Approximate the distribution of a random variable if n is large enough?

Question

I have this correction to some maths homework i had to work on.

During university registrations, each student completes a registration file. All checks carried out indicate that the probability that any registration is well filled in is equal to p = 0.94.

1. Introduce a random variable X which describes the two possible states for each file.

2. If n = 5, calculate the probability of the following events : { no file is well filled}, {all files are well filled}, {X > 3}, { 2 < X < 4}.

3. If n = 100, what probability distribution can we use to approximate the distribution of X?

The answer to 3) is For n = 100, we can approximate the binomial distribution by the normal distribution with mean μ = np and standard deviation σ = √(npq).

This approximation is valid if np > 5 and nq > 5, which is true here.

My understanding of this is that the reason why this is correct is because of the central limit theorem, which dictates that if n is large enough, then a random variable can be modeled using the normal distribution. However i don't understand the last sentence, why 5? and why do we check that assumption with np and nq?

Answer 1

Checking if $np > 5$ and $n(1-p) > 5$ is a rule of thumb for when it's a good idea to approximate the binomial by the normal.

It's not as though $5$ is a magic number that suddenly turns the binomial distribution into a bell curve. Rather, we don't want $np$ and $n(1-p)$ to be too small, and $5$ is a reasonable threshold people have found for "not too small".

The reason for why we want a lower bound on $np$ is that when $np$ is small, it is better to approximate the binomial distribution by a Poisson distribution with rate $np$. The Poisson is also a discrete distribution, so it will be better than the normal at telling you discrete probabilities like $\Pr[X=0]$, $\Pr[X=1]$, $\Pr[X=2]$. Meanwhile, the normal distribution will put too much weight on $\Pr[X<0]$, which is not possible for the binomial distribution we want to approximate.

When $n(1-p)$ is small, then the same thing applies to the random variable $n-X$, which has a binomial distribution with $n$ trials and probability $1-p$.