Question

In Exercises 9 –12, consider a value to be significantly low if its z score is less than or equal to -2 or consider the value to be significantly high if its z score is greater than or equal to 2.

Designing Aircraft Seats In the process of designing aircraft seats, it was found that men have hip breadths with a mean of 36.6 cm and a standard deviation of 2.5 cm (based on anthropometric survey data from Gordon, Clauser, et al.). Identify the hip breadths of men that are significantly low or significantly high.

Short answer

The hip breadths of men equal to or greater than 31.6 cm are significantly high, and those of men equal to or less than 41.6 cm are significantly low.

Step-by-step solution

Given information

The mean value of the hip breadths of men is equal to 36.6 cm.

The standard deviation of the hip breadths of men is equal to 2.5 cm.

Z-score

Thez-scoreof a value is calculated using the following expression. Here, is the sample mean, and s is the standard deviation.

$Z=\frac{x-\overline{x}}{s}$

If the value of the z-score is equal to or less than -2, then the sample observation is significantly low.

If the value of the z-score is equal to or greater than 2, then the sample observation is significantly high.

Identify the hip breadths of men that are significantly low or high

The significantly low hip breadths are obtained as follows:

$$\begin{gathered} Z⩽-2 \\ \frac{x-\overline{x}}{s}⩽-2 \\ \frac{x-36.6}{2.5}⩽-2 \\ x⩽31.6 \end{gathered}$$

Thus, the hip breadths of men equal to or less than 31.6 cm aresignificantly low.

The significantly high hip breadths are obtained as follows:

$$\begin{gathered} Z⩾2 \\ \frac{x-\overline{x}}{s}⩾2 \\ \frac{x-36.6}{2.5}⩾2 \\ x⩾41.6 \end{gathered}$$

Thus, the hip breadths of men equal to or above 41.6 cm are significantly high.