PRUEBAS DE HIPÓTESIS

Promedios

Pruebas paramétricas: Media

Datos: 1 muestra

Condiciones de validez: n > 30 o normalidad (Prueba de normalidad de Shapiro-Wilk y gráfico de normalidad)

Función R: t.test(x,…)

  • Ejemplo de uso:

\(\mu\) = media poblacional

\(\bar{x}\) = media muestral

\[\bar{x} \rightarrow \mu\]

set.seed(2022)
rto_papa = rnorm(120, 2.9, 0.3)
hist(rto_papa,
     col = "#E1DEFC",    
     border = "#9A68A4",
     xlab = "Rendimiento papa",
     xlim=c(2,4))
abline(v=min(rto_papa),lwd = 2, col="#836FFF")
abline(v=mean(rto_papa), lwd = 2, col="#FF82AB")

mean(rto_papa)
## [1] 2.929819
sd(rto_papa)
## [1] 0.3010215
  • Comparando la media teórica y un valor de referencia (caso con una muestra)

\[H_0: \mu_{rto} \geq 3.0\\ H_a: \mu_{rto} < 3.0\]

Prueba T-Student para la media de una poblacion normal

sim_1 = replicate(500, rnorm(120, 2.9, 0.3))
dim(sim_1)
## [1] 120 500
medias = colMeans(sim_1) ## Promedios de las medias
medias
##   [1] 2.852480 2.857523 2.904095 2.921658 2.864966 2.943554 2.911181 2.944627
##   [9] 2.878933 2.899643 2.895902 2.900049 2.910886 2.903798 2.880202 2.907759
##  [17] 2.885118 2.906294 2.910229 2.949471 2.908284 2.888517 2.895148 2.827914
##  [25] 2.913895 2.876842 2.930488 2.923620 2.906003 2.872278 2.867354 2.883575
##  [33] 2.914215 2.924412 2.890405 2.905573 2.918566 2.898858 2.912550 2.866506
##  [41] 2.884521 2.881346 2.896736 2.909127 2.911756 2.868764 2.925057 2.879869
##  [49] 2.870543 2.902831 2.914550 2.937862 2.899686 2.918003 2.907354 2.894508
##  [57] 2.883466 2.874419 2.886930 2.879746 2.897005 2.906682 2.898795 2.898519
##  [65] 2.816395 2.856758 2.932404 2.913079 2.911210 2.885818 2.890381 2.876894
##  [73] 2.880806 2.894218 2.904016 2.893657 2.942440 2.875229 2.904400 2.864862
##  [81] 2.925146 2.909802 2.883389 2.920407 2.942851 2.866790 2.918461 2.862322
##  [89] 2.884404 2.913945 2.885395 2.922205 2.923837 2.869049 2.897655 2.934710
##  [97] 2.879479 2.872388 2.927403 2.864753 2.936206 2.945424 2.895500 2.905121
## [105] 2.919928 2.877194 2.893788 2.895334 2.848405 2.897830 2.894915 2.929670
## [113] 2.889799 2.862391 2.877606 2.932281 2.904594 2.911256 2.939702 2.948365
## [121] 2.835220 2.936156 2.884025 2.953937 2.923742 2.919344 2.882945 2.885082
## [129] 2.876918 2.918100 2.909953 2.895281 2.911298 2.926764 2.894166 2.895440
## [137] 2.935268 2.899017 2.937318 2.921524 2.899348 2.889689 2.909886 2.861539
## [145] 2.948967 2.889884 2.943783 2.888699 2.854646 2.892321 2.909197 2.939792
## [153] 2.936188 2.900887 2.907319 2.924544 2.943234 2.935228 2.896769 2.912620
## [161] 2.882580 2.917197 2.895597 2.890606 2.881413 2.863008 2.924727 2.881922
## [169] 2.900400 2.880862 2.868187 2.900614 2.871461 2.854973 2.907477 2.910641
## [177] 2.871905 2.867311 2.878251 2.936047 2.841734 2.913127 2.878331 2.830723
## [185] 2.896469 2.872888 2.886741 2.911736 2.903250 2.853649 2.927536 2.877286
## [193] 2.917046 2.933947 2.900607 2.943538 2.914673 2.881779 2.897655 2.883398
## [201] 2.928735 2.901650 2.906433 2.887902 2.914377 2.907057 2.956146 2.873832
## [209] 2.874872 2.888678 2.916236 2.910369 2.920265 2.915659 2.893687 2.878069
## [217] 2.891586 2.906256 2.966403 2.921084 2.880782 2.882675 2.932350 2.867034
## [225] 2.898326 2.869459 2.900260 2.851934 2.890706 2.912990 2.872454 2.883941
## [233] 2.921296 2.906794 2.871714 2.936133 2.902514 2.891433 2.929723 2.922259
## [241] 2.951124 2.894447 2.883053 2.917805 2.908285 2.893316 2.879611 2.903665
## [249] 2.897425 2.867346 2.909961 2.836206 2.843852 2.877720 2.921922 2.908087
## [257] 2.935840 2.909089 2.966292 2.884979 2.942184 2.882814 2.872483 2.945426
## [265] 2.929858 2.921986 2.858392 2.927292 2.936179 2.856633 2.871021 2.889542
## [273] 2.874546 2.911568 2.905747 2.905014 2.903861 2.923117 2.901543 2.901019
## [281] 2.913517 2.944741 2.899110 2.899389 2.887735 2.873798 2.893698 2.855934
## [289] 2.923801 2.921801 2.899039 2.937112 2.910212 2.933821 2.878721 2.948950
## [297] 2.927457 2.878440 2.847878 2.932575 2.902528 2.907058 2.953879 2.890406
## [305] 2.927058 2.888837 2.871754 2.868870 2.919161 2.850971 2.885664 2.923285
## [313] 2.905634 2.901425 2.906601 2.869240 2.915024 2.921379 2.924289 2.892344
## [321] 2.875242 2.907050 2.897527 2.987473 2.899005 2.911470 2.841704 2.915077
## [329] 2.895639 2.910427 2.889575 2.920000 2.928647 2.905341 2.885459 2.949587
## [337] 2.903399 2.909018 2.897827 2.876034 2.894836 2.889320 2.924484 2.889441
## [345] 2.899430 2.897967 2.910452 2.899798 2.836443 2.917758 2.916671 2.927431
## [353] 2.904793 2.867527 2.947801 2.927011 2.943314 2.909744 2.907896 2.890911
## [361] 2.910140 2.925913 2.942454 2.891149 2.921853 2.919686 2.930835 2.886547
## [369] 2.926031 2.866534 2.886476 2.854279 2.911808 2.914477 2.923638 2.886979
## [377] 2.921350 2.866589 2.853420 2.912991 2.914668 2.877124 2.899442 2.877254
## [385] 2.885901 2.862678 2.921020 2.867310 2.912812 2.893064 2.895749 2.940522
## [393] 2.913567 2.910378 2.908333 2.898107 2.878412 2.904853 2.892350 2.917968
## [401] 2.884361 2.904970 2.863536 2.870063 2.948900 2.900544 2.916557 2.905823
## [409] 2.914844 2.876269 2.841158 2.917304 2.911828 2.917491 2.896539 2.941732
## [417] 2.904888 2.896836 2.909131 2.896842 2.926281 2.864193 2.919254 2.875479
## [425] 2.924454 2.977812 2.955960 2.932428 2.842169 2.917329 2.866623 2.891842
## [433] 2.941860 2.917550 2.899950 2.918749 2.886522 2.910431 2.905279 2.910915
## [441] 2.938131 2.868387 2.942029 2.882510 2.918655 2.912828 2.906489 2.862331
## [449] 2.863772 2.933772 2.907670 2.902163 2.889998 2.875315 2.867967 2.915868
## [457] 2.852924 2.884017 2.918475 2.922588 2.958778 2.928451 2.917560 2.901759
## [465] 2.867315 2.875840 2.900332 2.926154 2.858232 2.900007 2.938942 2.877599
## [473] 2.951922 2.919469 2.915630 2.957336 2.900827 2.939340 2.915469 2.894189
## [481] 2.904332 2.866078 2.865022 2.906893 2.921726 2.932659 2.898934 2.838566
## [489] 2.946098 2.921276 2.923397 2.869462 2.897681 2.878696 2.874605 2.876193
## [497] 2.839582 2.881209 2.897439 2.930626
mean(medias)
## [1] 2.901196
hist(medias,
     col = "#E1DEFC",    
     border = "#9A68A4")

# Agrgar curva normal  
x <- medias
h<-hist(x, breaks=10, col="#E1DEFC",border = "#9A68A4", xlab="Medias",
   main="Histogram with Normal Curve")
xfit<-seq(min(x),max(x),length=40)
yfit<-dnorm(xfit,mean=mean(x),sd=sd(x))
yfit <- yfit*diff(h$mids[1:2])*length(x)
lines(xfit, yfit, col="#836FFF", lwd=2)
abline(v=mean(rto_papa), col = "#FF82AB")
abline(v=3, col="#CDB5CD")

library(rcompanion)

plotDensityHistogram(medias,adjust = 1,col="#E1DEFC",linecol = "#9A68A4" )
abline(v=mean(rto_papa), col = "#836FFF")
abline(v=3, col="#FF82AB")

#abline(v=x_c, col = "yellow")

Cuantile de la distribución T-Student

Nivel de confianza \(1-\alpha\)

Nivel de significación \(\alpha\)

cuantile_t = qt(p = 0.05,df = 119, lower.tail = TRUE)
cuantile_t
## [1] -1.657759

\[t_{cal}=\frac{\bar{x}-\mu}{s/\sqrt{n}}\]

Donde:

\(\bar{x}\) es la media de la muestra

\(n\) es el tamaño de la muestra

\(s\) es la desviación típica de la muestra con \(n-1\) grados de libertad

\(\mu\) es el valor teórico

  • Si T calculado es igual al T de la tabla (cuantile T), en este punto \(\alpha\) vale 0.05

  • Se puede despejar la media de la muestra (\(\bar{x}\)), para saber hasta donde me puedo “bajar”

\[t_{tab}{s/\sqrt{n}} + \mu=\bar{x}\] \[\bar{x}_{media~crítica}=t_{tab}~{s/\sqrt{n}} + \mu\]

cuantile_t = qt(p = 0.05,df = 499, lower.tail = T);cuantile_t
## [1] -1.647913
x_c = (cuantile_t * sd(rto_papa))/sqrt(120)+ 3;x_c
## [1] 2.954716
ifelse(x_c>1.647913, 'No rechazo Ho', 'Rechazo Ho')
## [1] "No rechazo Ho"