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`).
This material is intended as a summary. Use your textbook for detail explanation.
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