State the Central Limit Theorem for a proportion. (b) When is it safe to assume normality for a?

Question

State the Central Limit Theorem for a proportion. (b) When is it safe to assume normality for a?

Answer 1

The central limit theorem states that the sum of a whole bunch of distributions will be a normal distribution.

The catch is what, exactly, is a whole bunch.  And that depends.  If you have a decidedly non-normal distribution like a coin flip, it takes more than if the distributions were kind of normal to start with.  When flipping coins, you can see the normal start taking shape at about 10 flips, and it gets smoother and smoother.  So when it is "safe" to assume depends on how picky you are.  It will never be perfectly normal, but it will be pretty much normal if you have 10 distributions with roughly equal variations.  There is a lot that can foul this up, though.  If the distributions have high correlation, they will converge to normality slower.  And if one distribution constitutes a large fraction of the total variation, you will never converge to normality even if you have a large number of distributions adding to it that don't vary much.