The Rule of five:

The normal with means np and npq can be used to approximate the binomial distribution with parameters n and p if \(npq > 5\).

Here, \(n = 500\),\(p = 0.001\).

\(npq = 500 \times 0.001 \times 0.999= 0.4995<5\)</span>

Normal approximation cannot be used for the binomial distribution with \(n = 500\), \(p = 0.001\).

The direct approximation of the binomial by Poisson says that the binomial distribution with parameters n and p has the same distribution as the Poisson with the parameter np when np is large.

The value of np is obtained as shown below:

\(np = 500 \times 0.001 =0.5\)

Since the value of n is large and p is small and the value of np is small, the binomial distribution can be approximated to Poisson.

That is, the binomial distribution with \(n = 500\), \(p = 0.001\) cannot be approximated to a normal distribution and it can be approximated to a Poisson distribution.

The normal with means np and npq can be used to approximate the binomial distribution with parameters n and p if \(npq > 5\).

Here, \(n = 500\),\(p = 0.001\).

\(npq = 500 \times 0.001 \times 0.999= 0.4995<5\)</span>

Normal approximation cannot be used for the binomial distribution with \(n = 500\), \(p = 0.001\).

The direct approximation of the binomial by Poisson says that the binomial distribution with parameters n and p has the same distribution as the Poisson with the parameter np when np is large.

The value of np is obtained as shown below:

\(np = 500 \times 0.001 =0.5\)

Since the value of n is large and p is small and the value of np is small, the binomial distribution can be approximated to Poisson.

That is, the binomial distribution with \(n = 500\), \(p = 0.001\) cannot be approximated to a normal distribution and it can be approximated to a Poisson distribution.