*Example:*

*Characteristics:*

This is not a test, but a distribution. The above probabilities are
*one-sided* probabilities. The binomial distribution is most
popular when working with nominal data. When there are two observations
possible, with and without a certain characteristic, each with a probability
of 1/2, the two-sided variant of this "test" is identical to the
two-sided
Sign-test.

*H0:*

The probability of observing a certain characteristic is equal to *p*.

*Assumptions:*

The observations are binomial distributed.

*Scale:*

Nominal

*Procedure:*

Count the number of observations, *x*, from in the sample with a total
size *N*.

*Level of Significance:*

The *one-tailed* level of significance is calculated as (if
*x < p*N*):

p <= Sum (i=0 to x) {*N*!/(i!*(*N*-i)!)*p**i*(1-p)**(N-i)}

(with k! = k*(k-1)*(k-2)*...*1 is the factorial of k and 0! = 1)

If *x > p*N*, sum from *x* to *N*.

*Approximation:*

If N*p > 5 *and* N*(1-p) > 5, the distribution of:

Z = ( | *x - N*p* | - 0.5)/sqrt( *N * p* * (1-*p*) )

can be approximated with a
Standard Normal distribution.

*Remarks:*

As *p* is only rarely known, this test is of limited use only. However,
there is one application that can be very handy. Assume that a number of tests
are applied to a group of data-sets that for some reason cannot be pooled
readily (e.g., vowel formant measurements from a limited number of speakers),
and each test individualy only reaches, e.g., a significance level of p <= 0.1,
which is not convincing. If H0 is true, a "positive" test result at p <= 0.1
is expected to be observed with *p* <= 0.1. It can now be tested whether
the *number* of positive test results is large enough to reject H0 at
a significance level <= 0.05 or smaller.

This procedure has only an exploratory value. If it shows that H0 must be rejected,
there should be a better test that will prove it on the data-sets themselves.
Furthermore, this kind of statistics is not publishable.

In this example we calculate the exact probabilities upto *N* = 100.

Note that we give the *one-tailed* levels of significance.

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