## Median Test

The
median test is one of the simplest and most widely used procedure for testing
the Ho that the independent sample have been drawn from the population with
equal median value. This process is a non-parametric testing procedure.

### Assumptions of Median Test

- The date consist of two independent random samples x1,x2,...,xn and y1,y2,...,yn of size n1 and n2 respectively.
- Scale of measurement is at least ordinal.
- Samples are drawn independently.
- If the population have the same median, then for each population, the probability P is the same that an observed value will exceed the grand median.

### Hypothesis of Median Test

H0 : The
two population are identical.

Vs

H_{1} : Ho is not true.

### Test
Statistic Median Test

If the
two population have the same median,
we would expect about half of observation in each of the two samples to be
above the common median and about hlf to be low.

Under
the Ho the two population median are equal, we may estimate the common
parameter by estimating. the median of the sample values by combining two
samples of size (n_{1} + n_{2}). If Ho is true we expect about half of
observation in each sample to fall above the combine sample median. And about
half to fall below. The median test allow us to conclude on the basis of sample data whether it is likely that
the Ho is false.

The
usual test statistic is,

Here,

A=
number of observation from sample -1 falling above the median.

B=
number of obs from sample-2 falling above me

n1 = 1^{st}
Sample size

n2= 2^{nd}
sample size

### Conclusion

If the combined sample size N = (n1+n2) is such
that Np and N (1-P) are larger than 5 and if Ho is true is approximately
distributed as mean zero and variance
one therfore the conclusion can be made as usual by following the level of
significance q= 0.0s the CV is ± 1.96
and for Î±= 0.01 the CV is ± 2.58

### Mathematical Example of Median Test

The following data give the marks obtained in a test of a certain subject by two groups A and B consisting of 32 and 16 student respectively. Test whether these data provide sufficient evidence to indicate that the median of the two population from which two group of students are selected are same. Consider Î±= 0.05

Marks obtained by Group A Student.

25,25,17,26,18,30,24,21,13,30,20,23,26,19,20,37,9,17,37,20,11,32,16,31,46,20,25,17,36,54,8,26

31,21,38,19,38,41,62,28,43,42,30,20,29,13,32,30

**Solution: **

Ho : The
population median of marks of two groups are equal.

Vs

H_{1}
: Ho is not true.

Combining
two sample :

8,9,11,12,13,16,16,17,17,17,18,19,20,20,20,20,20,20,21,21,23,24,25,25,25,26,26,28,29,30,30,30,31,31,39,39,36,37,37,38,38,41,42,43,46,54,68

Median = (25+25)/2=25

From A
the number of marks above median mark is 12, A= 12

From B
the number of marks above median mark 12 ; B= 12

The calculated value is in the critical Region then we may reject our null hypothesis at 5% level of significance. Thus, Median are not equal.

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