Reply #36: My probability calculations and analysis of roundoff error

[TruthIsAll]

National Exit Poll Nov 3, 12:23AM

Sample-size: 13,047 voters
Downloaded by: Jonathan Simon at 12:33 am.

As stated by Edison/Mitofsky (see notes in graphic below):
1-Margin of Error (MOE): 1.0%
2-Randomly-selected sample

This analysis is very similar to that of Ron Baiman.
Baiman calculated the odds of the Kerry shift (-2.56%).
For completeness, I calculate the odds of the Bush shift (+3.01%).

The voting split:
Female: 54%
Male: 46%

The split was consistent with earlier samples from 4pm and 7:33pm.

This is how they voted:
Women for Kerry: 54%-45%
Men for Bush: 52-47%

Since there were more women voting than men, Kerry had the advantage.

Calculation of the national percentages based on the gender split:
Kerry = .54*.54+.47*.46 = 50.78%
Bush = .45*.54+ .52*.46 = 48.22%

Kerry led the national exit poll by 2.56%.
But Bush won the vote by 2.80%.

Exit Actual Diff
Kerry 50.78% 48.43% -2.35%
Bush 48.22% 51.23% 3.01%
Diff 2.56% -2.80% -5.36%

As stated by Mitofsky/Edison, the MOE = 1%.
Therefore, the population standard deviation is:
StDev = MOE /1.96 = 0.00510

This means that 95% of the time we would expect that the Bush actual vote would fall between 47.22% and 49.22%, within 1.0% of his exit poll sample mean of 48.22%.

What is the probability that the Bush tally would exceed his exit poll sample mean by 3.01%, and rise from from 48.22% to 51.23%?

Probability = NORMDIST(0.4822,0.5123,0.0051,TRUE)
Probability = 0.00000000183

The odds are 1 out of 1/.00000000183 or

**************** 1 out of 547,044,797 ****************


ROUNDING
The exit poll percentages are rounded to the nearest 1%.
It would be preferable to show at least 1 decimal place for
greater computational accuracy.

Therefore, I have also calculated the odds assuming
1) a Best Case scenario for Bush (numbers rounded UP 0.5%),
and
2) a Worst Case scenario (his numbers rounded DOWN 0.5%).

The 0.5% adjustments were made to the female/male split as
well as to Bush's percentage for each.


THIS IS THE BASE CASE CALCULATION FOR THE EXIT POLL DATA
SHOWN IN THE GRAPHIC WHICH WAS ROUNDED TO THE NEAREST 1%:

Kerry = .54*.54+.47*.46 = 50.78%
Bush = .45*.54+ .52*.46 = 48.22%

These parameters were entered into the normal distribution
function:

Probability = NORMDIST(0.4822,0.5123,0.0051,TRUE)
Probability = 0.00000000183

............................................................
HERE ARE THE CORRESPONDING CALCULATIONS FOR THE BEST AND
WORST CASE SCENARIOS FOR BUSH, AFTER INCREMENTING INPUT
PARAMETERS BY THE +/-0.5% ROUNDOFF ERRROR.
............................................................

WORST CASE: -0.5%

Split
Male 45.5%
Female 54.5%

Vote%
Male 51.5%
Female 44.5%

Weight 47.69%

Probability = NORMDIST(0.4769,0.5123,0.0051,TRUE)
WORST CASE PROBABILITY
Prob Odds: 1 out of
1.82532E-12 547,849,842,147
..............................................................

BEST CASE +0.5%

Split
Male 46.5%
Female 53.5%

Vote%
Male 52.5%
Female 45.5%

Weight 48.76%

48.76% = .465*.525 + .535*.455


Probability = NORMDIST(0.4876,0.5123,0.0051,TRUE)
Probability ODDS: 1 OUT OF
6.09049E-07 1,641,904

To summarize:
Best Case (+0.5%): 1 out of 1,641,904
Most likely(as reported): 1 out of 547,044,797
Worst Case(-0.5%): 1 out of 547,849,842,147