Binomial Test

Educational Research Applications

An educational researcher may believe that the probability of a child answering multiple-choice questions (with four options) correctly on a chemistry pretest is .25. That hypothesis could be tested using the binomial test, which would consider the number of successes (i.e., correctly answered questions) in a series of independently administered pretest questions. Another educational researcher may want to test the hypothesis that there is a .50 probability that an elementary school principal will support a newly proposed district policy. If so, principals could be independently sampled and interviewed to determine whether they supported the policy. A binomial test could be used to determine whether the number of successes (i.e., observed supporters) was (or was not) sufficient to reject the hypothesis that the probability of support was .50.

As a final example, consider an educational researcher who is interested in whether an intervention would increase the prosocial behavior of children with behavioral and emotional disturbances. The researcher could hypothesize that the probability of observing an increase in prosocial behavior for a child was .50. A binomial test could be used to determine whether the number of successes (i.e., number of children with observed improvements) in an independent sample was sufficient to reject the null hypothesis that the probability was .50. This final application of the binomial test would often be referred to as a sign test because it is based on counting up the number of positive and negative signed differences.

Hypothesis and Assumptions

The binomial test allows us to test the null hypothesis that the probability of success (π) is equal to some researcher specified value a. For a nondirectional test, the null hypothesis is H0: π = a, and for a directional test, the null hypothesis is either H0: π ≤ a or H0: π ≥ a. The probability calculations, which are based on the binomial distribution, assume:

• 1.
  Observations are sampled from a binary population (i.e., there are only two possible values for each observation, a success or a failure).
• 2.
  Each observation is independent implying that it is not affected by any of the other observations.
• 3.
  [Page 203]The probability of a success is fixed for the population.

Binomial Probabilities

For a binomial experiment where the probability of success in any one trial is π, the probability that there will be r successes in n trials is computed

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