Binomial test

In statistics, the binomial test is an exact test of the statistical significance of deviations from a theoretically expected distribution of observations into two categories.

One common use of the binomial test is in the case where the null hypothesis is that two categories are equally likely to occur (such as a coin toss). Tables are widely available to give the significance observed numbers of observations in the categories for this case. However, as the example below shows, the binomial test is not restricted to this case.

Where there are more than two categories, and an exact test is required, the multinomial test, based on the multinomial distribution, must be used instead of the binomial test.

For large samples such as the example below, the binomial distribution is well approximated by convenient continuous distributions, and these are used as the basis for alternative tests that are much quicker to compute, Pearson's chi-squared test and the G-test. However, for small samples these approximations break down, and there is no alternative to the binomial test.

Suppose we have a board game that depends on the roll of one die (see dice) and attaches special importance to rolling a 6. In a particular game, the die is rolled 235 times, and 6 comes up 51 times. If the die is fair, we would expect 6 to come up 235/6 = 39.17 times. Is the proportion of 6s significantly higher than would be expected by chance, on the null hypothesis of a fair die?

To find an answer to this question using the binomial test, we use the binomial distribution B(N=235, p=1/6) to calculate the probability of 51 or more 6s in a sample of 235 if the true probability of rolling a 6 on each trial is 1/6. This is done by adding up the probability of getting exactly 51 6s, exactly 52 6s, and so on up to 235.

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