The document describes using the Mann-Whitney U test to analyze whether students who had breakfast had better reaction times than those who did not. It ranks the reaction time data for students with and without breakfast, calculates the test statistics U_n and U_b, and compares the smaller value U_n to the critical value. Since U_n is greater than the critical value, the null hypothesis that the samples are from identical populations is accepted. Therefore, there is insufficient evidence at the 5% level that students who had breakfast had better reaction times.

1. Mann-Whitney U Test

2. Question
 A experiment was carried out to see if having breakfast effect the
reaction time of a student
 Use the Mann-Whitney U test to investigate at 5% significance
level if there is enough evidence to show that students who have
breakfast have a better dominant hand reaction time
Breakfast (sec) 10 3 5.5 14 15 6 5 8 7 7 5 4 6
No breakfast (sec) 2.5 9 5 6 7 8

3. 1.Hypostasis
▪ Ho: samples are taken from identical populations
▪ Hi: samples are taken from identical populations. People
who have breakfast have a better dominant hand reaction
time

4. 2.Rank all values
▪ Rank all the values from 1 assigning the lowest rank to the
smallest value
Breakfast (sec) 10 3 5.5 14 15 6 5 8 7 7 5 4 6
No breakfast (sec) 2.5 9 5 6 7 8

5. 2.Rank all values
▪ Rank all the values from 1 assigning the lowest rank to the
smallest value
Breakfast (sec) 10
17
3
2
5.5
7
14
18
15
19
6
9
5
5
8
14.5
7
12
7
12
5
5
4
3
6
9
No breakfast (sec) 2.5
1
9
16
5
5
6
9
7
12
8
14.5
Rank values

6. 3.Calculate Tn & Tb
▪ Tn= the sum values of the ranks of the no breakfast values
▪ Tb= the sum values of the ranks of the breakfast values
▪ Tn: 1 + 16 + 5 + 9 + 12 + 14.5 = 57.5
▪ Tb: 17 + 2 + 7 + 18 + 19 + 9 + 5 + 14.5 + 12 + 12 + 5 + 3
+9 = 132.5
Breakfast (sec) 10
17
3
2
5.5
7
14
18
15
19
6
9
5
5
8
14.5
7
12
7
12
5
5
4
3
6
9
No breakfast (sec) 2.5
1
9
16
5
5
6
9
7
12
8
14.5

7. 4.Calculate Mann-Whitney U values
▪ 𝑈𝑛 = 𝑇𝑛 −
1
2
𝑚 𝑚 + 1
– M = the number of students that had no breakfast
– M= 6
▪ = 57.5 −
1
2
× 6 6 + 1
▪ Un = 36.5
▪ 𝑈𝑏 = 𝑇𝑏 −
1
2
𝑛 𝑛 + 1
– N= the number of students that had breakfast
– N= 13
▪ = 132.5 −
1
2
× 13 13 + 1
▪ Ub = 41.5

8. 5.Check Mann-Whitney U values
▪ To check if you got the correct values in step 4 the following
equation is used
▪ 𝑈𝑛 + 𝑈𝑏 = 𝑚𝑛
▪ 36.5 + 41.5 = 78
▪ 6 × 13 = 78
▪ This means you got the correct values in step 4

9. 6.Test statistic
▪ The test statistic is the smaller value of U out of Un and Ub
▪ Un = 36.5
▪ Ub = 41.5
▪ So Un is used for the test statistic value as it is smaller
– U = 36.5

10. 7.Reject or accept the Ho
▪ Reject Ho if U ≤ critical value
▪ To calculate the critical value us the table provided in the formula
book
▪ Use the one tail 5% to find the critical value
▪ Critical value = 19
▪ i.e. reject Ho if U ≤ 19
▪ Because U = 36.5 which is bigger than 19 we accept the Ho
hypostasis

11. 8.Conclusion
▪ Because we accepted the Ho hypothesis the conclusion will be
– There is insufficient evidence at the 5% level to suggest that
students that have breakfast have a better dominant hand
reaction time.