Question

Standard Normal DistributionIn Exercises 17–36, assume that a randomly selected subject is given a bone density test. Those 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 probability of the given bone density test scores. If using technology instead of Table A-2, round answers

to four decimal places.

Between -2.55 and -2.00.

Short answer

The graph is represented as,

The probability that the bone density score is between -2.55 and -2.00 is 0.0174.

Step-by-step solution

Given information

The bone density test scores are normally distributed.

The mean score is $\mu =0$.

The standard deviation is$\sigma =1$.

The z-scores are provided as -2.55 and -2.00.

Draw a graph

Let x represent the bone density test score.

As the mean and standard deviation are0 and 1, respectively, x follows a standard normal distribution.

Steps to make a normal curve:

Step 1: Make a horizontal and a vertical axis.

Step 2: Mark the points -4, -2, 0, 2, and 4 on the horizontal axis and points 0.1, 0.2, 0.3, and 0.4 on the vertical axis.

Step 3: Provide titles to the horizontal and vertical axes as ‘z’ and ‘f(z)’, respectively.

Step 4: Shade the region between z=-2.55 and z=-2.00.

The shaded area represents the probability.

Compute the probability

Using table A-2,

• the area to the left of 2 is obtained from the table in the intersection cell with the row value 2.0 and the column value 0.00, which is obtained as 0.9772, and
• the area to the left of 2.55is obtained from the table in the intersection cell with the row value 2.5 and the column value 0.05, which is obtained as 0.9946.

The probability that the bone density score is between -2.55 and -2.00 is computed as follows.

$$\begin{gathered} P\left(-2.55<z<-2.00\right)=P\left(z<-2.00\right)-P\left(z<-2.55\right) \\ =\left(1-P\left(z<2\right)\right)-\left(1-P\left(z<2.55\right)\right) \\ =\left(1-0.9772\right)-\left(1-0.9946\right) \\ =0.0174 \end{gathered}$$

Thus, the probability that the bone density score is between -2.55 and -2.00 is 0.0174.