Question

A shoe company wants to test if three groups of workers have different salaries.

Group (1): 23, 41, 54, 66, 78

Group (2): 45, 55, 60, 70, 72

Group (3): 18, 30, 34, 40, 44

Use kruskall Wallis test with α = 0.05.

Answer 1

Solution:

Kruskal Wallis test with α = 0.05.

• Step 1: State the null and alternative hypothesis

H₀: The median of three groups of workers salaries are equal

H₁: The median of three groups of workers salaries are not all equal.

Step 2:Computation of test statistics

The test statistic for the Kruskal Wallis test is denoted H and is defined as follows:

,

where k=the number of comparison groups, N= the total sample size, n_j is the sample size in the j^th group and R_j is the sum of the ranks in the j^th group.   

• First we need to arrange the given data in ascending order(smallest to largest).

Here the given samples are already arranged in ascending order.

	Samples ordered from smallest to largest
Group 1	Group 2	Group 3
23	45	18
41	55	30
54	60	34
66	70	40
78	72	44

• Now we assign the ranks to the ordered values and sum the ranks in each group.

	Samples ordered from smallest to largest			Ranks
Group 1	Group 2	Group 3	Group 1	Group 2	Group 3
		18			1
23			2
		30			3
		34			4
		40			5
41			6
		44			7
	45			8
54			9
	55			10
	60			11
66			12
	70			13
	72			14
78			15
			R1=44	R2=56	R3=20

Here the sum of the ranks will be always equal to n(n+1)/2=15(15+1)/2=120 which is also equal to 44+56+20=120

=(12 /(15(15+1)) [(44²/5)+(56²/5)+(20²/5)]-[3*(15+1)]

=[0.05*1094.4]-48

=54.72-48

=6.72

Step 3: Determination of critical value

For k=3,n1=5,n2=5,n3=5 and =0.05. the critical value is 5.780.

Step 4: Conclusion

Since observed value=6.72 is greater than critical value=5.780, we reject the null hypothesis and conclude that the three groups of workers have different salaries.