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...)]