Date: Jul 18, 2012 8:34 PM
Author: David Bee
Subject: [ap-stat] Proofs for APStat Teachers and Others

Among the important results of APStat that teachers should know how
to prove mathematically are the following three:

1. The Central Limit Theorem (CLT)
2. The Chi-Square Test Stat Is Approximately Chi-Square Distributed
3. The T Test Stat Has A Student's t Distribution (exactly so when
sampling from a normal population)

Thus far, the first two were offered for those who would like to
read/study them during these summer months. For those who did not
study math-stat, the proof of the CLT gives a couple of preliminary
definitions and theorems before the CLT is proved so that the
proof could be followed fairly easily after first reading such.

With respect to the chi-square test stat proof, two different proofs
were offered: The first, Fisherian in nature, primarily involves
being able to follow an argument involving k-dimensional space,
which would correspond to a Calculus III course. The second proof
for it, which is Neyman-Pearsonian in nature,shows, in the course of
it, this usual test stat used in APStat and other such courses is
really just the first term of an infinite series, and so a better approximation does exist. [For those of you thinking goodness-of-
fit test stat based on the likelihood ratio, Give yourself Great
credit, as in G^2. (Hmmm---these jokes don't seem to work---where's
Chris O when you need him???!!!)]

So, for those who did not receive any of these three proofs (one of
the CLT and two for the chi-square test stat) who would find such
refreshing to read during summertime, then let me know which one(s)
would like to read and have as a Word file and I'll forward such to
you. (Of course let me know OFF-LIST...)
[Suggestion: You should go with the CLT proof first, not only
because of its importance but because it's the most elementary of
the three to follow.]


-- David Bee

PS: For those who already read the CLT proof, then there's a second
one, done by two applications of L'Hospital's Rule. Thus, if
such is intriguing, then just let me know and I'll forward it
to you. [Note: Although this second proof of the CLT is not
difficult to follow, a couple of suggestions: (1) You should
read the first one first (again, if necessary) as I didn't
repeat some things from the first proof used in the second, and
(2) You should read through it with paper and pencil nearby as
"all the steps of L'Hospital's Rule" were not shown. (For those
who haven't done the Rule in a long time, such may even be fun,
especially as this time it is applied to something quite vital,
namely proofing the CLT...)]

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