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Chapter 19: The Binomial Test. The Binomial Test. The binomial test provides a method for testing hypotheses about population proportions for populations consisting of binomial data.
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The Binomial Test • The binomial test provides a method for testing hypotheses about population proportions for populations consisting of binomial data. • Binomial data exist when the measurement procedure classifies individuals into exactly two distinct categories.
The Binomial Test (cont.) • Traditionally, the two categories are identified as A and B, and the population proportions are identified as p(A) = p and p(B) = q. • The null hypothesis specifies the values of p and q for the population. • For example, when testing whether or not a coin is balanced, the null hypothesis would state that the coin is balanced or, p(Heads) = p = .50, and p(Tails) = q = .50.
The Binomial Test (cont.) • The sample data for the binomial test consist of a sample of n individuals each of whom is classified in category A or B. • The sample statistic, X, is simply the number of individuals classified in category A. • The logic underlying the binomial test is identical to the logic for the original z‑score test or the t‑statistic hypothesis tests.
The Binomial Test (cont.) • The test statistic compares the sample data with the hypothesized value for the population. • If the data are consistent with the hypothesis, we conclude that the hypothesis is reasonable. • However, if there is a large discrepancy between the data and the hypothesis, we reject the hypothesis.
The Binomial Test (cont.) • When the values of pn and qn are both greater than or equal to 10, the binomial distribution is approximately normal with a mean of μ = pn and a standard deviation of σ = npq. • In this case, the binomial test can be conducted by transforming the X value from the sample into a z-score and then using the unit normal table to determine critical values. • If the z-score is only slightly into the critical region, you should check both real limits for X to ensure that the entire score is beyond the critical boundary.
The Sign Test • The sign test is a special application of the binomial test used to evaluate the results from a repeated-measures research design comparing two treatment conditions. • The difference score for each individual is classified as either an increase (+) or a decrease (–) and the binomial test evaluates a null hypothesis stating that increases and decreases are equally likely: p(+) = p(–) = 1/2.
