Re: Cargo Cult Science
To prove the results that applied stats uses needs very advanced maths,
Yes
To prove the results that applied stats uses needs very advanced maths, which only a mathematical specialist will master
No.
Based on my personal suffering from the same subject (I am not a math graduate either), I can conclude that same as the majority of non-math graduates you have suffered from Idioticus Statisticus. I am quoting the species name by translating a good friend of mine who used to run the prob and stats dept in a major Eu university for several decades. That is how he referred to those of his colleagues which sucked so bad that they had to go teach non-math majors. That is curable.
Go read Feller, "An Introduction to Probability Theory and Its Applications". Amazon finally has it paperback so you do not have to shell out the relatively obscene 100 or so quid for the harcover (I have both the original and the English translation). Once you have read it, you are likely to change your opinion about "only advanced can grok it". It still holds a premium spot on my bookshelf till this day (many years after I did anything in proper math or sciences). Most of it reads like a novel by the way, you can enjoy reading it (that says a lot about a math book). I am probably going to add the English version of Gantmacher next to the original to it at some point so I can tell the kids to sod off and go read it when they ask math questions.