Notes: Comparing Counts - Goodness of Fit
Objectives:
1) Identify when the Goodness of Fit Test should be used to test a hypothesis.
2) Calculate the GOF test statistic (Chi-Square) and P-value with a graphing calculator.
3) Test a hypothesis using the Goodness of Fit Test.
Types of data you may test:
Does your zodiac sign predict how successful you will be later in life? Fortune magazine collected
the zodiac signs of 256 CEO’s of the largest 400 companies in America. Analyze the following data:
# of Births
23
20
18
23
20
19
18
21
19
22
24
29
Zodiac Sign
Aries (March 21 – April 19)
Taurus (April 20 – May 20)
Gemini (May 21 – June 20)
Cancer (June 21 – July 22)
Leo (July 23 – August 22)
Virgo (August 23 – September 22)
Libra (September 23 – October 22)
Scorpio (October 23 – November 21)
Sagittarius (November 22 – December 21)
Capricorn (December 22 – January 19)
Aquarius (January 20 – February 18)
Pisces (February 19 – March 20)
1. Are the counts independent of
each other?
2. Are the individuals from a
random sample?
3. Are there at least 5 individuals in
each count?
4. Is the sample less than 10% of
the population?
Think about the data:
 We have 256 CEO’s and 12 categories.
 The (ideal) H0 is that birth dates of CEO’s are divided equally among all the zodiac signs.
 That means that (1/12) of 256 = 21.333.
 So, the (ideal) H0 is expecting 21.333 people from each category.
 H1: Birth dates of CEO’s are not divided equally among all the zodiac signs.
 The test statistic looks at how closely the observed data match this ideal situation.
The Goodness of Fit Test (AKA Chi-Square Goodness of Fit Test)
 A way to test a hypothesis when you have many categories
 Scientists use this test to determine significance differences between theoretical data and
experimental data
 This is always a right tailed test
 Degrees of freedom = n – 1 (n is the number of categories)
When can you use the Goodness of Fit test?
1. Counts in the cells are independent of each other
2. Individuals are from a random sample
3. There are at least 5 individuals in each cell
4. The sample is less than 10% of the population
Test Statistic:
X2 
Calculator Steps:
(observed  exp erimental ) 2
exp erimental
Test the claim that birth dates of CEO’s are divided equally among all the zodiac signs.
Practice Problems – Goodness of Fit Test
Identify each H0 and H1. Use a calculator to generate the test-statistic and p-value. Test the
hypothesis. Show all work. Explain your conclusion.
1) Suppose we hypothesize that we have an unbiased six-sided die. To test
this hypothesis, we roll the die 300 times and observe the frequency of occurrence of each of the
faces. Because we hypothesized that the die is unbiased, we expect that the number on each face
will occur 50 times. However, suppose we observe frequencies of occurrence as follows:
Face value
Occurrence
1
42
2
55
3
38
4
57
5
64
6
44
Hints:
H0: The die is fair
H1: The die is not fair
2) In 200 flips of a coin, one would expect 100 heads and 100 tails. But what if 92 heads and 108
tails are observed? Would we reject the hypothesis that the coin is fair?
3) The president of a major university hypothesizes that at least 90 percent of the teaching and
research faculty will favor a new university policy on consulting with private and public agencies
within the state. Thus, for a random sample of 200 faculty members, the president would expect
0.90 x 200 = 180 to favor the new policy and 0.10 x 200 = 20 to oppose it. Suppose, however, for
this sample, 168 faculty members favor the new policy and 32 oppose it. Is the difference between
observed and expected frequencies sufficient to reject the president's hypothesis that 90 percent
would favor the policy?