[ExI] mersenne primes again

spike spike66 at att.net
Wed Mar 31 21:03:15 UTC 2010


 

> ...On Behalf Of Gordon Swobe
> Subject: Re: [ExI] mersenne primes again
> 
> Fascinating, spike.
> 
> I wonder if anyone has done a statistical analysis of the 
> series to determine if the perceived change has statistical 
> significance. Any idea?

Ja, anyone has, I have been doing these a number of different ways, and the
significance of the weirdness depends on how you calculate it.

Thanks for giving me a legitimate entry back into areas relevant to extropy.

> A statistician should find it possible to say something like 
> "The slope deviated at Mersenne prime number 38, and we know 
> this with x% confidence." 

Ja keep reading.

> 
> (Statistical significance generally requires an x equal to or 
> greater than 95.)... -gts

The concept of statistical significance worries me, for I have seen so many
examples of why it can actually lead to our fooling ourselves.  We have
arbitrarily set up the 95%ile threshold for statistical significance, or 2
standard deviations, but why 2 exactly?  Why not 1.9 or 2.1 or e sigmas?

Reason: our textbooks from college statistics arbitrarily chose 2 sigma
since back in the old old days when I was going there, before calculators,
we had those charts in the back of the book, and they were graduated in
tenths of a sigma, and the integers were bolded to make them easy to find
during the test.  This helped determine the perfectly arbitrary 95%ile
criterion for statistical significance, and I argue that this shortcut may
be more harmful than helpful.

Consider a discussion that went on here recently: New York blackout, nothing
to do, was there a baby boom 9 months later?  From a statistical
significance point of view there was not.  You can filter that data any way
you want, rolling average, interval slope, Butterworth filtering, quaternion
regression, it just doesn't matter: you will never find that particular
needle in that particular haystack.

Do the same thought experiment Drake equation style: a million homes without
power, about half of those homes with opposite gender couples, a quarter of
those without nature's own birth control devices (young children), half of
those with fertile couples, a tenth of those copulating because of nothing
else to do, a tenth of a percent of those resulting in an unexpected
pregnancy and a quarter of those deciding to carry to term, and you get
about 1 to 4 extra births per million, something that would never be noticed
by the local hospitals or statisticians, but hypothetically perfectly valid.

Next, consider the poker player who is dealt an ace high royal flush the
first deal of the evening.  The gambler is thinking "WOW!  I have been
playing poker with these guys every Saturday night for thirty years and I
have NEVER seen a royal flush!"  Not surprising because the probability of a
royal flush is about 1 in 650,000.  The next hand she gets a deuce of
spades, trey of hearts, 6 of clubs, 10 of spades and a jack of diaminks, she
thinks "Damn, nada."  But the second hand is just as unlikely as the first,
and she has never seen this particular hand either in all those thirty
years.  Point: we must be very careful to not fool ourselves.

I have a digital sim that suggests to me the weirdness of the last 8
Mersenne primes clustering the way they do is about 30 in 10,000, a three
sigma event, which should not be so surprising.  But I think I may have
underaccounted for the last two being not just anomalously close, but a
superanomaly.

One last point.  The last 8 Mersennes form a kind of hockey stick, but what
if the ratios of the last 8 primes been anomalously large instead of
anomalously small?  We conjecture that there are infinitely many Mersenne
primes, but had the hockey stick been pointed up instead of down, many of us
would (most probably incorrectly) conclude that there is a finite number of
Mersenne primes.

More later.  Assignment please: think about everything you know regarding
the definition of statistical significance, and what it means to you.

spike



   




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