1. Non parametric test - Median test for the following data
79,86,40,50,75,38,70,73,50,40,20,80,55,61,50,80,60,30,70,50
85,80,50,55,65,50,63,75,55,45,30,85,65,80,55,75,65,50,75,62, Significance Level `alpha=0.05` and One-tailed test

Solution:
Step-1:State the hypothesis
`H_0`: There is no difference between Sample A and Sample B.

`H_1`: There is difference between Sample A and Sample B.


Step-2: Ranking all Sample values
First we assign ranks to all observations using high to low ranking process in the combined sample.

Size in Descending Order	Rank	Name of related sample A for sample1, B for sample2	Rank for A	Rank for B
86	1	A	1
85	2.5	B		2.5
85	2.5	B		2.5
80	5.5	A	5.5
80	5.5	A	5.5
80	5.5	B		5.5
80	5.5	B		5.5
79	8	A	8
75	10.5	A	10.5
75	10.5	B		10.5
75	10.5	B		10.5
75	10.5	B		10.5
73	13	A	13
70	14.5	A	14.5
70	14.5	A	14.5
65	17	B		17
65	17	B		17
65	17	B		17
63	19	B		19
62	20	B		20
61	21	A	21
60	22	A	22
55	24.5	A	24.5
55	24.5	B		24.5
55	24.5	B		24.5
55	24.5	B		24.5
50	30	A	30
50	30	A	30
50	30	A	30
50	30	A	30
50	30	B		30
50	30	B		30
50	30	B		30
45	34	B		34
40	35.5	A	35.5
40	35.5	A	35.5
38	37	A	37
30	38.5	A	38.5
30	38.5	B		38.5
20	40	A	40
Total			446.5	373.5

Step-3: Find Grand Median
Median `= (20^(th) "term" + 21^(st) "term" )/2=(62 + 61)/2= 61.5`

	Sample A	Sample B	Total
`>` Median	8(a)	12(b)	20(a+b)
`<=` Median	12(c)	8(d)	20(c+d)
Total	20(a+c)	20(b+d)	40(a+b+c+d)=n

Step-4: Computation of test statistic
`chi^2=(n(|ad-bc|-n/2)^2)/((a+b)(c+d)(a+c)(b+d))`

`chi^2=(40(|64-144|-40/2)^2)/(20xx20xx2020)`

`=(40(80-20)^2)/(20xx20xx20xx20)`

`=0.9`

Step-5: Compute the degrees of freedom (df).
`df=2-1=1`

Step-6:
The Critical value of chi-square is `chi^2(0.05,1)=3.8415`

Since the computed `chi^2`(0.9) < critical `chi^2`(3.8415)

So we accept the null hypothesis (`H_0`) and conclude that the two samples are identical.