This document summarizes nonparametric hypothesis testing methods that can be used when sample sizes are small and variables are not normally distributed. It describes the sign test, which tests for differences in paired data by counting how many pairs differ, and the Mann-Whitney U test for unpaired data, which ranks all values together and tests for differences in average ranks between samples. An example compares the income of applicants for fixed-rate versus variable-rate mortgages using the Mann-Whitney U test, finding the probability of difference being due to chance is 18.3%, suggesting variable rate applicants have higher income.

1. Nonparametric Hypothesis Testing Guy Lion December 2005

2. Nonparametric tests handle variables that are not normally distributed.

5. Sign Test Binomial Distribution algorithm

6. Sign Test testing the creativity of an Ad campaign Classic situation where we need to use a nonparametric test. This is because the samples are small (17 observations), and the variables are not normally distributed (check Skewness and Kurtosis).

7. Sign Test testing the creativity of an Ad campaign (continued)

8. Different view of same data

10. Example: Testing income of mortg. applicants Testing if applicants for fixed-rate mortgages have higher income than applicants for variable-rate mortgages. The fixed-rate applicants have one high income value ($240,000). Kurtosis and Skewness of both samples confirm they are not normally distributed. The unpaired t test would not work well.

11. Sorting and Ranking Ranked in ascending order. The figures in yellow are identical ($36,500). They originally ranked 12 th , 13 th , and 14 th . So, they all received the tied ranking of 13 th .

12. P Value (probability difference is due to chance) Based on ranks (not values), there is an 18.3% probability the two samples come from same population. There is a 81.7% probability that the Variable Rate mortgage applicants have a higher income because they have a higher average rank (17.79 vs 13.50).

14. Appendix: The actual Mann-Whitney U Calculation