This document discusses using the chi-square statistic to test whether observed sample results differ significantly from expected results. It provides the formula for calculating chi-square and explains that larger chi-square values provide stronger evidence against the null hypothesis. It then works through an example using data on M&M colors to calculate chi-square and determine the P-value to assess significance. The key steps are outlined as calculating chi-square, determining the degrees of freedom, using a chi-square table to find the P-value based on the chi-square value and degrees of freedom, and comparing the P-value to the significance level to determine whether to reject or fail to reject the null hypothesis.

1. Inference – Dist. by Categories
How likely is it that differences this large or larger would occur
just by chance in random samples of size 60 from the population
distribution claimed by Mars?
To make the comparison, we use the chi-square statistic
χ2 = ∑ (observed – expected)2
Expected
This is the measure of the distance of the observe counts from the
expected counts
Large values of χ2 are strong evidence against H0 because the
observed counts are far away from expected
Small values of χ2 suggest that the data are consistent with the
null hypothesis

2. Inference – Dist. by Categories
Jerome’s class did the M&M Activity and here are his results:
Calculate the chi-square
statistic
We divide each category by its respective expected value so that
the largest relative difference contributes more heavily to the
evidence against the null
Color Observed Expected
Blue 9 14.4
Orange 8 12
Green 12 9.6
Yellow 15 8.4
Red 10 7.8
Brown 6 7.8

3. Inference – Dist. by Categories
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The chi-square distribution includes only positive and is skewed right.
Each chi-square distribution is specified by giving its degrees of
freedom. Degrees of freedom for a chi-square goodness-of-fit test =
number of categories – 1
As the degrees of freedom increase, the density curves become less
skewed, and larger values become more probable
The mean of a particular chi-square distribution is equal to its degrees
of freedom
When degrees of freedom are > 2, the peak of the chi-square density
curve is at df – 2

4. Inference – Dist. by Categories
To find the P-value from a chi-square distribution, we use Table
C
Use the degree of freedom row on the left of the table
Locate the approximate values the χ2 lies between and the P-value
lies between the two values at the top of those two columns.

5. Inference – Dist. by Categories
In the last example, we computed the chi-square statistic for
Jerome’s random sample of 60 M&M’s Milk Chocolate Candies:
χ2 = 10.180. Now let’s find the P-value. Because all the
expected counts are at least 5, the χ2 statistic will follow a chi-
square distribution reasonably well when H0 is true. There are 6
color categories for M&M’s Milk Chocolate Candies, so
df = 6 -1 = 5.
The P-value is the probability of getting a value of χ2 as large as
or larger than 10.180 when H0 is true.
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6. Inference – Dist. by Categories
Technology can give us a more precise P-value
DISTR – χ2 cdf
χ2 cdf ( χ2 , large # (1000), df )
Since our P-value is 0.07 is greater than our significance level α =
0.05, we fail to reject H0.
We don’t have sufficient evidence to conclude that the company’s
claimed color distribution is incorrect.
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