Overview The binomial test is an exact probability test, based on the rules of probability, and is used to examine the distribution of a single dichotomy when the researcher has a small sample. It tests the difference between a sample proportion and a given proportion, for one-sample tests.

Contents Key concepts and terms Assumptions Frequently asked questions Bibliography




Key Concepts and Terms

• Computation of p(r)_binomial. In a binomial test we determine the probability of getting r observations in one category of a dichotomy and (n - r) observations in the other category, given sample size n. Let p = the probability of getting the first category and let q = 1 - p = the probability of getting the other. Recall from probability that _nC_r is the number of combinations of n things taken r at a time. The binomial formula is:
  p(r)_binomial = _nC_r*p^r*q^n-r = (n!p^rq^n-r)/(r!(n-r)!)
  • Example. Assume we know that a particular city is 60% Democratic (p = .60) and 40% non-Democratic (q = .40), and we sample a particular fraternal organization in that city and find 70 Democrats (r = 70) in a sample of 100 persons (n = 100). We may then ask if the 70:30 split found in our sample is significantly greater than the 60:40 split known to exist in the population. This question can be reformulated as asking, "What is the probability of getting a sample distribution as strong or stronger than the observed distribution?" The answer is the sum of p(70)_binomial, p(71)_binomial, ....p(100)_binomial. For instance, p(70)_binomial = (100!.60⁷⁰ .40³⁰)/(70!30!). It can be seen that when n is at all large, computation requires a computer. Some statistics books also print a table of p()_binomial for various levels of n, r, and p.

    p = .60
    q = .40
    r = 70
    n = 100
    npq = 24
    np = 60
    

• Normal approximation of the binomial test. When n is greater than 25 and p is near .50, and the product of npq is at least 9, then the binomial distribution approximates the normal distribution. In this situation, a normal curve z-test may be used as an approximation of the binomial test, using this formula (Siegel, 1956: 41):
  z = ((r[+,-].5) - np)/SQRT(npq)

  For the formula above, r[+.-].5 means .5 is added or subtracted as a continuity correction, penalizing for the binomial distribution being discrete whereas a true normal distribution is continuous. The correction reduces the difference between r and np. If r is greater than np then the difference is positive and .5 is subtracted. If r is less than np, then the difference is negative and .5 is added. Here, for the example above, z = (69.5 - 60)/SQRT(24) = 1.94. Thus the area under a normal curve as or more extreme than 1.94 corresponds to the chance of getting a 70:30 split or greater. Using a table of areas under the normal curve, for this example the area under the normal curve for z = 1.94 is .0264. Therefore we can say that the hypothesis that the fraternal organization has more Democrats than would be expected for the city is significant at the .0264 level, which is below the conventional .05 cutoff used in social science. (Not that this is a one-tailed hypothesis test. For the two-tailed hypothesis having to do with the fraternal organization being that different from the city proportion, larger or smaller, the level is doubled to .0528, which just misses being considered significant at the .05 level.) Note that the normal approximation is useful only when manual methods are necessary. SPSS, for instance, always computes the exact binomial test.




Assumptions

• Dichotomous distribution. The binomial test assumes the variable of interest is a dichotomy whose two values are mutually exclusive and exhaustive for all cases.

• Data distribution. The binomial test is non-parametric (it does not assume the normal distribution).

• Random sampling. Like all significance tests, random sampling is assumed.




Frequently Asked Questions

• Where is the binomial test found in SPSS?
  The binomial test is available in the SPSS NONPAR TESTS module of SPSS BASE. From the menu, select Statistics, Nonparametric Statistics, Binomial. When the Binomial Test dialog box appears, select one or more numeric test variables and set the Test Proportion (which is p in the formula above). If you do not set a test proportion, the default of .50 is used instead. The Options button allows you to output descriptive and quartile statistics, and to choose how to treat missing values.




Bibliography

• Siegel, Sidney (1956). Nonparametric statistics for the behavioral sciences. NY: McGraw-Hill. A standard reference work.

Copyright 1998, 2008, 2011 by G. David Garson.
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Last updated 4/2/2011.