Question

How do you find the normal approximation without using the continuity correction?

Answer 1

Let's say you want to compute the probability $\mathbb P(a\le X\le b)$ where $X$ converges in distribution to $Y$, and $Y$ follows a normal distribution. The normal approximation (without the continuity correction) basically involves choosing $Y$ such that its mean and variance are the same as those for $X$.

Example: If $X$ is binomially distributed with $n=100$ and $p=0.1$, then $X$ has mean $np=10$ and variance $np(1-p)=9$. So you can approximate a probability in terms of $X$ with a probability in terms of $Y$:

$\mathbb P(a\le X\le b)\approx\mathbb P(a\le Y\le b)=\mathbb P\left(\dfrac{a-10}3\le\dfrac{Y-10}3\le\dfrac{b-10}3\right)=\mathbb P(a^*\le Z\le b^*)$

where $Z$ follows the standard normal distribution.