20 APRIL, 2007
MATH 311
LAB 3 – SAMPLING DISTRIBUTIONS
DUE: TUESDAY, APRIL 24TH AT 4:00 P.M.
Let’s imagine that you rolled a 34641-sided (trust me) die and let’s let X = the number of dots showing
on the top face.
Now let’s imagine that you rolled this die 10,000 times (note: this number is not important for the
following. The bigger we can make it, however, the clearer the pattern we’re looking for will be. So if
your computer can handle 50,000 or 100,000 – go for it!).
Question 1: What do you think the mean value of X (  X ) would be? Be careful – think about a 6-sided
die, the mean is not 3.
Okay, enough imagining, let’s make Minitab do some work: Generate at least 10,000 rows of data with
a lower endpoint of 1 and an upper endpoint of 34641 by following Calc>Random Data>Integer.
Question 2: Make a Histogram and compute the mean and standard deviation of the data. Include these
in your report. Are the data normally distributed? Test it.
Okay, great. Now let’s imagine that instead of rolling the die just once you rolled it 4 times and let Y =
the mean of the four rolls. Y is, essentially, the mean of a sample of size 4.
Further, let’s imagine that you computed this value of Y 10,000 times (10,000 samples of size 4).
Question 3: What do you think the mean value of this new random variable Y is?
Okay – so let’s make Minitab simulate this situation. Generate 10,000 rows and 4 columns (C1-C4) of
Integer data with a lower endpoint of 1 and an upper endpoint of 34641 as follows:
Select Calc>Random Data>Integer
In the Generate _______ rows of data box enter 10000
In the Store in Column(s) box, enter C1 – C4
In the Minimum Value box enter 1
In the Maximum Value box enter 34641
Notice that each row represents one sample of rolls. We can compute the mean of this sample of size 4
as follows:
Title column C5 “Mean: n = 4”
Select Calc>Row Statistics
Click Mean
In the Input variables: box enter C1-C4
In the Store result in: box enter C5
Select OK
The mean for each of the 10,000 rows should appear in C5.
Question 4: Make a Histogram and compute the mean and standard deviation of these data in C4 (the
values of Y). Include these in your report. Are the data normally distributed? (Test it.)
Alrighty, now imagine that you rolled the die 100 times (a sample of size n = 100) (whew!) and let Z =
the mean of the 100 rolls. And then you computed the value of Z 10,000 times. (Z is a sample mean
and, of course, will vary from sample to sample.) Tired yet?
Well, make Minitab compute 10,000 rows and 100 columns (C1-C100) of Integer data with a lower
endpoint of 1 and an upper endpoint of 34641. Compute the mean of each row and store it in C101 (you
should un-title column C5 and title C101 as “Mean: n = 100.”)
Question 5: Make a Histogram and compute the mean and standard deviation of these data in C101 (the
values of Z – remember Z? What was it?). Include these in your report. Are the data normally
distributed? (Test it.)
Have you noticed any patterns that the mean and standard deviations seem to be following for various
sample sizes? Look closely at the numbers and, perhaps, round to the “tens-place.” Still don’t see
anything? Try simulating 202 = 400 rolls. Then, maybe 332 = 1089 times (you might have to reduce the
number of rows used – it depends on how much memory the computer you’re using has). Ready to
generalize? Good, answer the last question:
Question 6: Lastly, let’s generalize (i.e. look at the pattern and come up with a formula): imagine that
you rolled the die n times and let X = mean of the n rolls.
Make a conjecture about the shape of the distribution of X, the mean (  X ), and the standard deviation
(  X ).