# Mann-Whitney U Test - Numerical

The numerical below is an example of the process of performing the Mann-Whitney U Test. This approach can be extended to similar cases by replacing the data.

## Test the hypothesis of no difference between the groups by using Mann- Whitney U test with the help of the following data:

 Scores obtained by educated women on attitude towards health 59, 60, 61, 64, 63, 51, 52, 55, 53, 57, 56, 54, 52, 64, 56, 54, 58, 56, 62, 60, 57 Scores obtained by uneducated women on attitude towards heath 53, 63, 63, 58, 60, 62, 66, 65, 64, 68

### How to perform the Mann-Whitney U Test?

#### Step 1: Hypotheses Formulation

• For the given problem, Null Hypothesis, H0, is that the two groups of women (educated and uneducated) do not differ systematically from each other, in their attitude towards health (as they were randomly selected from the same population).
• Alternative Hypothesis, H1 – the two groups of women (educated and uneducated) differ systematically from each other, in their attitude towards health.

#### Step 2: Ranking

• Rank the data taking both groups together (see table)
•  Educated Women Score Rank 59 16 60 18 61 20 64 27 63 24 51 1 52 2.5 55 8 53 4.5 57 12.5 56 10 54 6.5 52 2.5 64 27 56 10 54 6.5 58 14.5 56 10 62 21.5 60 18 57 12.5
 Uneducated women Score Rank 53 4.5 63 24 63 24 58 14.5 60 18 62 21.5 66 30 65 29 64 27 68 31

#### Step 3: Find sum of ranks for smaller sample

• ∑〖๐_๐ข=223.5〗

#### Step 4: Find sum of ranks for larger sample

• ∑〖๐_๐=272.5〗

#### Step 5: Calculate U for both groups

• ๐_๐ = ๐_๐ × ๐_๐ข + (๐_๐ × (๐_๐ + 1)) / 2 - ∑๐_๐ = 21 × 10 + (21 × (21 + 1)) / 2 - 272.5 = 210 + 231 - 272.5 = 168.5
• ๐_๐ข = ๐_๐ × ๐_๐ข+ (๐_๐ข × (๐_๐ข + 1)) / 2 - ∑๐_๐ข = 21 × 10 + (10 × (10 + 1)) / 2 - 223.5 = 210 + 55 - 223.5 = 41.5

#### Step 6: Determine the significance of U

• In this case the direction of the alternative Hypothesis is not important because we are only checking if there is a difference, therefore, we are making a two tailed decision. The critical value of U for ne=21 and nu=10, for a two tailed decision with ฮฑ=0.05 is not given in the table. This is because as the n for one of the groups increases beyond 20, the sampling distribution of U can be treated as normal. Therefore, we need to perform a z test as follows:
• ๐ง = (๐_๐ - ((๐_๐ × ๐_๐ข) / 2)) / √((๐_๐×๐_๐ข × (๐_๐ + ๐_๐ข + 1)) / 12) = (168.5 - (( 21 × 10 ) / 2)) / √((21×10×(21+10+1))/12) = 63.5/√560 = 2.68

#### Step 7: Interpretation of Result

• At ฮฑ=0.05, z value is 1.96 for two-tailed test. Since 2.68 > 1.96, the z is significant. Therefore, we decide to reject the null Hypothesis, as absolute value of obtained z is greater than the critical value of 1.96. Since we reject Null Hypothesis, we accept alternative hypothesis – the two groups of women (educated and uneducated) differ systematically from each other, in their attitude towards health.

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