The binomial test of significance is a kind of probability test that is based on various rules of probability. It is used to examine the distribution of a single dichotomous variable in the case of small samples. It involves the testing of the difference between a sample proportion and a given proportion.
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This document will discuss certain terms used in the binomial test of significance so that the reader can better understand.
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The calculation of the binomial test of significance is done in the following manner:
Let us assume that p(r) is to be calculated. p(r) is the probability that the researcher will obtain an ‘r’ observation in one category of a dichotomy and the researcher will obtain an ‘n – r’ observations in the other category, when the sample size is n. In the binomial test of significance, if ‘p’ is the probability that the researcher will obtain the first category, and ‘q’ is equal to ‘1 – p,’ then it denotes the probability that the researcher will obtain the second category. The formula is:
p(r) = nCr*pr*qn-r = (n!prqn-r)/(r!(n-r)!)
In this formula, nCr is denoted as the number of combinations of n things drawn from ‘r’ at a time.
The normal approximation of the binomial test of significance is made in following manner:
When the size of the sample ‘n’ is greater than 25, and the probability ‘p’ of obtaining the first category is around 0.50, then product of the term ‘npq’ is at least 9. In this case, the binomial distribution approximates the normal distribution in the binomial test of significance. Because of this approximation, a normal curve z-test is used as an approximation. This formula of the approximation of the binomial test of significance is given by the following:
z = ((r[+,-].5) – np)/SQRT(npq)
The binomial test of significance can be done in SPSS. This non parametric test is calculated in SPSS by selecting “Non Parametric test” from the “analyze” menu and then selecting “binomial test of significance.”
There are certain assumptions that are made in the binomial test of significance. The assumptions are the following:
A dichotomous kind of a distribution is assumed. In other words it is assumed that the variable of interest is considered to be dichotomous in nature where the two values are mutually exclusive and mutually exhaustive in all cases being considered. The word ‘binomial’ suggests that the variables of interest should be dichotomous in nature as the term ‘binomial’ means two.
Since this binomial test of significance does not involve any parameter and therefore is non parametric in nature, the assumption that is made about the distribution in the parametric test is therefore not assumed.
In the binomial test of significance, it is assumed that the sample that has been drawn from some population is done by the process of random sampling. The sample that is conducted by the researcher is therefore a random sample.