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





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