7.4 Normal Approximation of the Binomial Distribution

Draw a binomial distribution for the following information:

(a) (b) (c)

The results from both (a) and (b) are unimodal and skewed either positively or negatively. (c), however is unimodal and symmetry (or pretty close). When a set of data is unimodal and symmetric, it can be approximated using the normal distribution.

In order to approximate this distribution normally, we need to be able to discuss a mean and a standard deviation.

To approximate these, we use and . Notice that the formula for μ is the same as the formula for the expected value for a binomial distribution - which makes sense since the expected value is an average anyhow.

A binomial distribution can be normally approximated as long as both np and nq are both greater than 5.

Ex 1 A bank found that 24% of its loans to small business are delinquent. What is the probability that the bank will find at least 60 delinquent accounts if it chooses 200 small business accounts at random?

Ex 2 A soft drink company knows that it has 42% market share in a certain region. They conduct a blind taste test on 70 people at the mall.

(a) What is the probability that fewer than 25 people will choose this cola?

(b) What is the probability that exactly 25 people will choose this cola?

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