2. Non parametric test - Mann whitney U test for the following data
8,7,6,2,5,8,7,3
9,8,7,8,10,9,6,41, Significance Level `alpha=0.05` and One-tailed test
Solution:
Step-1: Take the hypothesis
Null Hypothesis `H_0` : The two populations are equal
Alternative Hypothesis `H_1` : The two populations are not equal
Step-2: Ranking all group values
First we assign ranks to all observations using low to high ranking process in the combined sample.
| Size in Ascending Order | Rank | Name of related sample
A for sample1, B for sample2 | Rank for A | Rank for B |
| 2 | 1 | A | 1 | |
| 3 | 2 | A | 2 | |
| 5 | 3 | A | 3 | |
| 6 | 4.5 | B | | 4.5 |
| 6 | 4.5 | A | 4.5 | |
| 7 | 7 | B | | 7 |
| 7 | 7 | A | 7 | |
| 7 | 7 | A | 7 | |
| 8 | 10.5 | B | | 10.5 |
| 8 | 10.5 | B | | 10.5 |
| 8 | 10.5 | A | 10.5 | |
| 8 | 10.5 | A | 10.5 | |
| 9 | 13.5 | B | | 13.5 |
| 9 | 13.5 | B | | 13.5 |
| 10 | 15 | B | | 15 |
| 41 | 16 | B | | 16 |
| Total | | | 45.5 | 90.5 |
The rank total for A is `R_1=45.5`
The rank total for B is `R_2=90.5`
We have `n_1=8` and `n_2=8`
Step-3: Compute test statistic
test statistic using `R_1`
`U_1 = n_1 * n_2 + (n_1(n_1+1))/2 - R_1`
`=8 * 8 + (8(8+1))/2 - 45.5`
`=64 + 36 - 45.5`
`=54.5`
`mu_U=(n_1 n_2)/2=(8*8)/2=32`
`sigma_U=sqrt((n_1 n_2 (n_1+n_2+1))/12)=sqrt((8*8*(8+8+1))/12)=9.4868`
According the limits of acceptance region, keeping in view 10% level of significance.
As the z value for 0.45 of the area under the normal curve is 1.64, we have the following limits of acceptance region.
Upper limit = `mu_U + 1.64 sigma_U=32 + 1.64 * 9.4868=47.5584`
Lower limit = `mu_U - 1.64 sigma_U=32 - 1.64 * 9.4868=16.4416`
The value of `U_1` is 54.5 which is not in the acceptance region, we reject the null hypothesis and conclude that the two samples do not come from identical populations at 10% level.
test statistic using `R_2`
`U_2 = n_1 * n_2 + (n_2(n_2+1))/2 - R_2`
`=8 * 8 + (8(8+1))/2 - 90.5`
`=64 + 36 - 90.5`
`=9.5`
The value of `U_2` does not lies in the acceptance region and as such our conclusion remians the same.
This material is intended as a summary. Use your textbook for detail explanation.
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