Biometric Security In designing a security system based on eye (iris) recognition, we must consider the standing eye heights of women, which are normally distributed with a mean of 59.7 in. and a standard deviation of 2.5 in. (based on anthropometric survey data from Gordon, Churchill, et al.).

a. If an eye recognition security system is positioned at a height that is uncomfortable for women with standing eye heights less than 54 in., what percentage of women will find that height uncomfortable?

b. In positioning the eye recognition security system, we want it to be suitable for the lowest 95% of standing eye heights of women. What standing eye height of women separates the lowest 95% of standing eye heights from the highest 5%.

It is given that the population of the standing heights of women is normally distributed with a mean value equal to 59.7 inches and a standard deviation equal to 2.5 inches. Women with a standing height less than 54 inches find the height of the eye recognition security system to be uncomfortable.

Here, the population mean value is equal to $\mu =59.7$.

The population standard deviation is equal to $\sigma =2.5$.

The sample value given is equal to x=54 inches.

The following formula is used to convert a given sample value (x=54) to a z-score:

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{54-59.7}{2.5} \\ =-2.28 \end{gathered}$$

By referring to the standard normal table, the required probability value can be computed using the value of the z-score.

a.

The probability of getting a standing height less than 54 inches is computed below.

$$\begin{gathered} P\left(x<54\right)=P\left(z<-2.28\right) \\ =0.0113 \end{gathered}$$

By converting the probability value to a percentage, the following value is obtained:

$$\begin{gathered} \text{Percentage}=0.0113\times 100\% \\ =1.13\% \end{gathered}$$

Therefore, the percentage of women who will find the height of the eye recognition system uncomfortable is equal to 1.13%.

b.

Let X denote the standing height of men.

Now, it is given that the positioning of the eye recognition system suits the shortest 95% of women.

Thus, the value that separates the bottom 95%of the standing eye height from the highest5% has the following expression:

$P\left(Z<z\right)=0.95$

Now, the corresponding z-score for the left-tailed probability value equal to 0.95 is seen from the table and is approximately equal to 1.645.

Thus, $P\left(z<1.645\right)=0.95$.

The value of the standing eye height corresponding to the z-score of 1.645 is computed below.

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ x=\mu +z\sigma \\ =59.7+\left(1.645\right)\left(2.5\right) \\ =63.8 \end{gathered}$$

Therefore, the standing height of women that separates the lowest 95% of standing eye heights from the top 5% is equal to 63.8 inches.