A sample has 9 observations: 2.4, 5.3, 2.4, 4.0, 1.2, 3.6, 4.0, 4.3, and 4.0.
Observation | Rank
(assuming no ties) |
Rank | |
---|---|---|---|
1.2 | 1 | 1 | |
Tied | 2.4 | 2 | 2.5 |
2.4 | 3 | 2.5 | |
3.6 | 4 | 4 | |
Tied | 4.0 | 5 | 6 |
4.0 | 6 | 6 | |
4.0 | 7 | 6 | |
4.3 | 8 | 8 | |
5.3 | 9 | 9 |
The Mann-Whitney test uses a normal approximation method to determine the p-value of the test.
is approximately distributed as a normal distribution with a mean of 0 and a standard deviation of 1, N(0,1).
Alternative hypothesis | P-value |
---|---|
H_{1}: η_{1} > η_{2} | |
H_{1}: η_{1} < η_{2} | |
H_{1}: η_{1} ≠ η_{2} |
The adjusted p-value is usually more accurate than the unadjusted p-value. However, the unadjusted p-value is the more conservative estimate because it is always greater than the adjusted p-value for a specific pair of samples.
Term | Description |
---|---|
W | Mann-Whitney test statistics |
n | size of sample 1 |
m | size of sample 2 |
η_{1} | median of sample 1 |
η_{2} | median of sample 2 |
k | |
i | 1, 2, …, I |
I | number of sets of ties |
t_{i} | number of tied values in the i^{th} set of ties |
The approximation algorithm that Minitab uses to calculate the point estimate of η_{1} – η_{2}is described in this article: J.W. McKean and T.A. Ryan, Jr. (1977). "An Algorithm for Obtaining Confidence Intervals and Point Estimates Based on Ranks in the Two Sample Location Problem", Transactions on Mathematical Software, 183–185.
The confidence interval of η_{1} – η_{2} is defined as the range of values of η_{1} – η_{2} for which the null hypothesis is not rejected.
The method that Minitab uses to calculate the confidence interval is described in this article: J.W. McKean and T.A. Ryan, Jr. (1977). "An Algorithm for Obtaining Confidence Intervals and Point Estimates Based on Ranks in the Two Sample Location Problem", Transactions on Mathematical Software, pp.183-185.