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

• ∑〖𝑅_𝑢=223.5〗

Step 4: Find sum of ranks for larger sample

• ∑〖𝑅_𝑒=272.5〗

Step 5: Calculate U for both groups

• 𝑈_𝑒 = 𝑛_𝑒 × 𝑛_𝑢 + (𝑛_𝑒 × (𝑛_𝑒 + 1)) / 2 - ∑𝑅_𝑒 = 21 × 10 + (21 × (21 + 1)) / 2 - 272.5= 210 + 231 - 272.5= 168.5
• 𝑈_𝑢 = 𝑛_𝑒 × 𝑛_𝑢+ (𝑛_𝑢 × (𝑛_𝑢 + 1)) / 2 - ∑𝑅_𝑢 = 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 α=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:
• 𝑧 = (𝑈_𝑒 - ((𝑛_𝑒 × 𝑛_𝑢) / 2)) / √((𝑛_𝑒×𝑛_𝑢 × (𝑛_𝑒 + 𝑛_𝑢 + 1)) / 12) = (168.5 - (( 21 × 10 ) / 2)) / √((21×10×(21+10+1))/12) = 63.5/√560 = 2.68

Step 7: Interpretation of Result

• At α=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.