The example is more of a statistical exercise that  shows the true significance and the density curve of simulated random normal data.  The code can be changed to generate data using either a different mean and standard deviation or a different distribution altogether.

This extends the idea of estimating pi by generating random normal data to determine the actual significance level. First, there is a simple function to calculate a pooled t statistic. Then repeat that function N times. The second part of the R code will produce a density graph of the t statistic (normalized to 1). By changing the distribution, mean, standard deviation one can see what happens to the true significance level and the density graph.  Just keep that in mind the next time someone assumes a standard normal distribution for hypothesis testing when it’s really a different distribution altogether and they are set on alpha = 0.05

###
#Function to calculate the t statistic
###
tstatfunc=function(x,y){
m <- length(x)
n <- length(y)
spooled <- sqrt(((m-1)*sd(x)^2+(n-1)*sd(y)^2)/(m+n-2))
tstat <- (mean(x)-mean(y))/(spooled*sqrt(1/m+1/n))
return(tstat)
}

###
#Set some constants
###
alpha <- .05
m <- 15
n <- 15
N <- 20000
n.reject <- 0

###
#Iterate N times
###
for (i in 1:N){
x <- rexp(m,1)
y <- rexp(n,1)
t.stat <- tstatfunc(x,y)
if (abs(t.stat)>qt(1-alpha/2,n+m-2)){
n.reject <- n.reject+1
}
}
true.sig.level <- n.reject/N
true.sig.level

###
#Function to simulate t-statistic vector
###
tsim=function(){
tstatfunc(rnorm(m,mean=0,sd=1), rnorm(n,mean=0,sd=1))
}

###
#Set up the values to graph
###
tstat.vec <- replicate(10000, tsim())
dens.y <- density(tstat.vec)$y/max(density(tstat.vec)$y)
dens.x <- density(tstat.vec)$x

###
#Graph the density for each
###
plot(
NULL, NULL, xlim=c(-5,5),ylim=c(0,1), main="Simulated and True Density"
)
lines(dens.x,dens.y, lty=1, col=3, lwd=3)

curve(
dt(x,df=n+m-2)/max(dt(x,df=n+m-2)),add=TRUE
)