# Normal Approximation to Binomial and Distribution of proportions

The distribution of p is closely related to the binomial distribution. The binomial distribution is the distribution of the total number of successes (favoring Candidate A, for example) whereas the distribution of p is the distribution of the mean number of successes. The mean, of course, is the total divided by the sample size, N. Therefore, the sampling distribution of p and the binomial distribution differ in that p is the mean of the scores (0.70) and the binomial distribution is dealing with the total number of successes (7).

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If the chance of success p and we have selected n instancess what is the chance of success.

if np>5 and n(1-p)>5 the distribution is like a normal distribution with:

$\mu&space;=&space;np$

$\sigma&space;=\sqrt{np(1-p)}$

the chances are based n a z

$Z=\frac{x-\mu&space;}{\sqrt{np(1-p)}}$

$Z=\frac{\frac{x}{n}-\frac{\mu}{n}}{\frac{\sqrt{np(1-p)}}{n}}$

$Z=\frac{\frac{x}{n}-\frac{np}{n}}{\sqrt{\frac{p(1-p)}{n}}}$

$Z=\frac{\bar{p}-p}{\sqrt{\frac{p(1-p)}{n}}}$