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Topic: [ap-stat] A Midsummer Night's Dream-Proof: T Distribution ---> Standard Normal Distribution
Replies: 1   Last Post: Aug 4, 2012 1:28 PM

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 David Bee Posts: 4,194 Registered: 12/6/04
[ap-stat] A Midsummer Night's Dream-Proof: T Distribution ---> Standard Normal Distribution
Posted: Aug 1, 2012 8:30 PM

Good evening / Good morning

Probably practically all APStat teachers and most of their students
become aware that as the number of degrees of freedom increases, the
t distribution approaches the standard normal distribution.

For example, we know for the SND that P_97.5 = 1.960[1.959963...].
Using the calculator for df = 10,100,200,300,400,500,1000,2000 gives
corresponding 97.5th percentiles for the t of 2.228,1.984,1.972,
1.968,1.966,1.965,1.962,1.961.

Anyway, for those who want to see this convergence for all t
mathematically, namely that the t distribution approaches the SND
as df-->infinity, then feel free to let me know---off-List of
course---and I'll forward a proof of such to you.

This proof, unlike the previous ones, is fairly straightforward,
a proof for those of you who in early calculus days liked to see/
read those step-by-step proofs where the imminence of the result
held you sort of spellbound until the finale, a truly aha moment!
(As in the much-more-important CLT proofs, a few Preliminaries are
given, but in this proof these are not from math-stat but just from
calculus, with perhaps the key one from Calculus I: The limit of a
product is the product of the limits.)
[BTW, for those of you who think Stirling's Formula (for large n in
n!) is now completely unimportant because of modern technology, you
may want to reconsider such after reading this proof.]

Good night /Good day

-- David Bee

PS: As with a couple of the previous much-more-important proofs,
this proof comes from the Murray R. Spiegel Schaum's book
Probability And Statistics. However, such is only in the
original 1975 edition as a Supplementary Problem (Problem
4.161 for anyone who happens to have it), and so should you
find any errors in this proof, then simply blame me as it's
my proof for this Supp. Problem. (Also, should someone find
another/better proof, then let me/us know.)

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Date Subject Author
8/1/12 David Bee
8/4/12 David Bee