2.879054
##  [81] 3.107956 2.870101 2.800804 3.058745 3.362391 3.266896 2.847534 3.078457
##  [89] 3.170523 2.595810 3.306572 2.746397 2.814937 2.779156 3.224973 3.105152
##  [97] 3.242098 2.727035 2.839146 3.223226 3.157536 3.039797 2.672172 3.242219
## [105] 3.151506 3.001401 2.733635 2.933059 2.937393 2.894789 2.841081 2.760499
## [113] 2.754494 2.701615 3.017321 3.298352 3.038228 2.796516 3.317218 2.914673
## [121] 3.361724 3.143495 2.772269 3.009822 2.609608 3.098954 2.950675 3.076369
## [129] 2.983256 3.191926 2.969420 3.442259 2.977584 2.744347 2.850296 2.890152
## [137] 2.815152 2.718048 3.125170 2.703799 2.855945 2.567231 3.148707 2.916793
## [145] 3.115619 2.770077 2.959347 3.118253 3.558312 3.130840 3.499280 2.841520
## [153] 3.020979 2.641074 3.300261 3.021693 2.873304 3.274427 3.111018 3.015555
## [161] 2.919806 3.241943 2.825561 3.412176 3.222882 3.165540 3.453369 2.974039
## [169] 3.000513 3.076567 3.154840 2.617969 2.865961 3.053532 3.247263 2.795903
## [177] 2.721469 2.759077 3.339878 2.969347 3.007836 2.931981 3.132595 2.968720
## [185] 2.955118 2.933089 2.756536 2.887680 2.689556 2.767238 3.128240 3.180219
## [193] 3.136656 2.815256 3.133913 2.787824 3.258399 3.405694 3.001377 2.951500
par(mfrow=c(1,2))
hist(criolla, col='pink')
abline(v=mean(criolla), col='red', lwd=3)
hist(pastusa, col ='blue')
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. 
##   2.280   2.668   2.805   2.798   2.940   3.171
summary(pastusa)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.567   2.850   3.001   3.013   3.151   3.558
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  2.8 0.19    2.8     2.8 0.2 2.28 3.17  0.89 -0.25    -0.44 0.01
psych::describe(pastusa)
##    vars   n mean   sd median trimmed  mad  min  max range skew kurtosis   se
## X1    1 200 3.01 0.21      3    3.01 0.22 2.57 3.56  0.99 0.24    -0.48 0.02

Disgresion

medA = 3.5; sdA = 0.35
medB = 3.2; sdB = 0.20
# ¿Cual seleccionar?
# Coeficiente de variación cv = 100 *  sd/mean
cvA = 100 * sdA/medA
cvB = 100 * sdB/medB

cvA; cvB
## [1] 10
## [1] 6.25
100* sd(criolla/mean(criolla))
## [1] 6.822921
100* sd(pastusa/mean(pastusa))
## [1] 7.09808

Conclusión desde el analisi descriptivo

  • Coeficiente de variacion bajos
  • Se puede omitri el problema de diferente variabilidad
  • Se selecciona la Variedad criolla por mayor rendimiento

Analisis inferencial a traves de pruebas de hipotesis

\[H_0: \mu_{pastusa} = \mu_{criolla}\\ H_a: \mu_{pastusa} \neq \mu_{criolla}\]

Prueba t-student

Modalidad 1: varianzas iguales Modalidad 2: varianzas desiguales

Prueba para comparacion de dos varianzas

\[H_0: \sigma^2_{pastusa} = \sigma^2_{criolla} \\ Ha: \ \sigma^2_{pastusa} \neq \sigma^2_{criolla}\]

var(pastusa)
## [1] 0.04574639
var(criolla)
## [1] 0.03643638
vt= var.test(pastusa, criolla)
vt$p.value
## [1] 0.1205414
ifelse(vt$p.value <0.025, 'Rechazo Ho', 'No rechazo')
## [1] "No rechazo"

Prueba t-Student para comparar las dos medias con varianzas iguales

pt = t.test(pastusa, criolla, alternative = 't',
       var.equal = TRUE)

ifelse(pt$p.value < 0.025, 'Rechazo Ho', 'No rechazo Ho')
## [1] "Rechazo Ho"

Conclusion final: los datos proporciona evidencia estadistica en contra de la hipotesis nula, es decir, que estadisticamente se consideran ambas variedades como desiguales o de diferente rendimiento.