Binomial test

The binomial test is a hypothesis test used when there is a categorical variable with two categories, e.g., gender with "male" and "female". The binomial test is used to 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%.

Equal proportions are a special case in which the expected probability is set to 50%.

The binomial test can therefore be used to test whether the proportion in a sample is consistent with a specified population proportion.

The test assumes a fixed number of independent trials, two possible outcomes for each trial, and the same probability of success on every trial. The sample should also be selected in a way that supports inference about the intended population.

Hypotheses in the 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 perform a binomial test, you need the sample size, the number of successes, and the expected probability of occurrence in the population. The exact test calculates p-values directly from the binomial distribution rather than using a normal approximation.

Binomial test example

A possible example of a binomial test is the question of whether the gender ratio among students specializing in marketing at University XY differs from that of all business students at University XY (the population).

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

Binomial test with numiqo

If you already have the number of events and total trials instead of individual observations, use the One-Sample Proportion Test Calculator.

Calculate the example with the numiqo statistics calculator. Simply copy the table above, including the first row, into the hypothesis test calculator.

numiqo gives you the following result for this example data:

Interpretation of a Binomial Test

With an expected test value of 55%, the two-sided exact p-value is 0.528. This is the probability, under the null hypothesis, of obtaining a result at least as incompatible with the expected proportion as the observed result. The p-value is above the significance level of 5%, so the result is not statistically significant.

Consequently, the null hypothesis is not rejected. The data do not provide sufficient evidence that the gender ratio in the marketing specialization (the sample) differs from that of all business administration students at XY University (the population). This does not prove that the two proportions are equal.

A confidence interval for the population proportion answers a different question: it shows a range of values compatible with the data and indicates the precision of the estimated proportion.