Chapter 2, Problem 12PT

(a)

To determine

To find: the percentile of the soldier's head circumference

Answer to Problem 12PT

Thesoldier's head circumference is in approximately the 84th percentile.

Explanation of Solution

Given:

Mean, $\mu =22.8$ inches

Standard deviation, $\sigma =1.1$ inches.

Calculation:

By definitions, the 2-score that corresponds to the soldier whose head circumference is 23.9 inches is given as follows:

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{23.9-22.8}{1.1}\\ =1\end{array}$

Using standard normal tables, we observe the proportion corresponding to the score z = 1 as 0.8413. That means that this soldier's head circumference is in approximately the 84th percentile. This is shown in the below graph.

  

Conclusion:

Therefore, the soldier's head circumference is in approximately the 84th percentile.

(b)

To determine

To find: percent of male soldiers who require custom helmets

Answer to Problem 12PT

Approximately 0.7% of soldiers require a customized helmet order.

Explanation of Solution

Calculation:

The army's helmet supplier regularly stocks helmets that fit male soldiers with head circumferences between 20 and 26 inches. Anyone with a head circumference outside that interval requires a customized helmet order.

By definitions, the Z-score that corresponds to the soldier whose head circumference is 20 inches is given as follows:

  $\begin{array}{l}z=\frac{20-22.8}{1.1}\\ =-2.55\end{array}$

By definitions, the Z-score that corresponds to the soldier whose head circumference is 20 inches is given as follows:

  $\begin{array}{l}z=\frac{26-22.8}{1.1}\\ =2.91\end{array}$

Using standard normal tables, we observe the proportion corresponding to the score $z=-2.55$ as 0.0054 and the proportion corresponding to the score $z=2.91$ as 0.9982. Thus the area above 2.91 is $1-0.9982=0.0018$ . This is shown in the below graph.

  

This means that the area in both tails is:

  $0.0054+0.0018=0.0072$

Therefore approximately 0.7% of soldiers require a customized helmet order.

Conclusion:

Therefore approximately 0.7% of soldiers require a customized helmet order.

(c)

To determine

To find: the interquartile range for the distribution of head circumference

Answer to Problem 12PT

Interquartile range is 1.474 inches.

Explanation of Solution

Calculation:

The quartiles of a Standard Normal distribution are -0.67 and 0.67. To find the quartiles of the head circumference distribution we solve the following equations for x .

  $\begin{array}{l}-0.67=\frac{x-228}{1.1}\\ x=22.063\\ 0.67=\frac{x-228}{1.1}\\ x=23.537\end{array}$

This means that $Q_1=22.063$ and $Q_3=22.537$ .

So,

  $\begin{array}{l}IQR=Q_3-Q_1\\ =23.537-22.063\\ =1.474\text{ inches}\end{array}$

Conclusion:

Therefore, the interquartile range is 1.474 inches.