# Math 108 Elementary Statistics S

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```					Math 108
Life is like a box of chocolates–you never know what you’re going to get.
Elementary Statistics                                                                    Forrest Gump in Forrest Gump, 1994.
Spring 2009

Project I: Statistical Analysis of M&Ms
The Project. There are many statistical questions that one can ask about packages of M&Ms. In this project you will
explore several of them and create appropriate graphs and displays to summarize your investigations.

Preparation. Each student in my three statistics classes has created a bar graph representing the contents of a single
tube of M&M minis. These bar graphs have been photographed and are available for viewing on the web at the site
http://www.wsc.ma.edu/math/faculty/carlip/StatisticsData.html

The Questions. Before we begin our analysis we need to think about what different studies we can conduct. You will
examine the three questions below in three sections of your project. For each question, you will ﬁrst need to identify
the appropriate experimental units and variables.
(a) Which colors are more common among the M&Ms? Which are least common?
– What are the experimental units (the “who”)?
– What is the variable (the “what”)?
– Is the variable categorical or quantitative?
(b) How many M&Ms are in a typical tube of M&Ms?
– What are the experimental units (the “who”)?
– What is the variable (the “what”)?
– Is the variable categorical or quantitative?
(c) How many green M&Ms are there in a typical tube of M&Ms? What proportion of the tube’s contents consists
of green M&Ms? (You can ask the same question for each of the other colors.)
– What are the experimental units (the “who”)?
– What is the variable (the “what”)?
– Is the variable categorical or quantitative?
Writing Assignment. This project will be due on Thursday, April 16. Your task is to prepare a carefully written
report to answer questions (a), (b), and (c) (using a second color assigned to you rather than green) as described below.
The report should be typed, with a cover page, and carefully prepared charts, either neatly done by hand on graph
paper (available from the math department) or printed using a computer program of your choice. The neatness and
You should address each of the questions in each part of the project, described below. Use full sentences, and describe
clearly your reasoning for each part.
You may use a computer to help prepare the charts. The program Microsoft Excel is available for free on campus, and
would make a reasonable choice for a program to use for the bar chart. I have observed that it is very difﬁcult to force
Excel to make a proper histogram! The CD for our text also includes a tool called DataDesk that could also be used.
On the other hand, it is may be more fun and informative to make nice charts by hand using colored pencils, markers,
crayons, or even black-and-white patterns. Neatness and clarity of presentation will be important. Graph paper is
available from the mathematics department. (Help yourself to paper on the table in the fourth-ﬂoor hall at the far end
of the corridor.)
You may discuss this project with other members of the class and divide up some of the work counting if you wish.
However . . .
• You should give explicit credit to anyone you discuss the project with, stating exactly what their role was.
• Your written work should be done completely on your own and in your own words. There will be serious
penalties for plagiarism!
Part I. The ﬁrst part should be devoted to the ﬁrst question.
Which colors are more common among the M&Ms? Which are least common?
You should start by gathering data. Prepare tables of the frequency and relative frequency of each color among the
M&Ms in your class and also in all three classes together. From this data prepare two bar charts:

• A chart describing the relative frequencies of the colors of M&Ms in your class (e.g., Math 108-009 or Math 108-
011).
• A chart describing the relative frequencies of the colors of M&Ms in all three classes together.
clearly:
• Which color is the most frequent in your class? Which is most frequent among all three classes? Did your class
have the same most frequent color as the three classes together? Be sure to include the relative frequency of the

• Which color is the least frequent in your class? Which color is least frequent among all three classes? Did your
class have the same least frequent color as the three classes together? Be sure to include the relative frequency
• Would you expect your own tube to have the same most frequent and least frequent color as the three classes
together? Explain. Did your tube actually match either your class or all three classes in this regard?

• I have a large tube of M&M minis (containing about twice the number of M&Ms as in your tube). Which colors
would you predict to be most common and least common? How conﬁdent are you in your prediction?
• If I go to the store and buy a random tube of M&M mini’s and select a candies at random, estimate the following
probabilities. You should assume that the candies in the store are distributed with the same relative frequencies
as the candies in all three classes together.
– The probability that the candy is red.
– The probability that the candy is either red, yellow, or orange.
– The probability that two candies selected are both blue.
– The probability that three candies selected include no blue candies.
Part II. The second part should be devoted to the second question:
How many M&Ms are in a typical tube of M&Ms?
Again, you should start by gathering data. Determine how many M&Ms are in each tube given out in your class
and given out in all three classes. This data is quantitative, so the appropriate displays are histograms. Prepare two
histograms
• A histogram describing number of M&Ms in each tube in your class (e.g., Math 108-009 or Math 108-011).
• A histogram describing number of M&Ms in each tube in all three classes together.
Your charts should be clearly labelled and easy to read. You should make a careful choice of how wide the bars should
be. Bars that are too wide hide too much information, while bars that are too narrow provide too much detail and hide
the overall pattern in the minor variation.
For quantitative data, we use statistics to describe the center and spread of the distribution. Compute the mean and
standard deviation for the data obtained for your class and for all three classes together.
Use your histograms and statistics to answer the following questions. You should use full sentences and present your

• What was the greatest number and least number of M&Ms in any of the tubes in your class and in the three
classes together?
• Is the shape of the distribution roughly unimodal and symmetric?
• Suppose that the mean number of M&Ms you found for all three classes is m and the standard deviation s. We
may guess that the Mars Corporation has designed their tube-ﬁlling machine to ﬁll the tubes with m candies,
but that the machine is inaccurate, sometimes placing more M&Ms and sometimes fewer in the tubes. It is a
reasonable hypothesis that the distribution of the number of M&Ms in a tube follows roughly the normal model
N (m, s) (where, again, m and s are the numbers you calculated above).
– Using the normal model, what z-score would correspond to a tube having exactly 100 candies?
– What proportion of tubes would you expect to have fewer than 100 candies?
– In the three classes together, what proportion of tubes actually had fewer than 100 candies?
– Would you expect any tubes with fewer than 100 candies if the class opened 10, 000 tubes?
– Using the normal model, what z-score would correspond to a tube having exactly 115 candies?
– What proportion of tubes would you expect to have more than 115 candies?
– In the three classes together, what proportion of tubes actually had more than 115 candies?
– What proportion of tubes would you expect to have between 105 and 115 candies?
– In the three classes together, what proportion of tubes actually had between 105 and 115 candies (including
both ends, 105 and 115)?

• The Mars Corporation advertises that the tubes always have at least 100 candies. Does your analysis support
their claim?
• If you buy a tube of M&Ms from the store, estimate the probability that you will get a tube containing more
than 115 candies in it. (Assume that all tubes are distributed similarly to those in the three classes.)
Part III. The second part should be devoted to the second question:
How many M&Ms of your color are there in a typical tube of M&Ms? What proportion of the tube’s contents
consists of M&Ms of your color?
First determine which color is your color for this study. The following table lists colors together with the ﬁrst letter
of your ﬁrst name. Thus, for example, if your ﬁrst name begins with A or B your color is red.
First Letter of First Name   Color
A,B               Red
C,D,E              Green
F,G,H,I,J           Yellow
K,L,M               Blue
N,O,P,Q,R            Orange
S,T,U,V,W,X,Y,Z          Brown
Again, you should start by gathering data. Determine how many M&Ms of your color are in each tube given out
in your class and given out in all three classes. This data is quantitative, so the appropriate displays are histograms.
Prepare two histograms
• A histogram describing number of M&Ms of your color in each tube in your class (e.g., Math 108-009 or
Math 108-011).
• A histogram describing number of M&Ms of your color in each tube in all three classes together.
Your charts should be clearly labelled and easy to read. You should make a careful choice of how wide the bars should
be. Bars that are too wide hide too much information, while bars that are too narrow provide too much detail and hide
the overall pattern in the minor variation.
For quantitative data, we use statistics to describe the center and spread of the distribution. Compute the mean and
standard deviation for the data obtained for your class and for all three classes together.
Use your histograms and statistics to answer the following questions. You should use full sentences and present your
• What was the greatest number and least number of your color M&Ms in any of the tubes in your class and in
the three classes together?
• Is the shape of the distribution roughly unimodal and symmetric?
• Suppose that the mean number of your color M&Ms you found for all three classes is m and the standard
deviation s. Again, we may hypothesize that the distribution of the number of your color M&Ms in a tube
follows roughly the normal model N (m, s) (where, again, m and s are the numbers you calculated above).

– Using the normal model, what z-score would correspond to a tube having exactly 10 candies of your color?
– What proportion of tubes would you expect to have fewer than 10 candies of your color?
– In the three classes together, what proportion of tubes actually had fewer than 10 candies of your color?
– If we examined 10, 000 tubes, would you predict that any of them would have no M&Ms of your color?
– Using the normal model, what z-score would correspond to a tube having exactly 33 candies of your
color?
– What proportion of tubes would you expect to have more than 33 candies of your color?
– In the three classes together, what proportion of tubes actually had more than 33 candies of your color?
– What proportion of tubes would you expect to have between 15 and 20 candies of your color?
– In the three classes together, what proportion of tubes actually had between 105 and 115 candies (including
both ends, 15 and 20)?
– If you buy a tube of M&Ms from the store, estimate the probability that you will get a tube containing
more than 33 candies of your color in it. (Assume that all tubes are distributed similarly to those in the
three classes.)

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