Chapter 3, Problem 3.54RE

To determine

To find: the head size gets custom-made helmets.

Answer to Problem 3.54RE

Head circumference of less than 19.965 inches or more than 24.6095 inches

Explanation of Solution

Given:

  $\begin{array}{l}\text{μ=22}\text{.8}\text{inches}\\ \text{σ=}\text{1}\text{.1}\text{inches}\end{array}$

Formula used:

  $z=\frac{x-\mu }{\sigma }$

Calculation:

Probability of 5% = 0.05 and 95% =0.95

Z-score for corresponding to 0.05 is -2.575

Z-score for corresponding to 0.95 is 1.645

For z = -2.575

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ -2.575=\frac{x-22.8}{1.1}\\ -2.8325=x-22.8\\ x=19.9675\end{array}$

For z= 1.645

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ 1.645=\frac{x-22.8}{1.1}\\ 1.8095=x-22.8\\ x=24.6095\end{array}$

Therefore the soldiers who have a head circumference of less than 19.965 inches or more than 24.6095 inches get a custom-made helmet.