Introduction to Chi-Square
Procedures
March 11, 2010
The Mars Candy Co. claims that the
distribution of colors of M&M’s are as
follows:
Red: 13%
Yellow: 14%
Green: 16%
Blue: 24%
Orange: 20%
Brown: 13%
Chi-Square Distribution
We are going to test whether or not this claim is
true.
To do so, we will encounter a new probability
density: the Chi-Square density with k degrees
of freedom (denoted χ2(k))
This family of densities will allow us to test
whether or not a population has a certain
distribution.
Gather Data
Each group should add up the number of each
color of M&M’s in their bags.
Now we will add up the total of each color in the
class:
Red Yellow Green Blue
Orange Brown Total
Observed vs. Expected
The previous table listed the observed counts
for each color. We also need to know the
expected count: how many there would be if the
claimed distribution is correct:
Red Yellow Green Blue
Orange Brown Total
Summary Table
R
Y
G
Bl
Or
Br
Total
O
E
(O-E)2/E
X2=
The Statistic X2
The statistic X2 has approximately a χ2
(read Chi-Square) distribution with k-1
degrees of freedom, where k is the
number of outcome categories.
The critical numbers are found in Table D
at the back of your book.
Significance Test
To carry out our inference, we need to
formulate the null and alternate
hypotheses:
H0:
Ha:
P-value
We now find our P-value.
This is P(χ2 > X2), which is found in Table
D, using 5 degrees of freedom.
Our P-value is:
We therefore conclude: