Question

A sports agent wants to find out whether the median annual income of players (athletes) in each different type of sport is uniform. The following table shows the annual income (in Million US $) of 52 players (athletes) for 4 types of sports

Football	1.2	1.6	2.2	3.4	1.4	1.6	5.0	4.0	3.0	2.9	7.6	4.8
Volley Ball	1.1	7.7	6.7	8.4	5.6	3.4	5.9	3.4	2.3	4.5	3.3	6.7	9.7	12.3	14.0
Basket Ball	4.0	2.3	1.1	0.4	0.9	2.3	1.5	3.4	5.6	3.9	5.6
Badminton	4.0	3.3	2.2	3.5	4.4	3.1	1.3	1.2	1.7	1.6	1.8	1.9	2.1	2.2

Question
1. How to design the hypothesis?
2. Find the critical value of the test at a significance level of 1%
3. What is the H statistic of the problem?
4. Using the available critical values, what is the conclusion from testing the hypothesis?

Answer 1

Kruskal-Wallis Rank Test for Differences in Medians

Sample	Value	Rank		Football	Volley Ball	Basket Ball	Badminton
Basket Ball	0.4	1		1.2	1.1	4	4
Basket Ball	0.9	2		1.6	7.7	2.3	3.3
Volley Ball	1.1	3.5		2.2	6.7	1.1	2.2
Basket Ball	1.1	3.5		3.4	8.4	0.4	3.5
Football	1.2	5.5		1.4	5.6	0.9	4.4
Badminton	1.2	5.5		1.6	3.4	2.3	3.1
Badminton	1.3	7		5	5.9	1.5	1.3
Football	1.4	8		4	3.4	3.4	1.2
Basket Ball	1.5	9		3	2.3	5.6	1.7
Football	1.6	11		2.9	4.5	3.9	1.6
Football	1.6	11		7.6	3.3	5.6	1.8
Badminton	1.6	11		4.8	6.7		1.9
Badminton	1.7	13			9.7		2.1
Badminton	1.8	14			12.3		2.2
Badminton	1.9	15			14
Badminton	2.1	16
Football	2.2	18
Badminton	2.2	18
Badminton	2.2	18
Volley Ball	2.3	21
Basket Ball	2.3	21
Basket Ball	2.3	21
Football	2.9	23
Football	3	24
Badminton	3.1	25
Volley Ball	3.3	26.5
Badminton	3.3	26.5
Football	3.4	29.5
Volley Ball	3.4	29.5
Volley Ball	3.4	29.5
Basket Ball	3.4	29.5
Badminton	3.5	32
Basket Ball	3.9	33
Football	4	35
Basket Ball	4	35
Badminton	4	35
Badminton	4.4	37
Volley Ball	4.5	38
Football	4.8	39
Football	5	40
Volley Ball	5.6	42
Basket Ball	5.6	42
Basket Ball	5.6	42
Volley Ball	5.9	44
Volley Ball	6.7	45.5
Volley Ball	6.7	45.5
Football	7.6	47
Volley Ball	7.7	48
Volley Ball	8.4	49
Volley Ball	9.7	50
Volley Ball	12.3	51
Volley Ball	14	52

1)

Ho: medianfootball=medianvolleyball=medianbasketball=medianBadminton

Ha: at least one median is different from other

------------

2)

Number of Groups   4
α=0.05
Critical Value   7.8147

3)

Kruskal-Wallis Rank Test for Differences in Medians
Data
Level of Significance	0.05		Group	Sample Size	Sum of Ranks	Mean Ranks
			r 4 types of sports	12	291	24.25
Intermediate Calculations			15	575	38.3333333
Sum of Squared Ranks/Sample Size	39614.73			11	239	21.7272727
Sum of Sample Sizes	52			14	273	19.5
Number of Groups	4
Test Result
H Test Statistic	13.4880

H Test Statistic =13.4880

4) since, test stat > critical value, reject the null hypothesis

there is enough evidence to conclude that median annual income of players (athletes) in each different type of sport is not uniform