Question

What are the conditions a sample needs to meet before you can assume it's binomial and that it approximates a normal distribution?

Answer 1

1)
`np\geq 5`

2)
`nq = n(1-p)\geq 5`

Other conditions that are important are:

3) n is large

4) p is close to 1/2 or 0.5

Explanation:

1) Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

2) Solution to the problem

Let X the random variable of interest, on this case we now that:

`X \sim Binom(n, p)`

The probability mass function for the Binomial distribution is given as:

`P(X)=(nCx)(p)^x (1-p)^(n-x)`

Where (nCx) means combinatory and it's given by this formula:

`nCx=(n!)/((n-x)! x!)`

In order to apply the normal apprximation we need to satisfy these two conditions:

1)
`np\geq 5`

2)
`nq = n(1-p)\geq 5`

Other conditions that are important are:

3) n is large

4) p is close to 1/2 or 0.5