Exercise 1 Page 570

We have been told that the random variable is normally distributed with a mean of $\mu =416$ and a standard deviation of $\sigma =55.$ We want to find the range of values that represent the middle $99.7\%$ of the distribution. The normal distribution is symmetrical with respect to its mean $\mu .$ Therefore the middle range is also symmetrical with respect to $\mu .$

$$\begin{array}{c}P(\mu -z<X<\mu +z)=99.7\%\end{array}$$

Let's recall the Empirical Rule. According to this rule approximately $99.7\%$ of normally distributed data falls within $3\sigma$ from the mean.

$$\begin{array}{c}P(\mu -3\sigma <X<\mu +3\sigma )=99.7\%\end{array}$$

Knowing that $\mu =416$ and $\sigma =55,$ let's find both ends of the range.

$\mu -3\sigma <X<\mu +3\sigma$

SubstituteII

$\mu =416$, $\sigma =55$

$416-3(55)<X<416+3(55)$

Multiply

Multiply

$416-165<X<416+165$

AddSubTerms

Add and subtract terms

$251<X<581$

We found the range that represents the middle $99.7\%$ of the distribution.