In Exercises 13–20, use the data in the table below for sitting adult malesand females (based on anthropometric survey data from Gordon, Churchill, et al.). Thesedata 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

Significance Instead of using 0.05 for identifying significant values, use the criteria that a value x is significantly high if P(x or greater) 0.025 and a value is significantly low if P(x or less) 0.025. Find the female back-to-knee length, separating significant values from those that are not significant. Using these criteria, is a female back-to-knee length of 20 in. significantly low?

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

The significantly high and low values are defined as $\text{P}\left(\text{x or greater}\right)⩽0.025$and $\text{P}\left(\text{x or less}\right)⩽0.025$

respectively.

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

Thus,

$$\begin{gathered} X~N\left(\mu ,{\sigma }^2\right) \\ ~N\left(22.7,{1.0}^2\right) \end{gathered}$$

Let $x_1$and $x_2$ be the minimum and maximum values such that, $\text{P}\left(X<x_1\right)=0.025$and$P\left(X>x_2\right)=0.025$

Further,

$$\begin{gathered} P\left(X>x_2\right)=0.025 \\ 1-P\left(X<x_2\right)=0.025 \\ P\left(X<x_2\right)=0.975 \end{gathered}$$

The two values are shown on the graph below:

The cumulative area to the left of $x_1$ and $x_2$ is 0.025 and 0.975, respectively. Let the corresponding z-scores be $z_1$and $z_2$.

Thus,

$$\begin{gathered} P\left(Z<z_1\right)=0.025 \\ P\left(Z<z_2\right)=0.975 \end{gathered}$$

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

• The intersection of row 1.9 and column 0.06 has the value 0.9750. Thus, $z_2=1.96$.
• The intersection of row -1.9 and column 0.06 has the value 0.0250. Thus, $z_1=1.96$.

Relationship between x and corresponding z-score,

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

Thus,

$$\begin{gathered} x_1=\mu +z_1\sigma \\ =22.7+\left(-1.96\right)\left(1.0\right) \\ =20.7 \end{gathered}$$

$$\begin{gathered} x_2=\mu +z_2\sigma \\ =22.7+\left(1.96\right)\left(1.0\right) \\ =24.7 \end{gathered}$$

Thus, it is concluded that the significantly high values are separated by 24.7 in, and the significantly low values are separated from the not significant by 20.7 in.

Thus, the significantly high values are larger than 24.7 in. Therefore,

$P\left(24.7\text{or}\text{greater}\right)=0.025$

Thus, the significantly low values are larger than 20.7 in. Therefore,

$P\left(20.7\text{or}\text{less}\right)=0.025$

The measure 20 in. lies to the left of 20.7 in, which implies it is a significantly low value.