1. ## Binomial trials variance

Hi all,

I am beginning to look at some probability problems, I am completely new to college level probability, so not sure if this is in the right sub forum (maybe it should be in the basic?- guess mods will decide.)

Given a scenario where there is a true statistic - proportion voting yes in a vote denoted by $\displaystyle p$, we sample $\displaystyle n$ people from the large population

For a binomial distribution for the sample where $\displaystyle x$ denotes the number of yes votes:

$\displaystyle b(x;n,p) = \binom {n}{x} p^{x} (1-p)^{n-x}$

$\displaystyle mean(b_i) = np$ ; $\displaystyle var(b_i) = np(1-p)$ since the binomial distribution is a 'sample' of the true statistic, we need to take into account the standard error:

$\displaystyle var(bi) = {\sigma(p)^2 \over \sqrt{n}}n = \sigma(p)^2 \sqrt{n}$

I have multiplied by $\displaystyle n$ to scale the variance up from the true average.... is this correct?

Does that mean the $\displaystyle z$ stat for the normal distribution approx becomes:

$\displaystyle z = {X- np \over \sqrt{\sigma(p)^2 \sqrt{n}}}$

Many thanks for reading! Any pointers wil be much appreciated.

2. I have simplifed the problem down to what I am contemplating... apoligies for the long winded garbled original post.