1. Non parametric test - Kruskal-wallis test for the following data
8,5,7,11,9,6
10,12,11,9,13,12
11,14,10,16,17,12
18,20,16,15,14,22, 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 D for sample-4	Rank for A	Rank for B	Rank for C	Rank for D
5	1	A	1
6	2	A	2
7	3	A	3
8	4	A	4
9	5.5	B		5.5
9	5.5	A	5.5
10	7.5	C			7.5
10	7.5	B		7.5
11	10	C			10
11	10	B		10
11	10	A	10
12	13	C			13
12	13	B		13
12	13	B		13
13	15	B		15
14	16.5	D				16.5
14	16.5	C			16.5
15	18	D				18
16	19.5	D				19.5
16	19.5	C			19.5
17	21	C			21
18	22	D				22
20	23	D				23
22	24	D				24
Total			25.5	64	87.5	123

The rank total for A is `R_1=25.5`

The rank total for B is `R_2=64`

The rank total for C is `R_3=87.5`

The rank total for D is `R_4=123`

Step-3: Compute test statistic
`sum R_j^2/n_j=(25.5)^2/6+(64)^2/6+(87.5)^2/6+(123)^2/6=4588.5833`

n = total number of samples = 24

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

`=12/(24(24+1)) (4588.5833) - 3(24+1)`

`=12/600 * (4588.5833) - 75`

`=16.7717`

Step-4: `alpha=0.05`

Step-5: Compute the degrees of freedom (df).
`df=(4-1)=3`

Step-6:
The Critical value of chi-square is `chi^2(0.05,3)=7.8147`

Since the computed `H`(16.7717) > critical `chi^2`(7.8147)

So we reject the null hypothesis (`H_0`).

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