Binomial Distribution/Binomial and Sign Tests

Binomial Distribution

The binomial distribution models experiments in which a repeated binary outcome is counted. Each binary outcome is called a Bernoulli trial, or simply a trial. For example, if we toss five coins, each binary outcome corresponds to H or T, and the outcome of the experiment could count the number of T out of these five trials.

Notations and Definitions

We call Y the random variable counting the number of outcomes of interest, N the total number of trials, P the probability of obtaining the outcome of interest on each trial, and C a given number of outcomes. For example, if we toss four coins and count the number of heads, Y counts the number of heads, N = 4, and P = ½. If we want to find the probability of getting two heads out of four, then C = 2.

With these notations, the probability of obtaining C outcomes out of N trials is given by the formula

The term (NC) gives the number of combinations of C elements from an ensemble of N; it is called the binomial of N and C and is computed as

For example, if the probability of obtaining two heads when tossing four coins is computed as

the mean and standard deviation of the binomial distribution are equal to

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The binomial distribution converges toward the normal distribution for large values of N (practically, for P = 1/2 and N = 20, the convergence is achieved).

Binomial Test

The binomial test uses the binomial distribution to decide whether the outcome of an experiment in which we count the number of times one of two alternatives has occurred is significant. For example, suppose we ask 10 children to attribute the name “keewee” or “koowoo” to a pair of dolls identical except for their size and that we predict that children will choose keewee for the smaller doll. We found that 9 children out of 10 chose keewee. Can we conclude that children choose systematically? To answer this question, we need to evaluate the probability of obtaining 9 keewees or more than 9 keewees if the children were choosing randomly. If we denoted this probability by p, we find (from Equation 1) that

Assuming an alpha level of α = .05, we can conclude that the children did not answer randomly.

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