Question

In Exercises 9–12, find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

Short answer

The area of shaded region is 0.8508.

Step-by-step solution

Given information

The graph for standard normal distribution of bone density scores is given with a shaded region.

State the relationship between area and probability 

The area has one-to-one related with probability in case of density curves with area equal to 1.

The left-tailed areas or cumulative probabilities can be obtained using the standard normal table.In the case of right-tailed areas of the z-score, the area is equivalent to the difference of cumulative probability at the respective z-score from the total area (equal to 1).

Find the area of the shaded region

It is required to compute the area to the right of z-score -1.04.

Mathematically,

$$\begin{gathered} \text{Area}\text{to}\text{the}\text{right}\text{of}-1.04=1-\text{Area}\text{to}\text{the}\text{left}\text{of}-1.04 \\ =1-P\left(Z<-1.04\right)...\left(1\right) \end{gathered}$$

By using the standard, normal table, the area is obtained from the table in the cell with row value -1.0 and the column value 0.04, which is obtained as 0.1492.

Mathematically, it is expressed as follows:

$P\left(Z<-1.04\right)=0.1492 ...\left(2\right)$

Substitute (2) into (1).

$$\begin{gathered} \text{Area}\text{to}\text{the}\text{right}\text{of}-1.04=1-0.1492 \\ =0.8508 \end{gathered}$$

Thus, the shaded area to the right of -1.04 is equal to 0.8508.