Example 44

EXAMPLE 44The normal approximation to the binomial distribution

The Centers for Disease Control and Prevention reported that 20% of preschool chil dren lack required immunizations, thereby putting themselves and their classmates at risk. For a group of $n = 64$ children with $p = 0.2$ , the binomial probability distribution is shown in Figure 72.

Verify that this distribution can be approximated by a normal distribution.
Find the mean and standard deviation of this normal distribution.

Solution

The normal approximation is valid if $n \cdot p \geq 5$ and $n \cdot q \geq 5$ . Substituting $n = 64$ and $p = 0.2$ , we get $\begin{matrix} n \cdot p = (64) (0.2) = 12.8 \geq 5 & and & n \cdot q = (64) (0.8) = 51.2 \geq 5 \end{matrix}$
Thus, the normal approximation is valid.
The mean and standard deviation of the normal distribution are

$\begin{array}{l} μ_{x} = n \cdot p & = & (64) (0.2) = 12.8 \\ σ_{x} & = & \sqrt{n \cdot p \cdot q} & = & \sqrt{(64) (0.2) (0.8) = 3.2} \end{array}$

NOW YOU CAN DO

Exercises 3–8.