In Exercises 13–20, use the data in the table below for sitting adult males and females (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats. (Hint: Draw a graph in each case.)

Sitting Back-to-Knee Length (Inches)

	Mean	St. Dev	Distribution
Males	23.5 in	1.1 in	Normal
Females	22.7 in	1.0 in	Normal

Instead of using 0.05 for identifying significant values, use the criteria that a value x is significantly high if P(x or greater) ≤ 0.01 and a value is significantly low if P(x or less) ≤ 0.01. Find the back-to-knee lengths for males, separating significant values from those that are not significant. Using these criteria, is a male back-to-knee length of 26 in. significantly high?

The table states the distribution of back-to-knee length for males and females.

The significantly high and low values are defined as $\text{P}\left(\text{x or greater}\right)⩽0.01$and $\text{P}\left(\text{x or less}\right)⩽0.01$ respectively.

Let X be the distribution of back-to-knee lengths of males.

Thus,

$$\begin{gathered} X~N\left(\mu ,{\sigma }^2\right) \\ ~N\left(23.5,{1.1}^2\right) \end{gathered}$$

Let $x_1$ and $x_2$ be the minimum and maximum values such that, $\text{P}\left(X<x_1\right)=0.01$and $P\left(X>x_2\right)=0.01$.

Further,

$$\begin{gathered} P\left(X>x_2\right)=0.01 \\ 1-P\left(X<x_2\right)=0.01 \\ P\left(X<x_2\right)=0.99 \end{gathered}$$

The two values are shown on the graph below:

The cumulative area to the left of $x_1$ and $x_2$ is 0.01 and 0.99, respectively. Let the corresponding z-scores be $z_1$and $z_2$.

Thus,

$$\begin{gathered} P\left(Z<z_1\right)=0.01 \\ P\left(Z<z_2\right)=0.99 \end{gathered}$$

The z-score is obtained from the standard normal table:

• The intersection of row 2.3 and column 0.03 has the value 0.9901 (closest to 0.99). Thus, $z_2=2.33$.
• The intersection of row -2.3 and column 0.03 has the value 0.0099 (closest to 0.01). Thus, $z_1=-2.33$.

Relationship between x and corresponding z-score:

$z=\frac{x-\mu }{\sigma }$

Thus,

$$\begin{gathered} x_1=\mu +z_1\sigma \\ =23.5+\left(-2.33\right)\left(1.1\right) \\ =20.9 \end{gathered}$$

$$\begin{gathered} x_2=\mu +z_2\sigma \\ =23.5+\left(2.33\right)\left(1.1\right) \\ =26.1 \end{gathered}$$

Thus, it is concluded that the significantly high values are separated by 26.1 in., and significantly low are separated from not significant by 20.9 in.

Thus, the significantly high values are larger than 26.1 in. Therefore,

$P\left(26.1\text{or}\text{greater}\right)=0.01$

Thus, the significantly low values are larger than 20.9 in. Therefore,

$P\left(20.9\text{or}\text{less}\right)=0.01$

The measure of 26 in. lies to the left of 26.1 in., which implies it is not a significantly high value.