the distribution of m&m’s

The Distribution of M&M’s

Bryan Wu/ Hongbo Liu

Question: Does the actual distribution of Peanut M&M colors follow the proportions claimed by the manufacturer?

This following study was conducted for the purpose of an AP Statistic Class investigation. There is no guarantee on the validity on any or all parts of the study. The conductor(s) of the study take no responsibility in the accuracy of the results. 

Background Information

• Small, sugar-coated, milk chocolate pieces, produced by Mars, Incorporated

• Variations- milk chocolate, peanut, mint chocolate, dulce de leche, toffee, crispy, almond, peanut butter, white chocolate and dark chocolate.

• Six colors: red, orange, yellow, green, brown, and blue

Procedure

ACME- 272 East Butler Avenue, Ambler, PA Numbered 8 bags

from 1-8

randInt (1,8,1)=2

Counted the number of M&M’s for each color

Data

168

45

25

22

19

2730

Using the Chi-Square Test for Goodness Fit

Is there is a relationship between two categorical variables?

Variables>Proportions from the sample we purchased >Proportions provided by the manufacturer

Assumptions for Chi-Square

1. Simple Random Sample• -Yes; the sample was selected

using random allocation2. All individual expected values are at

least 1• -Yes3. No more than 20% of the expected

counts are less than 5• -Yes; none are less than 5

Hypotheses

• Null Hypothesis: H0: The actual population proportions of M & M Colors are equal to the proportions claimed by the manufacturer

• Alternative Hypothesis: Ha: The actual population proportions of M & M Colors are NOT equal to the proportions claimed by the manufacturer

Before conducting the test  Red Blue Orange Green Brown Yellow TotalObserved 25 27 45 22 30 19 168

ExpectedExpected proportions

20.16 38.64 38.64 25.2 20.16 25.2 168

0.12 0.23 0.23 0.15 0.12 0.15 1

The Colors of M & Ms

0

10

20

30

40

50

Color

Pro

po

rtio

ns

Observed

Expected

The Colors of M & Ms

0%

20%

40%

60%

80%

100%

Observed Expected

Observed and Expected

Pro

po

rtio

n o

f C

olo

r Yellow

Brown

Green

Orange

Blue

Red

Chi-Square Test for Goodness of Fit

Formula

Observed (O) Expected (E) ((O-E)^2)/E

25 20.16 1.162

27 38.64 3.5065

45 38.64 1.0468

22 25.2 0.40635

30 20.16 4.8029

19 25.2 1.5254X²= 1.162+3.5065+1.0468+.40635+4.8029+1.5254= 12.4499 Test Statistic: X²=12.4499 Degrees of Freedom: n-1=5

P-Value & Rejection of Null Hypothesis

•P(X²≥X²)=.0291

•.0291<.05

Reject the null hypothesis at 0.05 significance level

ConclusionThe actual population proportions of M & M

Colors are NOT equal to the proportions claimed by the manufacturer.

HOWEVER… • Larger sample chosen from more varied

sources should be used.• Statisticians should repeat this investigation

multiple times to evaluate the validity of the manufacture’s claims

Future study

• Examine color distributions from different locations

• Conduct the same test on other flavors (almond, white chocolate, dark chocolate)

SOURCES

• http://en.wikipedia.org/wiki/M%26m%27s

• http://us.mms.com/us/about/products/peanut