*Example:*

**
Prob( p1 = p2 | **

*Characteristics:*

One of the prime characteristics of this test is that there are better
tests whenever it could be applied, most notably, tests based on
>
Chi-square statistics. This test is only included for completeness.

*H0:*

*p1* = *p2*

*Assumptions:*

Both samples are independent and the test parameter has a
Binomial distribution.

*Scale:*

Nominal

*Procedure:*

Count the elements in both samples that have the distinctive characteristic
(i.e., *x1* and *x2*).

*Level of significance:*

Determine: *p1* = *x1* / *N1*,
*p2* = *x2* / *N2* , and
*p* = ( *x1* + *x2* ) / ( *N1* + *N2* )

The level of significance can only be approximated.

*Approximation:*

If *x1*, *x2*, ( *N1* - *x1* ), and
(*N2* - *x2* ) are all larger than 5, then the distribution of:

Z = (*p1* - *p2*)/
sqrt( *p* * ( 1 - *p*) * ( 1 / *N1* + 1 / *N2* ))

can be approximated with a
Standard Normal distribution (e.g., Z =

*Remarks:*

As we cannot calculate the probabilities of a test result when the
*Standard Normal* approximation is invalid, we will use this
approximation for all values of x and N. However, the probabilities are
labeled with a * when the *Standard Normal* approximation is not
valid.

When the samples are *not* independent, use another test (e.g.,
McNemar's test).

Anyhow, this is one test you will not need. If you are tempted to use this
test of *Binomial Proportions*, pause and think again. It is *very*
unlikely that you cannot think of a better test to apply, e.g., one based
on
>
Chi-square statistics.

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