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Red Shift: Z-scores and Probabilities. Take a deep breath. [View All]

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TruthIsAll Donating Member (1000+ posts) Send PM | Profile | Ignore Sat Apr-23-05 02:02 AM
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Red Shift: Z-scores and Probabilities. Take a deep breath.
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Edited on Sat Apr-23-05 02:26 AM by TruthIsAll
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

					
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