Activity: Minority Coaches in College Football vs. the NFL
The Black Coaches and Administrators (BCA) released a report in 2008 to evaluate hiring
practices for head coaches in college athletics. According to this report, there were 8 minority
coaches out of 120 head coaches at FBS schools in 2008. In the NFL, there were 7 minority
coaches out of 32 head coaches.
Source: Who You Know & Who Knows You: The Hiring Process and Practice of NCAA FBS & FCS Head Coaching
Positions (BCA HRC #5), 2008.
The data are summarized in the following table:
Minority Head Coach Non-Minority Head Coach Total
NFL
7
25
32
FBS (college)
8
112
120
Total
15
137
152
Question of Interest: Do the data appear to support the claim that the NFL is more diverse
than college football in terms of head coaches?
Questions:
1. Calculate the proportion of head coaches who are of minority status for both the FBS
and the NFL. Does the NFL have a higher sample proportion than the FBS?
2. Note that the overall proportion of minority head coaches is 15/152 = 9.87%. If there is
no difference in the diversity of head coaches between the FBS and NFL, then this value
is used to compute the expected number of minority head coaches in each group. Given
that there are 120 FBS head coaches, how many do you expect of minority status in the
FBS? How many minority coaches do you expect in the NFL? Explain.
3. How much does the observed data (8/7 split) deviate from what is expected under the
assumption of no difference in diversity of head coaches between the FBS and NFL?
4. If there is no difference in diversity of head coaches between the FBS and NFL, then is it
possible to observe some deviation from the expected values? Explain.
5. Do you think the observed data (8/7 split) deviates “too much” from what we expect to
see if the NFL and FBS were equally diverse? How much confidence do you have in your
decision?
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Repeated Sampling – Hands-on Approach
Repeated sampling will be used to investigate how unlikely we are to observe the 8/7 split
under the assumption of no difference in diversity between the NFL and FBS. We will set up a
situation where we know that the NFL and FBS are equally diverse: start with 15 minority
coaches and 137 non-minority coaches, and we’ll randomly assign 25 to the NFL and the
remaining 112 to the FBS.
We could use 152 playing cards to carry out the random assignment:


Let 15 black cards represent minority head coaches
Let 137 red cards represent non-minority head coaches
Shuffle these cards together, and randomly deal out 32 cards to represent the head coaches in
the NFL. Then, count the number of black cards and enter your results in the following table to
show how many minority coaches ended up in each group.
Results for Outcome 1:
NFL
FBS (college)
Total
Minority Head Coach Non-Minority Head Coach Total
32
120
15
137
152
Questions:
6. Is the result of this random assignment as extreme as the 8/7 split observed in 2008?
Explain.
7. Shuffle the cards, and repeat this procedure a second time and enter your results in the
following table. Is the result of the random assignment in the second outcome as
extreme as the actual data observed in 2008?
Results for Outcome 2:
NFL
FBS (college)
Total
Minority Head Coach Non-Minority Head Coach Total
32
120
15
137
152
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Note that this procedure ensures that the row and column totals are the same: we will
always have 15 minority and 137 non-minority coaches, and we will always have 32 in
the NFL and 120 in the FBS. Given that the row and column totals are fixed, we need
only focus on the number of minority coaches in the NFL.
Plot the number of minority coaches in the NFL for each of your first two outcomes.
Share your results with the entire class. Add your classmates’ results to the above
dotplot.
8. Based on the data collected from the entire class, what proportion of the outcomes
resulted in exactly 7 NFL minority coaches?
9. If one conjectures that the NFL is more diverse than the FBS and believes that observing
7 NFL minority coaches is extreme, then observing more than 7 would also be
considered extreme. Based on the data collected from the entire class, what proportion
of the outcomes resulted in 7 or more NFL minority coaches?
10. Is this proportion small enough that you would consider such an outcome unlikely to
occur when the coaches are randomly assigned to each group? Remember that this
random assignment mimics the situation where the NFL and FBS are equally diverse.
11. Would you say the data provides [ no | moderate | strong ] evidence that the NFL is
more diverse than the FBS in terms of head coaches? Explain.
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Repeated Sampling – Technology Approach
Technology can be used to assist us in this process of repeated sampling. We can use the
applet found on Rossman and Chance’s website:
(http://www.rossmanchance.com/applets/Friendly1/friendly1.html).
To use this applet for our data, we must do the following:
1. Enter the observed data in the table.
2. Request Number of repetitions =100.
3. Click randomize.
4. If you would like to see the frequency for each group, you can click on the Show tallies
button:
The applet returns the following:
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Here is a dotplot based on another 100 repetitions of this random assignment process:
Questions:
12. In what proportion of the 100 simulated random assignments did we observe 7 or more
minority NFL coaches?
13. Is the proportion small enough that you would consider such an outcome unlikely to
occur by chance alone?
14. Would you say the data provides [ no | moderate | strong ] evidence that the NFL is
more diverse than the FBS in terms of head coaches? Explain.
15. Do you have [ less | same | more ] confidence in your decision when using a larger
number of repetitions?
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If sampling were repeated continuously, we would obtain a distribution known as the
hypergeometric. This distribution describes how the number of minority coaches in the NFL will
vary under the assumption that the NFL and FBS are equally diverse.
The exact probabilities for the hypergeometric distribution can be calculated in various
software packages (e.g. Excel).
These exact probability values (represented by the vertical lines) are overlaid on the dotplot
from our simulation.
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Questions:
16. Under the assumption that the NFL and FBS are equally diverse, what is the probability
of observing 7 or more NFL minority coaches?
17. Is this probability small enough that you would consider such an outcome unlikely to
occur by chance alone?
18. Would you say the data provides [ no | moderate | strong ] evidence that the NFL is
more diverse than the FBS in terms of head coaches? Explain.
19. Do you have [ less | same | more ] confidence in your decision when using a distribution
based on an infinite number of outcomes?
Comment: The following is standard output for Fisher’s Exact test for 2x2 tables. The Left Prob
value matches the probability computed above from the hypergeometric distribution.
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