91Ó°ÊÓ

The data for sitting back-to-knee length for adult males and females are provided.

The left tailed area is equal to the cumulative probabilities, which are obtained by using the standard normal table (Table A-2) for z-scores.

In the case of finding the right-tailed areas, the difference of these cumulative probabilities from 1 gives the required area towards the right of the z-score.

Let X represent the male back-to-length.

Let x be the value of length separating the bottom 90% of lengths from the top 10%, with a corresponding z-score z.

The shaded area in the graph shows the 90% bottom region corresponding to value x.

Then,

$$\begin{gathered} P\left(X<x\right)=0.90 \\ P\left(Z<z\right)=0.90 \end{gathered}$$

Where

$z=\frac{x-\mathrm{μ}}{\mathrm{σ}}$

Use the standard normal table;the area of 0.90 is observed corresponding to the row value 1.2 and the column value 0.08. This implies that the z score is 1.28.

Mathematically,

$P\left(Z<1.28\right)=0.90$

Thus, the value of z is 1.28.

The length is computed as:

$$\begin{gathered} \frac{x-23.5}{1.1}=1.3 \\ x=\left(1.3×1.1\right)+23.5 \\ =24.93 \\ ≈24.9 \end{gathered}$$

Therefore, the length separating the bottom 90% from the top 10% is $P_{90}=24.9\text{in}$.