Basis for the Range Rule of Thumb and the Empirical Rule. In Exercises 45–48, find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.

About______ % of the area is between z = -3.5 and z = 3.5 (or within 3.5 standard deviation of the mean).

The z-scores are -3.5 and 3.5.

Let Z be the random variable following the standard normal distribution.

Thus,

$$\begin{gathered} Z~N\left(\mu ,{\sigma }^2\right) \\ ~N\left(0,1^2\right) \end{gathered}$$

Steps to draw a normal curve:

1. Make a horizontal axis and a vertical axis.
2. Mark the points -3.5, -3.0, -2.0 up to 3 on the horizontal axis and points 0, 0.05, 0.10 up to 0.50 on the vertical axis.
3. Provide titles to the horizontal and vertical axes as ‘z’ and ‘P(z)’, respectively.
4. Shade the region between -3.5 and 3.5.

The shaded area of the graph indicates the probability that the z-score is between -3.5 and 3.5.

As probability has a one-to-one correspondence with the area under the curve, the probability between -3.5 and 3.5 is computed as

$P\left(-3.5<Z<3.5\right)=P\left(Z<3.5\right)-P\left(Z<-3.5\right) ...\left(1\right)$

Referring to the standard normal table,

• the cumulative probability of -3.5 is obtained from the cell intersection for rows -3.50 and below and the column value 0.00, which is 0.0001, and
• the cumulative probability of 3.5 is obtained from the cell intersection for rows 3.50 and up and the column value 0.00, which is 0.9999.

Thus,

$$\begin{gathered} P\left(Z<3.5\right)=0.9999 \\ P\left(Z<-3.5\right)=0.0001 \end{gathered}$$

Substituting the values in equation (1),

$$\begin{gathered} P\left(-3.5<Z<3.5\right)=P\left(Z<3.5\right)-P\left(Z<-3.5\right) \\ =0.9999-0.0001 \\ =0.9998 \end{gathered}$$

Expressing the result as a percentage,about 99.98 % of the area is between -3.5 and 3.5.