The **binomial test** is an exact test base on the binomial distribution while the **proportion z-test** uses the normal approximation.

For a small sample please use the binomial test.

Hypotheses

H_{0}: p ≥ = ≤ P_{0}

H_{1}: p < ≠ > P_{0}

Test statistic

Normal distribution

__Sample data__

validation message

Binomial distribution - the probability for event within each group is identical | |

Expected difference d between the populations's average is known |

p the sample probabilities or x the number of successes | |

n - Sample size |

The following R code should produce the same results:

The prop.test uses the Chi-squared statistic with one degree of freedom, which is **exactly** as using the Z test: z^{2}=χ^{2}(1)