## Wald- Wolfowitz Run Test for two variable

In many
situation, we are interested to whether the data collected for statistical analysis is random or not. In most all statistical inference we assume that the
data is in a random sample. If the randomness of a sample is suspected, we test
it’s randomness before, going to statistical analysis. In detail, There are
many situations where we may investigate the assumptions of randomness. The
data of statistical quality control analysis and regression analysis are the
most important situations in while we must be sure about randomness. Run test is an non-parametric test.

Procedure for investigating the randomness are based on the number and nature of runs present in a data of interest.

**Length
of run: **The run which contains maximum number of item or symbols is
called length of run.

### Assumptions of run test

The data
given for statistical analysis consists of a sequence of observations recorded
in the order of their occurrence can be
classified into two mutually exclusive types.

Let us
consider,

n = the
sample size

n1= number of observations in one type.

n2= number of observations in anathoter type.

### Hypothesis of run test

H0 : The
pattern of accuracy of the two types of
observation is determined by a random process.

For two
tail : H_{1} : The pattern..... is not random.

For one
tail : H_{1} : The pattern..... is not random.

(because
there are too few runs to chance)

H_{1}
: The pattern........ is not random

(because
there are too may runs to be attributed to chance)

### Test statistic of Run test

Test statistic is r.

r = the
number of run

**Conclusion**: If r is
less than or equal to lower critical value or greater than or equal to upper critical
value then we reject H0.

Run test f or
large sample (n_{1} or n_{2} > 20)

The mean
of r from the distribution is,

**Comment:** If the calculated value falls in critical region then we may reject H0 ;
other wise we may accept it.

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