## Wednesday, April 2, 2014

### Footnote to the QB choice hypothesis testing problem.

Calculation of Wilcoxon test statistic for QB choice problem

Below is a quick illustration of the calculation of the Wilcoxon statistic for the QB choice problem.

 Calculation of Wilcoxon Test Statistic for Difference in First QB and Second QB Career Touchdowns c1 c2 c3 c4 c5 c6 c7 First Choice QB Second Choice QB D Abs(D) DUM=1 if D is positive Rank ABS D DUM*Rank 65 79 -14 14 0 2 0 128 363 -235 235 0 30 0 102 0 102 102 1 13 13 64 234 -170 170 0 26 0 491 14 477 477 1 33 33 1 161 -160 160 0 22 0 77 11 66 66 1 9 9 174 208 -34 34 0 7 0 15 113 -98 98 0 12 0 251 50 201 201 1 29 29 16 48 -32 32 0 6 0 2 8 -6 6 0 1 0 154 5 149 149 1 17 17 165 0 165 165 1 23 23 170 0 170 170 1 26 26 275 7 268 268 1 31 31 203 19 184 184 1 28 28 207 40 167 167 1 25 25 247 94 153 153 1 18 18 300 29 271 271 1 32 32 3 100 -97 97 0 11 0 3 136 -133 133 0 16 0 86 60 26 26 1 4 4 33 199 -166 166 0 24 0 100 28 72 72 1 10 10 3 159 -156 156 0 20 0 124 0 124 124 1 15 15 156 182 -26 26 0 4 0 155 0 155 155 1 19 19 124 16 108 108 1 14 14 3 17 -14 14 0 2 0 164 125 39 39 1 8 8 212 55 157 157 1 21 21

The Wilcoxon W is obtained from the SUMPRODUCT command for c5 and c6 or just add up c7.

The expected value of W is

EW =  n x (n+1) /4

Where n is the sample size minus ties.

The STDW is

STDW  = ((n x (n+1) x  (2n+1))/24)0.5

The p-value for the Wilcoxon test was obtained from z

Z=(W-EW)/STDW

The function NORM.DIST in excel can be used to calculate the p value for this Z.

Footnote:  The Di used to calculate the Wilcoxon are not normally distributed.   The upper tail is very large because of Manning and Elway two very successful first-choice picks and the lower tail is very large because of Simms and Testaverde, two great second-choice picks.  The tests may be more affected by the size of the tails than by the location of the means or medians.

The hypothesis testing problem:
http://dailymathproblem.blogspot.com/2014/04/hypothesis-tests-for-qb-performance.html