Binomial test

The binomial test is a hypothesis test used when there is a categorical variable with two expressions, e.g., gender with "male" and "female". The binomial test can then check whether the frequency distribution of the variable corresponds to an expected distribution, e.g.:

• Men and women are equally represented.
• The proportion of women is 54%.

This is a special case when you want to test whether the frequency distribution of the variables is random or not. In this case, the probability of occurrence is set to 50%.

The binomial test can therefore be used to test whether or not the frequency distribution of a sample is the same as that of the population.

Hypotheses in binomial test

The binomial test is a statistical test used to determine whether the observed proportion of successes in a fixed number of trials is consistent with an expected probability. The hypotheses for the binomial test are:

Null Hypothesis:

The probability of success is equal to a specified value, p₀.

Alternative Hypothesis

The probability of success is different from (two-tailed), greater than (right-tailed), or less than (left-tailed) the specified value.

Two-tailed: Tests whether the observed proportion is significantly different from p₀ ​ in either direction.

Right-tailed: Tests whether the observed proportion is significantly greater than p₀

Left-tailed: Tests whether the observed proportion is significantly less than p₀ .

Binomial test calculation

To calculate a binomial test you need the sample size, the number of cases that are positive of it, and the probability of occurrence in the population.

Binomial test example

A possible example for a binomial test would be the question whether the gender ratio in the specialization marketing at the university XY differs significantly from that of all business students at the university XY (population).

Listed below are the students majoring in marketing; women make up 55% of the total business degree program.

DATAtab gives you the following result for this example data:

Interpretation of a Binomial Test

With an expected test value of 55%, the p-value is 0.528. This means that the p-value is above the signification level of 5% and the result is therefore not significant. Consequently, the null hypothesis must not be rejected. In terms of content, this means that the gender ratio of the marketing specialization (=sample) does not differ significantly from that of all business administration students at XY University (=population).