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Topic: [ap-stat] Two Proofs APStat Teachers Should Be Familiar With...
Replies: 1   Last Post: Aug 9, 2012 8:30 PM

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David Bee

Posts: 4,194
Registered: 12/6/04
[ap-stat] Two Proofs APStat Teachers Should Be Familiar With...
Posted: Aug 8, 2012 8:02 PM
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Probably the two most importanttheorems used in an APStat course are
the Central Limit Theorem and the Theorem stating the usual T test
stat has a t distribution when sampling from a normal population and
otherwise has approximately such in conjunction with the CLT.

Two proofs of the CLT have been offered already --- the first the
standard one (with a few preliminary concepts, especially for those
who have not taken a math-stat course but a good concise review for
those who have) and the second one for those familiar with the
standard proof but would like to see an intriguing alternative.

With respect to the proof that the T test stat does indeed have a t
distribution when sampling from a normal population, such is offered
now, which is titled "Ten Steps Proving the Usual T Statistic Has a
Student's t Distribution With n-1 Degrees of Freedom When Sampling
>From a Normal Distribution".

Steps 1 and 2 are preliminary concepts and Steps 3-10 are theorems
(with proofs), with Step 10 of course being the Theorem itself,
namely: If random samples of size n are taken from a normally
distributed population, then T = (Xbar - mu)/(S/sqrt(n)) has a t
distribution with n-1 degrees of freedom.

Although such is of course found in math-stat books, the preliminary
concepts/theorems are for the most part not in the same chapter,
especially as most of the preliminary prerequisite theorems are
probabilistic in nature (and so would be earlier in the book or
covered in the prerequisite course in mathematical probability).

In addition, a couple of byproducts of the prerequisite theorems
are concepts arising in APStat, such as the square of a standard normal random variable is a chi-square random variable with one degree of freedom. [Step 4] (This is seen in APStat when dealing with a two-by-two table and doing a chi-square test for independence or homogeneity.)

So, for those with interest in reading (and time to read) these
ten steps (which may not be feasible to do in a single sitting,
especially for those who have not taken a math-stat course yet),
feel free to let me know off-List and I'll forward a copy to you.
(The Theorem and its proof run only six lines but the prerequisites
run six pages!)

For those requesting this, just be sure you are ready to read/study
it as it will take time, need focus, and require familiarity and
comfort with several concepts in calculus.

-- David Bee

PS: For those considering this, you should read the proof of the
CLT first. (Feel free to request such also if not already done

PPS: For those who intend to use the CLT proof with students who
already took BCCalc, this one would be appropriate too...

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