Standardizing a binomial distribution Let $Y$ be a binomial random variable with parameters $n = 72$ and $p = 1/3$.
a) Find the exact probability $P(22 ≤ Y ≤ 28)$.
b) Is the condition $n ≥ 9 \left( \frac{\max(p, 1−p)}{\min(p, 1−p)} \right)$ required to approximate the distribution of Y by a normal distribution (textbook p.380) satisfied? If yes, approximate
$P(22 ≤ Y ≤ 28)$ with a normal probability.
I found a. I got $0.6$ but when I apply the central limit theorem and standardize it, I get $8.485$ and $-4.2$.
This already looks really wrong but I put it in R anyways by doing pnorm(8.485) - pnorm(-4.2) and I get $.999$.
The weird thing is if i code
pnorm(28,24,4)-pnorm(22,24,4)  I get $.53$.
Then I remembered the .5 rule so I did pnorm(28.5,24,4)-pnorm(21.5,24,4)  which is $.65$ and doing pnorm(28,24,4)-pnorm(21,24,4) gave me $.61$ which is REALLY close now to the answer. So I'm pretty sure the normal distribution DOES work in this case but I do not know why everytime I standardize it I get a big mess.

$$np=24$$
$$npq=16$$
CLT in the version of De Moivre- Laplace is
$$\frac{\Sigma_i X_i-np}{\sqrt{npq}}\sim \Phi$$
Thus
$$\mathbb{P}[22\leq Y\leq 28]=\mathbb{P}[21.5<Z<28.5]= $$
$$=\Phi\left[\frac{28.5-24}{4}\right]- \Phi\left[\frac{21.5-24}{4}\right] = 0.87-0.27=0.6 $$
