Question

To experimentally verify the central limit theorem, take 1000 random samples of size n=33 from a population that is distributed exponentially with λ=3. To this end, you must show

1. The histogram of the distribution is approximately normal.
2. Its expected value is μ
3. Its variance is sigma^2/33

Please complete in Rstudio

The center thickness (in mils) of a sample of twenty-five contact lenses are given below

0.3978             0.4019             0.4031             0.4044             0.3984             0.3972             0.3981

0.3947             0.4012             0.4043             0.4051             0.4016             0.3994             0.3999

0.4062             0.4048             0.4071             0.4015             0.3991             0.4021             0.4009

0.3988             0.3994             0.4016             0.4010

Answer 1

According to central limit theorem the sample mean of 1000 random samples of size n=33 will have a normal distrubution with mean = mean of the exponential distribution and variance = variance of exponential distribution /33

Now we know when X~ then E(X)=1/ and Var(X)=1/

Here which implies mean of the exponential distribution is 1/3 and variance of the exponential ditrubution is 1/9.

Therefore, according to CLT mean of the sample means would be 1/3 =0.3333

and variance of the sample means would be 1/(9*33) =0.003367

R codes:

meu=NULL
for(i in 1:1000)
{
   data=rexp(33,3)
   meu[i]=mean(data)
}
hist(meu,prob=T)

output:

So the histogram of the distribution is approximately bell shaped indicating it follows a normal distribution.

now for mean and variance

R code:

mean(meu)

result: 0.3331588

var(meu)
result : 0.003161583

Both the values matches with what we expect from the theory of CLT (Central limit theorem)