e-Statistics

Normal Approximation

Suppose that $ X$ is a binomial random variable with $ n =$ and $ p =$ . If the size $ n$ is adequately large, then the distribution of $ X$ can be approximated by the normal distribution with mean $ np$ = and standard deviation $ \sqrt{np(1-p)}$ = . That is, a normal distribution approximates a binomial distribution. A general rule for ``adequately large'' $ n$ is to satisfy $ np \ge 5$ and $ n(1-p) \ge 5$.

Let $ Y$ be a normal random variable whose distribution approximates the binomial distribution of a random variable $ X$. Then the probability involving $ X$ can be approximated by that of $ Y$. Having taken into account the fact that $ Y$ is a continuous random variable, the approximated probability

$ P\Big(i =$ $ \le X \le j =$ $ \Big) \approx P(i-0.5 \le Y \le j+0.5)$ = .

is calculated with continuity correction.