see also types of error

see first sampling distribution and z-test

…

> rm(list=ls())
> 
> rnorm2 <- function(n,mean,sd){ 
+   mean+sd*scale(rnorm(n)) 
+ }
> 
> n.p <- 100000
> m.p <- 100
> sd.p <- 10
> p1 <- rnorm2(n.p, m.p, sd.p)
> m.p1 <- mean(p1)
> sd.p1 <- sd(p1)
> 
> p2 <- rnorm2(n.p, m.p+4, sd.p)
> m.p2 <- mean(p2)
> sd.p2 <- sd(p2)
> 
> n.s <- 25
> se.z1 <- c(sqrt(var(p1)/n.s))
> se.z2 <- c(sqrt(var(p2)/n.s))
> se.z1
[1] 2
> se.z2
[1] 2
> 
> x.p1 <- seq(mean(p1)-5*se.z1, 
+             mean(p2)+5*se.z1, 
+             length.out = 500)
> x.p2 <- seq(mean(p2)-5*se.z1, 
+             mean(p2)+5*se.z1, 
+             length.out = 500)
> 
> # Calculate the probability 
> # density for a normal distribution
> y.p1 <- dnorm(x.p1, mean(p1), se.z1)
> y.p2 <- dnorm(x.p2, mean(p2), se.z2)
> 
> # Plot the theoretical PDF
> plot(x.p1, y.p1, type = "l", 
+      lwd=3, 
+      main = paste0("sample (n = ",n.s, ") means from p1 and p2 (unknown)"),
+      xlab = "Value", ylab = "Density")
> lines(x.p2, y.p2, lty=2, lwd=3)
> 
> 
> m.p1 <- mean(p1)
> se1 <- c(m.p1-se.z1, m.p1+se.z1)
> se2 <- c(m.p1-2*se.z1, m.p1+2*se.z1)
> se3 <- c(m.p1-3*se.z1, m.p1+3*se.z1)
> abline(v=c(m.p1,se1,se2,se3), 
+        col=c('black', 'orange', 'orange', 
+              'green', 'green', 
+              'blue', 'blue'), 
+        lwd=1)
> 
> treated.s <- sample(p2, n.s)
> m.treated.s <- mean(treated.s)
> abline(v=m.treated.s, col='red', lwd=2)
> 
> diff <- m.treated.s-mean(p1)
> diff/se.z1
[1] 0.5253907
> zscore <- abs(diff/se.z1)
> pnorm(zscore, lower.tail = F)*2
[1] 0.5993116
> tscore <- zscore
> pt(tscore, df=n.s-1, lower.tail = F)*2
[1] 0.6041323
> 
> # usual way - using sample's variance 
> # instead of p1's variance to get
> # standard error value
> se.s <- sqrt(var(treated.s)/n.s)
> se.s
[1] 1.83652
> tscore <- diff/se.s
> tscore
[1] 0.572159
> 
> 
> se1 <- c(m.p1-se.s, m.p1+se.s)
> se2 <- c(m.p1-2*se.s, m.p1+2*se.s)
> se3 <- c(m.p1-3*se.s, m.p1+3*se.s)
> abline(v=c(se1,se2,se3), 
+        col=c('darkorange', 'darkorange',
+              'darkgreen', 'darkgreen', 
+              'darkblue', 'darkblue'), 
+        lwd=2)
> 
> 
> 
> 
> plot(x.p1, y.p1, type = "l", 
+      lwd=3, 
+      main = "Sample means from p1 and p2 (imaginary)",
+      xlab = "Value", ylab = "Density")
> lines(x.p2, y.p2, lty=2, lwd=3)
> 
> 
> m.p1 <- mean(p1)
> se1 <- c(m.p1-se.s, m.p1+se.s)
> se2 <- c(m.p1-2*se.s, m.p1+2*se.s)
> se3 <- c(m.p1-3*se.s, m.p1+3*se.s)
> abline(v=c(m.p1,se1,se2,se3), 
+        col=c('black', 'darkorange', 'darkorange',
+              'darkgreen', 'darkgreen', 
+              'darkblue', 'darkblue'), 
+        lwd=2)
> abline(v=m.treated.s, col='red', lwd=3)
> se.s
[1] 1.83652
> se.z1
[1] 2
> 
> c(m.treated.s-2*se.s, m.treated.s+2*se.s)
[1]  97.37774 104.72382
> c <- qt(0.975, n.s-1)
> c
[1] 2.063899
> c(m.treated.s-c*se.s, m.treated.s+c*se.s)
[1]  97.26039 104.84117
> m.p2
[1] 104
> 
> x.p.est <- seq(m.treated.s-5*se.s, 
+                m.treated.s+5*se.z1, 
+                length.out = 500)
> y.p.est <- dnorm(x.p.est, m.treated.s, se.s)
> 
> plot(x.p.est, y.p.est, type = "l", 
+      lwd=3, 
+      main = paste0("mu (", m.p2, ") estimation \n", "from a sample ", "mean=", round(m.treated.s,2)),
+      xlab = paste0("Value"), ylab = "Density")
> se1 <- c(m.treated.s-se.s, m.treated.s+se.s)
> se2 <- c(m.treated.s-2*se.s, m.treated.s+2*se.s)
> se3 <- c(m.treated.s-3*se.s, m.treated.s+3*se.s)
> abline(v=c(m.treated.s,se1,se2,se3), 
+        col=c('black', 'darkorange', 'darkorange',
+              'darkgreen', 'darkgreen', 
+              'darkblue', 'darkblue'), 
+        lwd=2)
> 
> tscore <- abs(diff/se.s)
> pt(tscore, df=n.s-1, lower.tail = F) * 2
[1] 0.5725352
> t.test(treated.s, mu=m.p1, var.equal = T)

	One Sample t-test

data:  treated.s
t = 0.57216, df = 24, p-value = 0.5725
alternative hypothesis: true mean is not equal to 100
95 percent confidence interval:
  97.26039 104.84117
sample estimates:
mean of x 
 101.0508 

> 

> 
> 
> rm(list=ls())
> 
> rnorm2 <- function(n,mean,sd){ 
+   mean+sd*scale(rnorm(n)) 
+ }
> 
> n.p <- 10000
> m.p <- 100
> sd.p <- 10
> p1 <- rnorm2(n.p, m.p, sd.p)
> m.p1 <- mean(p1)
> sd.p1 <- sd(p1)
> 
> p2 <- rnorm2(n.p, m.p+10, sd.p)
> m.p2 <- mean(p2)
> sd.p2 <- sd(p2)
> 
> n.s <- 100
> se.z1 <- c(sqrt(var(p1)/n.s))
> se.z2 <- c(sqrt(var(p2)/n.s))
> 
> x.p1 <- seq(mean(p1)-5*se.z1, 
+                 mean(p2)+5*se.z1, 
+                 length.out = 500)
> x.p2 <- seq(mean(p2)-5*se.z1, 
+             mean(p2)+5*se.z1, 
+             length.out = 500)
> 
> # Calculate the probability 
> # density for a normal distribution
> y.p1 <- dnorm(x.p1, mean(p1), se.z1)
> y.p2 <- dnorm(x.p2, mean(p2), se.z2)
> 
> # Plot the theoretical PDF
> plot(x.p1, y.p1, type = "l", 
+      lwd=3, 
+      main = "Sample means from p1 and p2 (imaginary)",
+      xlab = "Value", ylab = "Density")
> lines(x.p2, y.p2, lty=2, lwd=3)
> 
> 
> m.p1 <- mean(p1)
> se1 <- c(m.p1-se.z1, m.p1+se.z1)
> se2 <- c(m.p1-2*se.z1, m.p1+2*se.z1)
> se3 <- c(m.p1-3*se.z1, m.p1+3*se.z1)
> abline(v=c(m.p1,se1,se2,se3), 
+        col=c('black', 'orange', 'orange', 
+              'green', 'green', 
+              'blue', 'blue'), 
+        lwd=1)
> 
> treated.s <- sample(p2, n.s)
> m.treated.s <- mean(treated.s)
> abline(v=m.treated.s, col='red', lwd=2)
> 

> se.z1
[1] 1
> 
> diff <- m.treated.s-mean(p1)
> diff/se.z1
[1] 9.057418
> 
> # usual way - using sample's variance 
> # instead of p1's variance to get
> # standard error value
> se.s <- sqrt(var(treated.s)/n.s)
> se.s
[1] 1.015243
> diff/se.s
[1] 8.921425
> 
> pt(diff/se.s, df=n.s-1, lower.tail = F) * 2
[1] 2.455388e-14
> t.test(treated.s, mu=m.p1, var.equal = T)

	One Sample t-test

data:  treated.s
t = 8.9214, df = 99, p-value = 2.455e-14
alternative hypothesis: true mean is not equal to 100
95 percent confidence interval:
 107.0430 111.0719
sample estimates:
mean of x 
 109.0574 

> 

n.ajstu <- 100000
mean.ajstu <- 100
sd.ajstu <- 10

set.seed(1024)
ajstu <- rnorm2(n.ajstu, mean=mean.ajstu, sd=sd.ajstu)

mean(ajstu)
sd(ajstu)
var(ajstu)

iter <- 10000 # # of sampling 

n.4 <- 4
means4 <- rep (NA, iter)
for(i in 1:iter){
  means4[i] = mean(sample(ajstu, n.4))
}

n.25 <- 25
means25 <- rep (NA, iter)
for(i in 1:iter){
  means25[i] = mean(sample(ajstu, n.25))
}

n.100 <- 100
means100 <- rep (NA, iter)
for(i in 1:iter){
  means100[i] = mean(sample(ajstu, n.100))
}

n.400 <- 400
means400 <- rep (NA, iter)
for(i in 1:iter){
  means400[i] = mean(sample(ajstu, n.400))
}

n.900 <- 900
means900 <- rep (NA, iter)
for(i in 1:iter){
  means900[i] = mean(sample(ajstu, n.900))
}

n.1600 <- 1600
means1600 <- rep (NA, iter)
for(i in 1:iter){
  means1600[i] = mean(sample(ajstu, n.1600))
}

n.2500 <- 2500
means2500 <- rep (NA, iter)
for(i in 1:iter){
  means2500[i] = mean(sample(ajstu, n.2500))
}

h4 <- hist(means4)
h25 <- hist(means25)
h100 <- hist(means100)
h400 <- hist(means400)
h900 <- hist(means900)
h1600 <- hist(means1600)
h2500 <- hist(means2500)


plot(h4, ylim=c(0,3000), col="red")
plot(h25, add = T, col="blue")
plot(h100, add = T, col="green")
plot(h400, add = T, col="grey")
plot(h900, add = T, col="yellow")

m4 <- mean(means4)
m25 <- mean(means25)
m100 <- mean(means100)
m400 <- mean(means400)
m900 <- mean(means900)
m1600 <- mean(means1600)
m2500 <- mean(means2500)

s4 <- sd(means4)
s25 <- sd(means25)
s100 <- sd(means100)
s400 <- sd(means400)
s900 <- sd(means900)
s1600 <- sd(means1600)
s2500 <- sd(means2500)

sss <- c(4,25,100,400,900,1600,2500) # sss sample sizes
means <- c(m4, m25, m100, m400, m900, m1600, m2500)
sds <- c(s4, s25, s100, s400, s900, s1600, s2500)

temp <- data.frame(sss,
                   means,
                   sds)

temp

ses <- rep (NA, length(sss)) # std error memory
for(i in 1:length(sss)){
  ses[i] = sqrt(var(ajstu)/sss[i])  # std errors by theorem
}

data.frame(ses)
se.1 <- ses
se.2 <- 2 * ses 

lower.s2 <- mean(ajstu)-se.2
upper.s2 <- mean(ajstu)+se.2
data.frame(cbind(sss, ses, lower.s2, upper.s2))

# 12/2 lecture 
# note that we draw the statistical calculation 
# by "diff/se" = "diff/random_error"
n <- 80
mean.sample <- 103

sample <- rnorm2(n, mean.sample, sd.ajstu)
mean(sample)
sd(sample)

diff <- mean.sample - mean.ajstu # this is actual difference
se <- sd.ajstu / sqrt(n) # this is random error 
t.cal <- diff/se
t.cal
qnorm(0.025, lower.tail = F)
qnorm(0.01/2, lower.tail = F)
qt(0.05/2, n-1, lower.tail=F)

t.test(sample, mu=mean.ajstu)

# or we obtain the exact p value
p.value <- pt(t.cal, n-1, lower.tail = F)
p.value*2