This is another state exit poll analysis. The chances are even
more remote then previously calculated. Probabilities are
calculated based on various Z-score thresholds, rather than
the MoE.
The probability of at least N states exceeding a given Z-score
(number of standard deviations from the mean) is based on the
number and magnitude of Bush exit poll deviations from the
recorded vote.
The Z-score is a function of: exit poll sample size, Kerry and
Bush exit poll percentages, and the Bush deviation from exit
poll to the vote.
The standard deviation for a polling n-sample size is:
Stdev = Sqrt (P*Q/n), where
n = state exit poll sample-size
P = Kerry poll percentage
Q = 1-P = Bush poll percentage
The Z-score is simply:
Z = Bush deviation/Stdev
Using Florida as an example:
n= 2846
P= .5 (assume a polling dead heat)
Q= .5
Stdev = .00937 = Sqrt (.5*.5/2846) = .5/53.3479
The Bush vote deviation from the FL exit poll was 2.46%, so
Z = 2.63 = .0246/.00937
Using Excel, the probability Pr1 of the Bush FL deviation is:
Pr1 = 1 - NORMSDIST(Z) (not NORMDIST)
Pr1 = 1 - NORMSDIST(2.63) = .0043 or 1 in 231
The FL exit poll margin of error is:
MoE = 1.84% = 1.96 * Stdev = 1.96* .00937
Z Probability
1.00 .1587 (68% confidence level)
1.96 .0250 (95% confidence level)
2.00 .0228
2.33 .0099
2.58 .0049 (99% confidence level)
3.00 .0012
Let PrN = probability (Z>=X) in at least N states.
PrN = Prob (N>=Z) = 1- BINOMDIST (N-1, 50, Pr1, TRUE)
Z-scores for Bush exceeded:
1) 2.20 standard deviations in N = 16 states
2) 2.60 standard deviations in N = 10 states
3) 3.35 standard deviations in N = 4 states
4) 3.75 standard deviations in N = 3 states
3) 4.10 standard deviations in N = 2 states
Here are the corresponding probabilities:
N Z Pr1 Pr (N>=Z)
16 2.20 1 in 72 1 in 160,000 trillion
10 2.60 216 23 trillion
4 3.35 2475 165 million
3 3.75 11307 74 million
2 4.10 48382 1.9 million
Here are the individual Z-scores and probabilities for each
state.
N St Favor Z Pr1
1 NH Bush 4.16 1 in 63547
2 NY Bush 4.10 48382
3 SC Bush 3.75 11307
4 NC Bush 3.35 2475
5 VT Bush 2.95 628
6 OH Bush 2.94 602
7 PA Bush 2.89 521
8 MN Bush 2.67 263
9 FL Bush 2.63 231
10 DE Bush 2.60 216
11 AK Bush 2.44 138
12 MA Bush 2.38 114
13 NJ Bush 2.36 108
14 NE Bush 2.33 102
15 RI Bush 2.23 77
16 AL Bush 2.20 72
17 CT Bush 1.94 38
18 MS Bush 1.88 33
19 NV Bush 1.83 30
20 UT Bush 1.77 26
21 VA Bush 1.75 25
22 CO Bush 1.73 24
23 NM Bush 1.70 23
24 AR Bush 1.64 20
25 IL Bush 1.61 19
26 LA Bush 1.54 16
27 WV Bush 1.43 13
28 WA Bush 1.36 12
29 AZ Bush 1.36 11
30 WY Bush 1.33 11
31 ID Bush 1.32 11
32 GA Bush 1.21 9
33 IA Bush 1.13 8
34 MO Bush 1.07 7
35 DC Bush 1.02 6
36 IN Bush 0.94 6
37 MI Bush 0.81 5
38 MD Bush 0.50 3
39 KY Bush 0.50 3
40 CA Bush 0.46 3
41 ME Bush 0.32 3
42 OK Bush 0.24 2
43 WI Bush 0.01 2
44 MT Kerry 0.12 2
45 HI Kerry 0.47 3
46 OR Kerry 0.49 3
47 KS Kerry 1.27 10
48 SD Kerry 1.34 11
49 ND Kerry 1.35 11
50 TN Kerry 1.39 12
51 TX Kerry 1.40 12