MINITAB Assignment 3
(due Friday March 17)
Probability Histograms from Sample Data
< Text (such as this)
which is enclosed in angle brackets, < >,
is communication about what to do in the assignment,
or commentary about it. I will also color such text
green. Text which is not green and bracketed is
text you should type or paste into the session window,
perhaps with obvious changes, such as replacing
Your Name with your name. This will make your
minitab work more meaningful on review and in execution.
Please try to understand the exercise.>
< What you must turn in for this assignment is the
completed session window, together with the annotated
graphs you generate. >
Minitab Assignment 3
In this assignment we generate probability histograms
for random data to better understand the meaning of
a probability distribution.
I. Probability Histogram
From a list of numbers, Minitab can
create a probability histogram (density histogram).
For this, Minitab chooses appropriate bin lengths;
fraction of the numbers falling in each bin; and
draws over the bin a rectangle whose area equals
the fraction of the numbers falling in the bin.
To see an example of the probability histogram,
we first have Minitab (as in assignment 2) generate a
sequence of 10 numbers from the Bernoulli(.5)
distribution. (We can think of Minitab as having a
crew which flips fair coins, recording 1 when the
coin falls heads, and recording 0 when the coin
falls tails.) For this, we use
Calc > Random Data > Bernoulli ;
type in 10 for the number of rows;
and choose C1 for the column in which to save the data.
< Then you must also click something
like "ok" to get minitab to execute. I will omit this
type of obvious command. >
We generate the probability (density) histogram
with the following commands:
we follow Graph > Histogram ;
from options we choose "density";
we click footnote and type in My Name;
we click title and type in Bernoulli(.5) Distribution, 10
< Now use the printer icon on the top bar menu to print out
your histogram. Examine the histogram and understand its relation to
the data in C1. For your later graphs, at home it is probably best to
issue a print command after each graph is created, in the OWL
you might want to print them in larger groups.
Next we do the same thing as above, but using C2 with 50 rows
and C3 with 5000 rows, and changing the titles of the graphs
< Execute your commands. >
Comment on the histograms:
< Here type in your comment. You are likely to
see the density histograms better reflect
the Bernoulli(.5) distribution as the number of trials
(rows) increases. >
< Now you are done with the data in C1-C3. It is probably
safest to simply delete it: highlight the column headings
C1-C3 in the data window, this will highlight the columns,
then hit the delete key. Likewise, later you will have
to delete or overwrite to avoid exceding Student
Minitab's 5000 cell limit; I won't mention this again.
Part II: Uniform [0,1]
We will look at examples in which we see the probability
histograms of sample data from continuous distributions looking
more like the graphs of the corresponding density functions as
the sample size increases.
First, in this Part II,
we look at uniform distribution on the unit interval.
The density function for this distribution has the following
< Type in a definition for this density
Minitab again can generate sample numbers representative of
this distribution. (We can think of the Minitab Crew as repeatedly
picking numbers from [0,1] and writing them down. The picks
are completely accidental, no more likely from one location than
We put into columns C1, C2, C3 sample data from the)
uniform distribution. We use 50, 500 and 4400 rows respectively for
these columns. We do this just as we did in Step I, but choosing
Uniform instead of Bernoulli.
< Execute your commands. >
Next we generate density histograms for these data.
< Execute your commands.
Of course the title of the graphs should change. For example,
let the first graph be
Uniform Distribution on [0,1], 50 Trials >
Comment on these density histograms:
< Type in your comments.
It is likely that the tops of the rectangles will be more the
same as the number of trials increases -- that is, the
discrete distribution described by the probability histogram
will be better approximated by the uniform distribution on
the unit interval [0,1].
There will probably be two exceptional rectangles, the leftmost
and rightmost, for many trials. Explain in your comments
why these are only
half as high as the others. >
Part III: The Standard Normal Distribution
The standard normal distribution is the normal distribution
with mean 0 and standard deviation 1; its density function is
the standard bell shaped
cuve. In this exercise, we do for the normal distribution what we just
did with the uniform distribution (except we don't type in
the definition of the density function for the normal distribution).
The procedure is the same (even the
numbers of trials for columns C1,C2,C3), with "normal" replacing
< Execute your commands and
give your comments. If Minitab is doing its job, then the
probability histograms from the sample data should look
pretty close to the region under a bell shaped curve when
the sample size is large. >
< Of course the title of the graphs should change.
let the first graph be titled
Normal Distribution, 50 Trials >
Part IV: The binomial distribution
For the binomial(n,p) distribution with n=10 and p=.5,
we generate sample numbers in columns C1,C2,C3 respectively
with 50, 500 and 4000 rows; and we generate probability
histograms for these three data sets.
Execute your commands.
This goes essentially as before. You follow
Calc > Random Data > Binomial
and then you can see how to set the parameters corresponding
to n=10 and p=.5 , and so on. Let your first graph title be
Binomial (n=10,p=.5) Distribution, 50 trials
Finally, repeate the exercise above for binomial(n,p)
with n=10 and p=.1 . Include in your comment a
comparison to the previous binomial case, if you notice
Go back over your session window to erase unneeded errors
and garbage, print it out, staple it to your correctly ordered
and titled graphs, and hand this in.