Question

1. A store sells three different clothing designs and has recorded sales of each over five different 24-hour periods:

Design A         Design B         Design C

     16                    33                    23

     18                    31                    27

     19                    37                    21

     17                    29                    28

     13                    34                    25

Use the Kruskal-Wallis H test and the Chi-Square table at the 0.05 level to compare the three designs.

Answer 1

Design A	Rank A	Design B	Rank B	Design C	Rank C
16	2	33	13	23	7
18	4	31	12	27	9
19	5	37	15	21	6
17	3	29	11	28	10
13	1	34	14	25	8
Total	15	Total	65	Total	40

Null Hypothesis(H0):

All three samples come from identical populations.

Alternative Hypothesis(H1):

At least one sample comes from the different population than the others.

Test statistic:

Total sample size, n =15

n₁ =n₂ =n₃ =5

Formula: Test statistic, H =[12/(n(n+1))*​​​​​] - 3(n+1)

(where, i =1, 2, 3)

H =[12/(15*16)*((15²+65²+40²)/5)] - 3(16) =(0.05*1210) - 48 =12.5

Thus, the test statistic, H =12.5

Critical value:

Kruskal-Wallis H statistic follows Chi-square distribution.

Degrees of freedom, df =Number of samples - 1 =3 - 1 =2

Significance level given =0.05

At 0.05 significance level and at df =2, the Chi-square critical value is: =5.991

Conclusion:

H: 12.5 > : 5.991

Since the test statistic is greater than the critical value, we reject the null hypothesis(H0) at 0.05 significance level.

Thus, there is a sufficient evidence to claim that at least one sample comes from the different population than the others.