# 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.

### Definition

The binomial test checks whether the frequency distribution of a variable with two values/categories in the sample corresponds to the distribution in the population.

## Hypotheses in binomial test

The hypothesis of the binomial test results in the one-tailed case to

• Null hypothesis: The frequency distribution of the sample corresponds to that of the population.
• Alternative hypothesis: The frequency distribution of the sample does not correspond to that of the population.

Thus, the non-directional hypothesis only tests whether there is a difference or not, but not in which direction this difference goes.

In the two sided case, the aim is to investigate whether the probability of occurrence of an expression in the sample is greater or less than a given or true percentage.

In this case, an expression is defined as "success" and it is checked whether the true "probability of success" is smaller or larger than that in the sample.

The alternative hypothesis then results in:

• Alternative hypothesis: True probability of success is smaller/larger than specified value

## 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.

Alternative hypothesis p

## 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.

Marketing student Gender
1 female
2 male
3 female
4 female
5 female
6 male
7 female
8 male
9 female
10 female

### Binomial test with DATAtab:

Calculate the example in the statistics calculator. Simply add the upper table including the first row into the hypothesis test calculator.

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 be retained. 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).

Cite DATAtab: DATAtab Team (2023). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net