Question

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 and z = 3 (or within 3 standard deviation of the mean).

Short answer

The shaded area between -3 and 3 under the curve of the standard normal distribution is shown below.

About 99.74% of the area is between z = -3 and z = 3.

Step-by-step solution

Given information

The z-scores are -3 and 3.

Describe the distribution

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

Thus,

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

Draw the area under the curve for the standard normal distribution

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 and 3.

The shaded area of the graph indicates the probability that the z-score is between -3 and 3.

Find the shaded area

Due to a one-to-one correspondence between the area and probability, the area between -3 and 3 can be expressed as follows.

The probability between -3 and 3 is computed as

$P\left(-3<Z<3\right)=P\left(Z<3\right)-P\left(Z<-3\right) ...\left(1\right)$

Referring to the standard normal table,

• the cumulative probability of -3 is obtained from the cell intersection for rows -3.0 and the column value 0.00, which is 0.0013, and
• the cumulative probability of 3 is obtained from the cell intersection for rows 3.0 and the column value 0.00, which is 0.9987.

Thus,

$$\begin{gathered} P\left(Z<3\right)=0.9987 \\ P\left(Z<-3\right)=0.0013 \end{gathered}$$

.

Substituting the values in equation (1),

$$\begin{gathered} P\left(-3<Z<3\right)=P\left(Z<3\right)-P\left(Z<-3\right) \\ =0.9987-0.0013 \\ =0.9974 \end{gathered}$$

Expressing the result as a percentage, about 99.74 % of the area is between z = -3 and z = 3.