Math 362 - Computer Lab #9 - Fall 2002
Introduction
to Probability Theory
Computer Lab 9: The normal distribution
and the Central Limit Theorem
In this lab, we will get familiar with the normal distribution and use MINITAB to explore the Central
Limit Theorem.
- The normal distribution
- Use MINITAB to plot the p.d.f. of a normal distribution with:
- Mean 0 and standard deviation 1
- Mean 1 and standard deviation 1
- Mean 0 and standard deviation 0.3
- Then answer the following questions:
- What happens to the p.d.f. if you change the mean of the random variable but keep the
standard deviation constant?
- What happens to the p.d.f. when you change the standard deviation but keep the mean constant?
- The p.d.f. of a normal random variable is bell-shaped.
- For what value of x does the p.d.f. reach its maximum?
- Where are the two inflection points of the p.d.f.?
- The Central Limit Theorem
The Central Limit Theorem says that the sample mean of a random sample of size n
taken from a distribution of mean m and standard deviation
s has a distribution which is approximately normal
(i.e. which tends towards a normal distribution as n goes to infinity) with mean
m and variance s2/n.
We will use MINITAB to explore this statement.
- Assume that X1, X2, ..., Xn
are normally distributed with mean m and standard deviation
s. Then, the sample mean (X1 + X2 +
... + Xn)/n is also normally distributed with mean
m and variance s2/n. In
the case of normally distributed random variables, this is true for any integer n.
With MINITAB, create 10000 random samples of size 3 from a normal distribution with mean 0 and variance
3. Calculate the sample mean in each case and plot a histogram approximating the distribution
of the sample mean. Check that the approximate p.d.f. has a mean of 0 and a variance of 1.
- Repeat the above for random samples taken from a uniform distribution with mean 0 and variance 3.
Is the approximate p.d.f. almost normal? Why or why not?
- Perform the same experiment but with a binomial random variable of variance 3. Note that if X
is binomial with parameters n and p, then E(X) = n p and Var(X)
= n p (1 - p).
- Repeat all of the above with n ≥ 10. What do you conclude? How is this an illustration of
the Central Limit Theorem?
- Reading Assignment: Sections 5.6 and 5.7
The macro Normal.MTB performs the above operations with n = 3 and
n = 20. You may try and modify this file at your convenience.
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