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Binomial Test of Significance

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

The binomial test of significance involves the determination of the probability of getting r observations in one category of a dichotomy, and (n - r) observations in the other category when a sample of size ‘n’ is given.
 Let p = the probability of getting the first category and let q = 1 - p = the probability of getting the other category. The binomial test of significance formula is:
 
p(r)binomial = nCr*pr*qn-r = (n!prqn-r)/(r!(n-r)!)
 
Let us assume that a particular city is 60% Democratic (p = 0.60) and 40% non-Democratic ( q = 0 .40). A particular fraternal organization in that city is being sampled and it was found that in a sample of 100 people (n = 100), there are 70 Democrats (r = 70). Then the probability of getting a sample distribution was calculated as strong as or stronger than the observed distribution by the binomial test of significance. The answer will be the sum of p(70)binomial, p(71)binomial, ....p(100)binomial where p(70)binomial = (100!.6070 .4030)/(70!30!).
 
Here, nCr is the number of combinations of ‘n’ units taken ‘r’ times.
In the binomial test of significance, when n is greater than 25, p is approximately 0.50 and the product of npq is at least 9, and 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 of significance. The formula is given by:
 
z = ((r[+,-].5) - np)/SQRT(npq) , here , r[+.-].5 means 0 .5 is added to r if r is smaller than np and is subtracted if r is larger than np.
 
For the above given example, 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. By using the table of areas under the normal curve, it is found that the value for z = 1.94 is .0264. Therefore, it can be stated that the hypothesis about the fraternal organization having more Democrats than would be expected for the city is significant at .0264 level, which is below the conventional .05 cutoff used in social science.
The following are some assumptions which are made while conducting a binomial test of significance:
A distribution should always be a Dichotomous distribution.
The binomial test of significance assumes that the variable under consideration is dichotomous, and has two values which are mutually exclusive and exhaustive for all cases. Events are said to be mutually exclusive if they do not occur at the same time.
There is a random sampling of the sample.
In the binomial test of significance, like all significance tests, random sampling is assumed.
In SPSS, the binomial test of significance is operated by selecting “Statistics,” “Nonparametric Statistics,” and “Binomial” from the menu. When the Binomial Test of Significance is selected, a dialog box appears, which we select and is where the Test Proportion (which is p in the formula above) is being set. If a test proportion is not being set, then the default p of .50 is used.

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

The binomial test of significance involves the determination of the probability of getting r observations in one category of a dichotomy, and (n - r) observations in the other category when a sample of size ‘n’ is given.

 Let p = the probability of getting the first category and let q = 1 - p = the probability of getting the other category.  The binomial test of significance formula is:

p(r)binomial = nCr*pr*qn-r = (n!prqn-r)/(r!(n-r)!)

Let us assume that a particular city is 60% Democratic (p = 0.60) and 40% non-Democratic ( q = 0 .40). A particular fraternal organization in that city is being sampled and it was found that in a sample of 100 people (n = 100), there are 70 Democrats (r = 70).  Then the probability of getting a sample distribution was calculated as strong as or stronger than the observed distribution by the binomial test of significance. The answer  will be the sum of p(70)binomial, p(71)binomial, ....p(100)binomial where p(70)binomial = (100!.6070 .4030)/(70!30!).

Here, nCr is the number of combinations of ‘n’ units taken ‘r’ times.

In the binomial test of significance, when n is greater than 25, p is approximately 0.50 and the product of npq is at least 9, and 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 of significance. The formula is given by:

z = ((r[+,-].5) - np)/SQRT(npq) , here , r[+.-].5 means 0 .5 is added to r if r is smaller than np and is subtracted if r is larger than np.

For the above given example, 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. By using the table of areas under the normal curve, it is found that the value for z = 1.94 is .0264. Therefore, it can be stated that the hypothesis about the fraternal organization having more Democrats than would be expected for the city is significant at .0264 level, which is below the conventional .05 cutoff used in social science.

 

 

The following are some assumptions which are made while conducting a binomial test of significance:

A distribution should always be a Dichotomous distribution.

The binomial test of significance assumes that the variable under consideration is dichotomous, and has two values which are mutually exclusive and exhaustive for all cases. Events are said to be mutually exclusive if they do not occur at the same time.

There is a random sampling of the samples.

In the binomial test of significance, like all significance tests, random sampling is assumed.

In SPSS, the binomial test of significance is operated by selecting “Statistics,” “Nonparametric Statistics,” and “Binomial” from the menu. When the Binomial Test of Significance is selected, a dialog box appears, which we select and is where the Test Proportion (which is p in the formula above) is being set. If a test proportion is not being set, then the default p of .50 is used.

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