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