2. Non parametric test - Kruskal-wallis test for the following data
24,32,39,34,30,40,36,38
28,31,35,22,38,28,32,28
34,31,28,24,31,26,28,34, Significance Level `alpha=0.05` and One-tailed test

Solution:
Step-1: Take the hypothesis
Null Hypothesis `H_0` : All groups are equal

Alternative Hypothesis `H_1` : Atleast one group is not equal

Step-2: Ranking all group values

Size in Ascending Order	Rank	Name of related sample A for sample-1 B for sample-2 C for sample-3	Rank for A	Rank for B	Rank for C
22	1	B		1
24	2.5	C			2.5
24	2.5	A	2.5
26	4	C			4
28	7	C			7
28	7	C			7
28	7	B		7
28	7	B		7
28	7	B		7
30	10	A	10
31	12	C			12
31	12	C			12
31	12	B		12
32	14.5	B		14.5
32	14.5	A	14.5
34	17	C			17
34	17	C			17
34	17	A	17
35	19	B		19
36	20	A	20
38	21.5	B		21.5
38	21.5	A	21.5
39	23	A	23
40	24	A	24
Total			132.5	89	78.5

The rank total for A is `R_1=132.5`

The rank total for B is `R_2=89`

The rank total for C is `R_3=78.5`

Step-3: Compute test statistic
`sum R_j^2/n_j=(132.5)^2/8+(89)^2/8+(78.5)^2/8=3954.9375`

n = total number of samples = 24

`H=12/(n(n+1)) sum R_j^2/n_j - 3(n+1)`

`=12/(24(24+1)) (3954.9375) - 3(24+1)`

`=12/600 * (3954.9375) - 75`

`=4.0987`

Step-4: `alpha=0.05`

Step-5: Compute the degrees of freedom (df).
`df=(3-1)=2`

Step-6:
The Critical value of chi-square is `chi^2(0.05,2)=5.9915`

Since the computed `H`(4.0987) < critical `chi^2`(5.9915)

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

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