3.6 - Normal Approximation to the Binomial

Remember binomial random variables from last week's discussion? A binomial random variable can also be approximated by using normal random variable methods discussed above. This approximation can take place as long as:

1. The population size must be at least 10 times the sample size.
2. np = 10 and n(1 − p) = 10. [These constraints take care of population shapes that are unbalanced because p is too close to 0 or to 1.]

The mean of a binomial random variable is easy to grasp intuitively: Say the probability of success for each observation is 0.2 and we make 10 observations. Then on the average we should have 10 * 0.2 = 2 successes. The spread of a binomial distribution is not so intuitive, so we will not justify our formula for standard deviation.

If sample count X of successes is a binomial random variable for n fixed observations with probability of success p for each observation, then X has a mean and standard deviation as discussed in section 8.4 of:

\(Mean=np\) and \(Standard\ Deviation=\sqrt {np(1-p)}\)

And as long as the above 2 requirements are for n and p are satisfied, we can approximate X with a normal random variable having the same mean and standard deviation and use the normal calculations discussed previously in these notes to solve for probabilities for X.