the distribution of m&m’s
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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?. - PowerPoint PPT PresentationTRANSCRIPT
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