Editing Binomial test (section)

==Example==
Suppose we have a [[board game]] that depends on the roll of one [[dice|die]] 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 

: <math>235\times1/6 = 39.17</math> 

times. We have now observed that the number of 6s is higher than what we would expect on average by pure chance had the die been a fair one. But, is the number significantly high enough for us to conclude anything about the fairness of the die? This question can be answered by the binomial test. Our [[null hypothesis]] would be that the die is fair (probability of each number coming up on the die is 1/6).

To find an answer to this question using the binomial test, we use the [[binomial distribution]] 

: <math>B(N=235, p=1/6)</math>  with [[Probability mass function|pmf]] <math>f(k,n,p) = \Pr(k;n,p) = \Pr(X = k) = \binom{n}{k}p^k(1-p)^{n-k}</math> .

As we have observed a value greater than the expected value, we could consider the probability of observing 51 6s or higher under the null, which would constitute a [[One- and two-tailed tests|one-tailed test]] (here we are basically testing whether this die is biased towards generating more 6s than expected). In order to calculate the probability of 51 or more 6s in a sample of 235 under the null hypothesis we add up the probabilities of getting exactly 51 6s, exactly 52 6s, and so on up to probability of getting exactly 235 6s:

: <math>\sum_{i=51}^{235} {235\choose i}p^i(1-p)^{235-i} = 0.02654</math>

If we have a significance level of 5%, then this result (0.02654 < 5%) indicates that we have evidence that is significant enough to reject the null hypothesis that the die is fair.

Normally, when we are testing for fairness of a die, we are also interested if the die is biased towards generating fewer 6s than expected, and not only more 6s as we considered in the one-tailed test above. In order to consider both the biases, we use a [[One- and two-tailed tests|two-tailed test]]. Note that to do this we cannot simply double the one-tailed p-value unless the probability of the event is 1/2. This is because the binomial distribution becomes asymmetric as that probability deviates from 1/2. There are two methods to define the two-tailed p-value. One method is to sum the probability that the total deviation in numbers of events in either direction from the expected value is either more than or less than the expected value. The probability of that occurring in our example is 0.0437. The second method involves computing the probability that the deviation from the expected value is as unlikely or more unlikely than the observed value, i.e. from a comparison of the probability density functions. This can create a subtle difference, but in this example yields the same probability of 0.0437. In both cases, the two-tailed test reveals significance at the 5% level, indicating that the number of 6s observed was significantly different for this die than the expected number at the 5% level.