Question

Standard normal distribution, 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 score. If using technology instead of Table A-2, round answers to four decimal places.

Less than 4.55

Short answer

The graph for the bone density test score less than 4.55 is as follows.

The probability of the bone density test score less than 4.55 is 0.9999.

Step-by-step solution

Given information

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

Describe the distribution

The distribution of bone density follows the standard normal distribution, and the variable for bone density is denoted by Z.

Thus,

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

Sketch a graph that the z-score is less than 4.55

Steps to draw a normal curve:

1. Make a horizontal axis and a vertical axis.
2. Mark the points -4, -3, -2 up to 6 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 less than 4.55.

The shaded area of the graph indicates the probability that the z-score is lesser than 4.55.

Find the cumulative area corresponding to the z-score

Referring to the standard normal table for the negative z-score, the cumulative probability of 4.55 is obtained from the cell intersection for row 4.5 and the column value of 0.05, which is 0.9999.

As the probability and area have a one-to-one correspondence, the probability that the bone density test score is less than 4.55 is computed as

$$\begin{gathered} \text{Area}\text{to}\text{left}\text{of}4.55=P\left(Z<4.55\right) \\ =0.9999 \end{gathered}$$

.

Thus, the probability of the bone density test score less than 4.55 is 0.9999.