Finding Bone Density Scores. In Exercises 37–40 assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.

Find the bone density scores that can be used as cutoff values separating the lowest 3% and highest 3%.

The bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1.

As the distribution of bone density follows a standard normal distribution, the random variable for bone density is expressed as Z.

Thus,

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

Sketch the standard normal curve as shown below, where the shaded regions represent the lowest 3% and highest 3% areas under the z-score.

In the graph, represents the cutoff for the lowest 3% z-score ofbone density scores.

$$\begin{gathered} {\text{Area to the left ofz}}_{\text{1}}=P\left(Z<z_1\right) \\ =0.03 \end{gathered}$$

In the standard normal table, the value closest to 0.03 is 0.0301, and the corresponding row value -1.8 and the column value 0.08 correspond to the z-score of -1.88.

Thus, the value of the cutoff score corresponding to the lowest 3% scores is -1.88.

In the graph, $z_2$represents the cutoff for the top 3% z-scores ofbone density scores.

$$\begin{gathered} {\text{Area to the right ofz}}_2=P\left(Z>z_2\right) \\ =0.03 \end{gathered}$$

The left-tailed area corresponding to the cutoff score is

$$\begin{gathered} P\left(Z>z_2\right)=0.03 \\ 1-P\left(Z<z_2\right)=0.03 \\ P\left(Z<z_2\right)=0.97 \end{gathered}$$

In the standard normal table, the value closest to 0.97 is 0.9699, and the corresponding row value 1.8 and the column value 0.08 correspond to the z-score of 1.88.

Thus, the value of the cutoff score corresponding to the top 3% scores is 1.88.

The shaded area of the graph indicates the probability that the z-score is lesser than -1.88 and greater than 1.88.

The two cutoff scores separating the bottom 3% and the top 3% scores of bone density are -1.88 and 1.88, respectively.