Editing Binomial test (section)

== Large samples ==
For large samples such as the example below, the [[binomial distribution]] is well approximated by convenient [[continuous distribution]]s, and these are used as the basis for alternative tests that are much quicker to compute, such as [[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.

The most usual (and easiest) approximation is through the standard normal distribution, in which a [[z-test]] is performed of the test statistic <math>Z</math>, given by

: <math>Z=\frac{k-n\pi}{\sqrt{n\pi(1-\pi)}}</math>

where <math>k</math> is the number of successes observed in a sample of size <math>n</math> and <math>\pi</math> is the probability of success according to the null hypothesis. An improvement on this approximation is possible by introducing a [[continuity correction]]:

: <math>Z=\frac{k-n\pi\pm \frac{1}{2}}{\sqrt{n\pi(1-\pi)}}</math>

For very large <math>n</math>, this continuity correction will be unimportant, but for intermediate values, where the exact binomial test doesn't work, it will yield a substantially more accurate result.

In notation in terms of a measured sample proportion <math>\hat{p}</math>, null hypothesis for the proportion <math>p_0</math>, and sample size <math>n</math>, where <math>\hat{p}=k/n</math> and <math>p_0=\pi</math>, one may rearrange and write the z-test above as

: <math> Z=\frac{ \hat{p}-p_0 } { \sqrt{ \frac{p_0(1-p_0)}{n} } }</math>

by dividing by <math>n</math> in both numerator and denominator, which is a form that may be more familiar to some readers.