[R] Doubt about Student t distribution simulation

[John Fox]

Dear Jose,

The problem is that you're using the population standard deviation (sigma)
rather than the sample SD of each sample [i.e., t[i]  = (mean(amo.i) - mu) /
(sd(amo.i) / sqrt(n)) ], so your values should be normally distributed, as
they appear to be.

A couple of smaller points: (1) Even after this correction, you're sampling
from a discrete population (albeit with replacement) and so the values won't
be exactly t-distributed. You could draw the samples directly from N(mu,
sigma) instead. (2) It would be preferable to make a quantile-comparison
plot against the t-distribution, since you'd get a better picture of what's
going on in the tails.

I hope this helps,
 John 

--------------------------------
John Fox
Department of Sociology
McMaster University
Hamilton, Ontario
Canada L8S 4M4
905-525-9140x23604
http://socserv.mcmaster.ca/jfox 
-------------------------------- 

> -----Original Message-----
> From: r-help-bounces at stat.math.ethz.ch 
> [mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Jose 
> Claudio Faria
> Sent: Friday, August 04, 2006 3:09 PM
> To: R-help at stat.math.ethz.ch
> Subject: [R] Doubt about Student t distribution simulation
> 
> 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
> Estatística Experimental/Prof. Adjunto
> mails: joseclaudio.faria at terra.com.br
>         jc_faria at uesc.br
>         jc_faria at uol.com.br
> 
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