In a 25-game stretch of the 2014-15 NFL seasons, the New England Patriots won 19 coin tosses, for a winning percentage of 76%. The Patriots have a reputation for ethical lapses. Could they have cheated at the coin toss, too?
In this walkthrough, we'll conduct a simple hypothesis test. For this you'll need the mosaic library, so make sure to load it first:
library(mosaic)
Our null hypothesis here is that the Patriots had a 50% chance of
winning each coin toss. Obviously they might not win exactly 50% in
any given sequence of tosses. We can see this by simulating a few runs
of 25 coin tosses, using the nflip command in the mosaic library:
nflip(n=25)
## [1] 15
nflip(n=25)
## [1] 11
nflip(n=25)
## [1] 13
This simulates three different sequences of 25 fair coin tosses, and counts how many times that one side (the Patriots) won the toss. Due to random variation, the answer is different each time. What we've done here is a simple Monte Carlo simulation, in which we use a computer to simulate a random process.
Well, three sequences is fine for getting a little bit of intuition, but
it's better to run a lot more than three. We can do this using the
do() function in the mosaic library:
do(20)*nflip(n=25)
## nflip
## 1 8
## 2 11
## 3 12
## 4 12
## 5 11
## 6 13
## 7 14
## 8 11
## 9 16
## 10 11
## 11 13
## 12 13
## 13 12
## 14 6
## 15 9
## 16 11
## 17 15
## 18 6
## 19 11
## 20 11
That's 20 sequences of 25 fair coin flips, with each line representing the number of flips won by the Patriots (assuming a 50% win probability) for a single simulation.
Let's now run a much larger Monte Carlo simulation (i.e. many more than
20 sequences of 25 flips), and save the result in an object called
sim1:
sim1 = do(2500)*nflip(n=25)
head(sim1)
## nflip
## 1 12
## 2 11
## 3 9
## 4 12
## 5 11
## 6 11
You can see that sim1 is a data frame with 1 variable, called nflip.
If we look at a histogram of sim1$nflip, we will see the probability
distribution of Patriots' wins in 25 fair coin tosses:
hist(sim1$nflip, breaks=20)
Reaching 19 wins looks pretty unusual under this probability distribution. If you want to quantify just how unlikely this would be, you can can simply count up the number of times that the team reached 19 wins or more, divided by the number of simulations (here 2500):
pval = sum(sim1$nflip >= 19)/2500
pval
## [1] 0.0076
This probability is called a p-value. This one is pretty small (less than 1%, although yours will be different than mine, because of Monte Carlo variability). We conclude that it would be pretty unlikely for a team to win at least 19 times in 25 coin tosses, assuming that the tosses were fair.
Despite the small probability of such an extreme result, it's hard to believe that the Patriots cheated on the coin toss, for a few reasons. First, how could they? The coin toss would be extremely hard to manipulate, even if you were inclined to do so. Moreover, the Patriots are just one team, and this is just one 25-game stretch. There are 32 NFL teams, so the the probability that _one of them would go on an unusual coin-toss winning streak over some 25-game stretch over a long time period is a lot larger than the number we've calculated. Finally, after this 25-game stretch, the Patriots reverted back to a more typical coin-toss winning percentage, closer to 50%. The 25-game stretch was probably just luck.
Nevertheless, I would encourage you to focus on the process here, rather than the result. There were four steps:
- We have a null hypothesis, that the pre-game coin toss in the Patriots' games was truly random.
- We use a test statistic, number of Patriots' coin-toss wins, to measure the evidence against the null hypothesis.
- There is a way of calculating the probability distribution of the test statistic, assuming that the null hypothesis is true. Here, we just ran a Monte Carlo simulation of coin flips, assuming an unbiased coin.
- Finally, we used this probability distribution to assess whether the null hypothesis looked believable in light of the data.
All hypothesis testing problems have these same four elements. Usually the hard part is step 3, for which we'll soon learn a cool technique called the permutation test.