A binomial test compares the number of successes observed in a given number of trials with a hypothesised probability of success. The test has the null hypothesis that the real probability of success is equal to some value denoted p, and the alternative hypothesis that it is not equal to p. The test can also be performed with a one-sided alternative hypothesis that the real probability of success is either greater than p or that it is less than p.
You can perform a binomial test with the
binom.test function. The command takes the general form:
nsuccesses is the number of successes observed,
ntrials is the total number of trials and
p is the hypothesised probability of success.
Alternatively you can give the number of successes and the number of failures observed, as shown below.
To perform a one-sided test, set the
alternative argument to
"greater" as required.
ntrials, p, alternative="greater")
The output includes a 95% confidence interval for the true probability. To adjust the size of this interval, use the
conf.level argument as shown.
ntrials, p, conf.level=0.99)
Example: Binomial test for die rolls
In a game, you suspect your opponent is using a die which is biased to roll a six greater than 1/6 of the time. Suppose you want to prove this by rolling the die 300 times and using a binomial test to determine whether the probability of rolling a six is equal to 1/6. A one-tailed test with a significance level of 0.05 will be used.
You roll the die 300 times and throw a total of 60 sixes. To perform the test, use the command:
> binom.test(60, 300, 1/6, alternative="greater")
Exact binomial test data: 60 and 300 number of successes = 60, number of trials = 300, p-value = 0.07299 alternative hypothesis: true probability of success is greater than 0.1666667 95 percent confidence interval: 0.1626847 1.0000000 sample estimates: probability of success 0.2
From the output you can see that the p-value is 0.07299. As this is not less that the significance level of 0.05, we cannot reject the null hypothesis that the probability of rolling a six is 1/6. This means that there is no evidence to prove that the die is not fair.