Application of z-Scores Solutions

• The z-table provides the area under the curve    that is less than    (to the left of) the z-score.

• The total area under a standard normal curve is    1.00 or 100%   .

• A negative z-score represents    a data point below the mean   , and a positive z-score represents    a data point above the mean   .

• If two z-scores are given and the area between those points needs to be determined,    calculate the difference between their areas   .

Example 4

Determine the area of the shaded and unshaded regions. Then find the area above $\mathit{z}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{3}$.

Example 5

The heights of college basketball players were measured. The average height was 195 cm, with a standard deviation of 6.5 cm. If sixteen players were randomly selected from all the college players, does the data appear to be normally distributed? Explain.  

Basketball Player Height Distribution
z	–2	–1	0	1	2
Area < z	0.02	0.16	0.5	0.84	0.98
Predicted: 16z	$0.32\approx 0$	$2.26\approx 2$	8	$13.44\approx 13$	$15.68\approx 16$
$\mathit{X}\mathbf{<}\mathbf{𝜇}\mathbf{\pm }\mathbf{2}\mathbf{𝜎}$	$$\begin{gathered} 195–13= \\ X<182 \end{gathered}$$	$$\begin{gathered} 195–6.5= \\ X<188.5 \end{gathered}$$	$X<195$	$$\begin{gathered} 195+6.5= \\ X<201.5 \end{gathered}$$	$$\begin{gathered} 195+13= \\ X<208 \end{gathered}$$
Actual Count	none	2	9	14	all 16

Example 6

According to the United States Mint, the average diameter of a nickel is 21.21 millimeters. The likelihood of a nickel having a diameter less than 21.288 mm is 94.7%.

1. Determine the standard deviation for the diameter of a nickel.
2. Determine $\mathrm{P}\left(21.21<X<21.288\right)$.

2. $$\begin{gathered} \begin{array}{ccc}z=\frac{21.21–21.21}{0.0481}& & 0.947–0.05=0.447\\ z=0& & \mathrm{P}\left(21.21<X<21.288\right)=0.477\end{array} \\ \mathrm{P}\left(X<21.21\right)=0.50 \end{gathered}$$