De donde viene el Pvalor?: -> El P-valor es una probabilidad, por lo que siempre va a estar entre 0 y 1 -> Esta relacionado con inferencia estadistica ->entre mas pequeño sea el p valor mas fuerte sera la evidencia en contra de la hipotesis nula -> el p valor se obtiene a partir del contraste de hipotesis de un estadistico, por ejemplo el promedio poblacional Los pasos para obtener el p valor son: -> identificar el tipo de contraste, bilateral o unilateral ej: \[Si:\qquad H_0 : \mu_0\leq\mu_\qquad entonces:\qquad H_1:\mu>\mu_0\qquad unilateral\ por\ la\ derecha\] \[Si:\qquad H_0 : \mu\geq\mu_0\qquad entonces:\qquad H_1:\mu<\mu_0\qquad unilateral\ por\ la\ izquierda\] \[Si:\qquad H_0 : \mu=\mu_0\qquad entonces:\qquad H_1:\mu\neq\mu_0\qquad bilateral\] ->se toma el estadistico y su estimador puntal, en el coso de la media poblacional, puede ser la media muestral que sera \(\mu_0\) ->Si se tiene una distribucion normal hay varias formas de calcularlo se tendra en cuenta la siguiente forma: $ z_{p}=z{p-valor}=$, sin embargo para emplearla se debe estandarizar la distribucion nomrmal

->\(si\quad\frac{{\bar{x}}-\mu_0}{\sigma/\sqrt{n}}\leq z_{1-\alpha}\) se acepta la hipotesis

plot()

x<-seq(-6,6,0.15)
y<-dnorm(x,0,1)
plot(x,y,ylim = c(0,0.4),col="blue", type = "l",lwd=2,main = "Funcion densidad")

mean(y)
## [1] 0.08230453
mean(x)
## [1] -2.041834e-16
sd(y)
## [1] 0.1290317
x<-seq(-6,6,0.15) 
m<-replicate(2,sort.int(dnorm(x,0,1),partial = 20))
k<-replicate(3,sort.int(dnorm(x,0,1),partial = 26))
d<-c(m,k)
n<-data.frame(m,k)
n
##              X1           X2         X1.1         X2.1           X3
## 1  6.075883e-09 6.075883e-09 6.075883e-09 6.075883e-09 6.075883e-09
## 2  6.075883e-09 6.075883e-09 6.075883e-09 6.075883e-09 6.075883e-09
## 3  1.477708e-08 1.477708e-08 1.477708e-08 1.477708e-08 1.477708e-08
## 4  1.477708e-08 1.477708e-08 1.477708e-08 1.477708e-08 1.477708e-08
## 5  3.513955e-08 3.513955e-08 3.513955e-08 3.513955e-08 3.513955e-08
## 6  3.513955e-08 3.513955e-08 3.513955e-08 3.513955e-08 3.513955e-08
## 7  8.170190e-08 8.170190e-08 8.170190e-08 8.170190e-08 8.170190e-08
## 8  8.972435e-07 8.972435e-07 8.972435e-07 8.972435e-07 8.972435e-07
## 9  4.128471e-07 4.128471e-07 4.128471e-07 4.128471e-07 4.128471e-07
## 10 1.857362e-07 1.857362e-07 1.857362e-07 1.857362e-07 1.857362e-07
## 11 8.170190e-08 8.170190e-08 8.170190e-08 8.170190e-08 8.170190e-08
## 12 1.857362e-07 1.857362e-07 1.857362e-07 1.857362e-07 1.857362e-07
## 13 4.128471e-07 4.128471e-07 4.128471e-07 4.128471e-07 4.128471e-07
## 14 8.972435e-07 8.972435e-07 8.972435e-07 8.972435e-07 8.972435e-07
## 15 1.906601e-06 1.906601e-06 5.894307e-05 5.894307e-05 5.894307e-05
## 16 1.906601e-06 1.906601e-06 3.104141e-05 3.104141e-05 3.104141e-05
## 17 3.961299e-06 3.961299e-06 1.598374e-05 1.598374e-05 1.598374e-05
## 18 8.047182e-06 8.047182e-06 8.047182e-06 8.047182e-06 8.047182e-06
## 19 3.961299e-06 3.961299e-06 3.961299e-06 3.961299e-06 3.961299e-06
## 20 8.047182e-06 8.047182e-06 1.906601e-06 1.906601e-06 1.906601e-06
## 21 1.598374e-05 1.598374e-05 1.906601e-06 1.906601e-06 1.906601e-06
## 22 1.598374e-05 1.598374e-05 3.961299e-06 3.961299e-06 3.961299e-06
## 23 3.104141e-05 3.104141e-05 8.047182e-06 8.047182e-06 8.047182e-06
## 24 5.894307e-05 5.894307e-05 1.598374e-05 1.598374e-05 1.598374e-05
## 25 3.104141e-05 3.104141e-05 3.104141e-05 3.104141e-05 3.104141e-05
## 26 5.894307e-05 5.894307e-05 5.894307e-05 5.894307e-05 5.894307e-05
## 27 1.094340e-04 1.094340e-04 1.094340e-04 1.094340e-04 1.094340e-04
## 28 1.722569e-03 1.722569e-03 2.239453e-02 2.239453e-02 2.239453e-02
## 29 1.038281e-03 1.038281e-03 1.544935e-02 1.544935e-02 1.544935e-02
## 30 6.119019e-04 6.119019e-04 1.042093e-02 1.042093e-02 1.042093e-02
## 31 3.525957e-04 3.525957e-04 6.872767e-03 6.872767e-03 6.872767e-03
## 32 1.986555e-04 1.986555e-04 4.431848e-03 4.431848e-03 4.431848e-03
## 33 1.094340e-04 1.094340e-04 2.794258e-03 2.794258e-03 2.794258e-03
## 34 1.986555e-04 1.986555e-04 1.722569e-03 1.722569e-03 1.722569e-03
## 35 3.525957e-04 3.525957e-04 1.038281e-03 1.038281e-03 1.038281e-03
## 36 6.119019e-04 6.119019e-04 6.119019e-04 6.119019e-04 6.119019e-04
## 37 1.038281e-03 1.038281e-03 3.525957e-04 3.525957e-04 3.525957e-04
## 38 1.722569e-03 1.722569e-03 1.986555e-04 1.986555e-04 1.986555e-04
## 39 2.794258e-03 2.794258e-03 1.094340e-04 1.094340e-04 1.094340e-04
## 40 3.944793e-01 3.944793e-01 1.986555e-04 1.986555e-04 1.986555e-04
## 41 3.989423e-01 3.989423e-01 3.525957e-04 3.525957e-04 3.525957e-04
## 42 3.944793e-01 3.944793e-01 6.119019e-04 6.119019e-04 6.119019e-04
## 43 3.813878e-01 3.813878e-01 1.038281e-03 1.038281e-03 1.038281e-03
## 44 3.605270e-01 3.605270e-01 1.722569e-03 1.722569e-03 1.722569e-03
## 45 3.332246e-01 3.332246e-01 2.794258e-03 2.794258e-03 2.794258e-03
## 46 3.011374e-01 3.011374e-01 4.431848e-03 4.431848e-03 4.431848e-03
## 47 2.660852e-01 2.660852e-01 6.872767e-03 6.872767e-03 6.872767e-03
## 48 2.298821e-01 2.298821e-01 1.042093e-02 1.042093e-02 1.042093e-02
## 49 1.941861e-01 1.941861e-01 1.544935e-02 1.544935e-02 1.544935e-02
## 50 1.603833e-01 1.603833e-01 2.239453e-02 2.239453e-02 2.239453e-02
## 51 1.295176e-01 1.295176e-01 3.173965e-02 3.173965e-02 3.173965e-02
## 52 1.022649e-01 1.022649e-01 1.022649e-01 1.022649e-01 1.022649e-01
## 53 7.895016e-02 7.895016e-02 7.895016e-02 7.895016e-02 7.895016e-02
## 54 5.959471e-02 5.959471e-02 5.959471e-02 5.959471e-02 5.959471e-02
## 55 4.398360e-02 4.398360e-02 4.398360e-02 4.398360e-02 4.398360e-02
## 56 3.173965e-02 3.173965e-02 1.295176e-01 1.295176e-01 1.295176e-01
## 57 2.239453e-02 2.239453e-02 1.603833e-01 1.603833e-01 1.603833e-01
## 58 1.544935e-02 1.544935e-02 1.941861e-01 1.941861e-01 1.941861e-01
## 59 1.042093e-02 1.042093e-02 2.298821e-01 2.298821e-01 2.298821e-01
## 60 6.872767e-03 6.872767e-03 2.660852e-01 2.660852e-01 2.660852e-01
## 61 4.431848e-03 4.431848e-03 3.011374e-01 3.011374e-01 3.011374e-01
## 62 3.813878e-01 3.813878e-01 3.332246e-01 3.332246e-01 3.332246e-01
## 63 3.605270e-01 3.605270e-01 3.605270e-01 3.605270e-01 3.605270e-01
## 64 3.332246e-01 3.332246e-01 3.813878e-01 3.813878e-01 3.813878e-01
## 65 3.011374e-01 3.011374e-01 3.944793e-01 3.944793e-01 3.944793e-01
## 66 2.660852e-01 2.660852e-01 3.989423e-01 3.989423e-01 3.989423e-01
## 67 2.298821e-01 2.298821e-01 3.944793e-01 3.944793e-01 3.944793e-01
## 68 1.941861e-01 1.941861e-01 3.813878e-01 3.813878e-01 3.813878e-01
## 69 1.603833e-01 1.603833e-01 3.605270e-01 3.605270e-01 3.605270e-01
## 70 1.295176e-01 1.295176e-01 3.332246e-01 3.332246e-01 3.332246e-01
## 71 1.022649e-01 1.022649e-01 3.011374e-01 3.011374e-01 3.011374e-01
## 72 7.895016e-02 7.895016e-02 2.660852e-01 2.660852e-01 2.660852e-01
## 73 5.959471e-02 5.959471e-02 2.298821e-01 2.298821e-01 2.298821e-01
## 74 4.398360e-02 4.398360e-02 1.941861e-01 1.941861e-01 1.941861e-01
## 75 3.173965e-02 3.173965e-02 1.603833e-01 1.603833e-01 1.603833e-01
## 76 2.239453e-02 2.239453e-02 1.295176e-01 1.295176e-01 1.295176e-01
## 77 1.544935e-02 1.544935e-02 1.022649e-01 1.022649e-01 1.022649e-01
## 78 1.042093e-02 1.042093e-02 7.895016e-02 7.895016e-02 7.895016e-02
## 79 6.872767e-03 6.872767e-03 5.959471e-02 5.959471e-02 5.959471e-02
## 80 4.431848e-03 4.431848e-03 4.398360e-02 4.398360e-02 4.398360e-02
## 81 2.794258e-03 2.794258e-03 3.173965e-02 3.173965e-02 3.173965e-02
plot(x,n$X1,ylim = c(0,0.4),col="blue", type = "l",lwd=2,main = "Funcion densidad 1")

plot(x,n$X2,ylim = c(0,0.4),col="blue", type = "l",lwd=2,main = "Funcion densidad 2")

plot(x,n$X1.1,ylim = c(0,0.4),col="blue", type = "l",lwd=2,main = "Funcion densidad 3")

plot(x,n$X2.1,ylim = c(0,0.4),col="blue", type = "l",lwd=2,main = "Funcion densidad 4")

plot(x,n$X3,ylim = c(0,0.4),col="blue", type = "l",lwd=2,main = "Funcion densidad 5")

d
##   [1] 6.075883e-09 6.075883e-09 1.477708e-08 1.477708e-08 3.513955e-08
##   [6] 3.513955e-08 8.170190e-08 8.972435e-07 4.128471e-07 1.857362e-07
##  [11] 8.170190e-08 1.857362e-07 4.128471e-07 8.972435e-07 1.906601e-06
##  [16] 1.906601e-06 3.961299e-06 8.047182e-06 3.961299e-06 8.047182e-06
##  [21] 1.598374e-05 1.598374e-05 3.104141e-05 5.894307e-05 3.104141e-05
##  [26] 5.894307e-05 1.094340e-04 1.722569e-03 1.038281e-03 6.119019e-04
##  [31] 3.525957e-04 1.986555e-04 1.094340e-04 1.986555e-04 3.525957e-04
##  [36] 6.119019e-04 1.038281e-03 1.722569e-03 2.794258e-03 3.944793e-01
##  [41] 3.989423e-01 3.944793e-01 3.813878e-01 3.605270e-01 3.332246e-01
##  [46] 3.011374e-01 2.660852e-01 2.298821e-01 1.941861e-01 1.603833e-01
##  [51] 1.295176e-01 1.022649e-01 7.895016e-02 5.959471e-02 4.398360e-02
##  [56] 3.173965e-02 2.239453e-02 1.544935e-02 1.042093e-02 6.872767e-03
##  [61] 4.431848e-03 3.813878e-01 3.605270e-01 3.332246e-01 3.011374e-01
##  [66] 2.660852e-01 2.298821e-01 1.941861e-01 1.603833e-01 1.295176e-01
##  [71] 1.022649e-01 7.895016e-02 5.959471e-02 4.398360e-02 3.173965e-02
##  [76] 2.239453e-02 1.544935e-02 1.042093e-02 6.872767e-03 4.431848e-03
##  [81] 2.794258e-03 6.075883e-09 6.075883e-09 1.477708e-08 1.477708e-08
##  [86] 3.513955e-08 3.513955e-08 8.170190e-08 8.972435e-07 4.128471e-07
##  [91] 1.857362e-07 8.170190e-08 1.857362e-07 4.128471e-07 8.972435e-07
##  [96] 1.906601e-06 1.906601e-06 3.961299e-06 8.047182e-06 3.961299e-06
## [101] 8.047182e-06 1.598374e-05 1.598374e-05 3.104141e-05 5.894307e-05
## [106] 3.104141e-05 5.894307e-05 1.094340e-04 1.722569e-03 1.038281e-03
## [111] 6.119019e-04 3.525957e-04 1.986555e-04 1.094340e-04 1.986555e-04
## [116] 3.525957e-04 6.119019e-04 1.038281e-03 1.722569e-03 2.794258e-03
## [121] 3.944793e-01 3.989423e-01 3.944793e-01 3.813878e-01 3.605270e-01
## [126] 3.332246e-01 3.011374e-01 2.660852e-01 2.298821e-01 1.941861e-01
## [131] 1.603833e-01 1.295176e-01 1.022649e-01 7.895016e-02 5.959471e-02
## [136] 4.398360e-02 3.173965e-02 2.239453e-02 1.544935e-02 1.042093e-02
## [141] 6.872767e-03 4.431848e-03 3.813878e-01 3.605270e-01 3.332246e-01
## [146] 3.011374e-01 2.660852e-01 2.298821e-01 1.941861e-01 1.603833e-01
## [151] 1.295176e-01 1.022649e-01 7.895016e-02 5.959471e-02 4.398360e-02
## [156] 3.173965e-02 2.239453e-02 1.544935e-02 1.042093e-02 6.872767e-03
## [161] 4.431848e-03 2.794258e-03 6.075883e-09 6.075883e-09 1.477708e-08
## [166] 1.477708e-08 3.513955e-08 3.513955e-08 8.170190e-08 8.972435e-07
## [171] 4.128471e-07 1.857362e-07 8.170190e-08 1.857362e-07 4.128471e-07
## [176] 8.972435e-07 5.894307e-05 3.104141e-05 1.598374e-05 8.047182e-06
## [181] 3.961299e-06 1.906601e-06 1.906601e-06 3.961299e-06 8.047182e-06
## [186] 1.598374e-05 3.104141e-05 5.894307e-05 1.094340e-04 2.239453e-02
## [191] 1.544935e-02 1.042093e-02 6.872767e-03 4.431848e-03 2.794258e-03
## [196] 1.722569e-03 1.038281e-03 6.119019e-04 3.525957e-04 1.986555e-04
## [201] 1.094340e-04 1.986555e-04 3.525957e-04 6.119019e-04 1.038281e-03
## [206] 1.722569e-03 2.794258e-03 4.431848e-03 6.872767e-03 1.042093e-02
## [211] 1.544935e-02 2.239453e-02 3.173965e-02 1.022649e-01 7.895016e-02
## [216] 5.959471e-02 4.398360e-02 1.295176e-01 1.603833e-01 1.941861e-01
## [221] 2.298821e-01 2.660852e-01 3.011374e-01 3.332246e-01 3.605270e-01
## [226] 3.813878e-01 3.944793e-01 3.989423e-01 3.944793e-01 3.813878e-01
## [231] 3.605270e-01 3.332246e-01 3.011374e-01 2.660852e-01 2.298821e-01
## [236] 1.941861e-01 1.603833e-01 1.295176e-01 1.022649e-01 7.895016e-02
## [241] 5.959471e-02 4.398360e-02 3.173965e-02 6.075883e-09 6.075883e-09
## [246] 1.477708e-08 1.477708e-08 3.513955e-08 3.513955e-08 8.170190e-08
## [251] 8.972435e-07 4.128471e-07 1.857362e-07 8.170190e-08 1.857362e-07
## [256] 4.128471e-07 8.972435e-07 5.894307e-05 3.104141e-05 1.598374e-05
## [261] 8.047182e-06 3.961299e-06 1.906601e-06 1.906601e-06 3.961299e-06
## [266] 8.047182e-06 1.598374e-05 3.104141e-05 5.894307e-05 1.094340e-04
## [271] 2.239453e-02 1.544935e-02 1.042093e-02 6.872767e-03 4.431848e-03
## [276] 2.794258e-03 1.722569e-03 1.038281e-03 6.119019e-04 3.525957e-04
## [281] 1.986555e-04 1.094340e-04 1.986555e-04 3.525957e-04 6.119019e-04
## [286] 1.038281e-03 1.722569e-03 2.794258e-03 4.431848e-03 6.872767e-03
## [291] 1.042093e-02 1.544935e-02 2.239453e-02 3.173965e-02 1.022649e-01
## [296] 7.895016e-02 5.959471e-02 4.398360e-02 1.295176e-01 1.603833e-01
## [301] 1.941861e-01 2.298821e-01 2.660852e-01 3.011374e-01 3.332246e-01
## [306] 3.605270e-01 3.813878e-01 3.944793e-01 3.989423e-01 3.944793e-01
## [311] 3.813878e-01 3.605270e-01 3.332246e-01 3.011374e-01 2.660852e-01
## [316] 2.298821e-01 1.941861e-01 1.603833e-01 1.295176e-01 1.022649e-01
## [321] 7.895016e-02 5.959471e-02 4.398360e-02 3.173965e-02 6.075883e-09
## [326] 6.075883e-09 1.477708e-08 1.477708e-08 3.513955e-08 3.513955e-08
## [331] 8.170190e-08 8.972435e-07 4.128471e-07 1.857362e-07 8.170190e-08
## [336] 1.857362e-07 4.128471e-07 8.972435e-07 5.894307e-05 3.104141e-05
## [341] 1.598374e-05 8.047182e-06 3.961299e-06 1.906601e-06 1.906601e-06
## [346] 3.961299e-06 8.047182e-06 1.598374e-05 3.104141e-05 5.894307e-05
## [351] 1.094340e-04 2.239453e-02 1.544935e-02 1.042093e-02 6.872767e-03
## [356] 4.431848e-03 2.794258e-03 1.722569e-03 1.038281e-03 6.119019e-04
## [361] 3.525957e-04 1.986555e-04 1.094340e-04 1.986555e-04 3.525957e-04
## [366] 6.119019e-04 1.038281e-03 1.722569e-03 2.794258e-03 4.431848e-03
## [371] 6.872767e-03 1.042093e-02 1.544935e-02 2.239453e-02 3.173965e-02
## [376] 1.022649e-01 7.895016e-02 5.959471e-02 4.398360e-02 1.295176e-01
## [381] 1.603833e-01 1.941861e-01 2.298821e-01 2.660852e-01 3.011374e-01
## [386] 3.332246e-01 3.605270e-01 3.813878e-01 3.944793e-01 3.989423e-01
## [391] 3.944793e-01 3.813878e-01 3.605270e-01 3.332246e-01 3.011374e-01
## [396] 2.660852e-01 2.298821e-01 1.941861e-01 1.603833e-01 1.295176e-01
## [401] 1.022649e-01 7.895016e-02 5.959471e-02 4.398360e-02 3.173965e-02
mean(d)
## [1] 0.08230453
t.test(x,d)
## 
##  Welch Two Sample t-test
## 
## data:  x and d
## t = -0.20987, df = 80.042, p-value = 0.8343
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.8627246  0.6981155
## sample estimates:
##     mean of x     mean of y 
## -2.041834e-16  8.230453e-02
mean(m)
## [1] 0.08230453

\[ z_{\alpha_p}=z_{p-valor}=\frac{{\bar{x}}-\mu_0}{\sigma/\sqrt{n}}\]

en este caso \(z_p=0\) lo cual significa que el valor P sera 0.5 esto indica que la hipotesis nula se acepta, ya que el P valor es menor que \(z_{1-\alpha}\)