# [R] Doubt about Student t distribution simulation

Jose Claudio Faria joseclaudio.faria at terra.com.br
Fri Aug 4 22:09:23 CEST 2006

```Dear R list,

I would like to illustrate the origin of the Student t distribution using R.

So, if (sample.mean - pop.mean) / standard.error(sample.mean) has t
distribution with (sample.size - 1) degree free, what is wrong with the
simulation below? I think that the theoretical curve should agree with
the relative frequencies of the t values calculated:

#== begin options=====
# parameters
mu    = 10
sigma =  5

# size of sample
n = 3

# repetitions
nsim = 10000

# histogram parameter
nchist = 150
#== end options=======

t   = numeric()
pop = rnorm(10000, mean = mu, sd = sigma)

for (i in 1:nsim) {
amo.i = sample(pop, n, replace = TRUE)
t[i]  = (mean(amo.i) - mu) / (sigma / sqrt(n))
}

win.graph(w = 5, h = 7)
split.screen(c(2,1))
screen(1)
hist(t,
main     = "histogram",
breaks   = nchist,
col      = "lightgray",
xlab     = '', ylab = "Fi",
font.lab = 2, font = 2)

screen(2)
hist(t,
probability = T,
main        = 'f.d.p and histogram',
breaks      = nchist,
col         = 'lightgray',
xlab        = 't', ylab = 'f(t)',
font.lab    = 2, font = 2)

x = t
curve(dt(x, df = n-1), add = T, col = "red", lwd = 2)

Many thanks for any help,
___
Jose Claudio Faria
Brasil/Bahia/Ilheus/UESC/DCET