Question

Biometric Security Standing eye heights of men are normally distributed with a mean of 64.3 in. and a standard deviation of 2.6 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 men with standing eye heights greater than 70 in., what percentage of men will find that height uncomfortable?

b. In positioning the eye recognition security system, we want it to be suitable for the tallest 98% of standing eye heights of men. What standing eye height of men separates the tallest 98% of standing eye heights from the lowest 2%?

Short answer

a. The percentage of men who will find the height of the eye recognition system uncomfortable is equal to 1.43%.

b. The standing height of men that separates the tallest 98% of heights from the lowest 2% is equal to 59.0 inches.

Step-by-step solution

Given Information

It is given that the population of standing heights of men is normally distributed with a mean value equal to 64.3 inches and a standard deviation equal to 2.6 inches.Men with a standing height greater than 70 inches find the height of the eye recognition security system to be uncomfortable.

Conversion of sample value to z-score

Here, the population mean value is equal to $\mu =64.3$.

The population standard deviation is equal to $\sigma =2.6$.

The sample value given is equal to x=70 inches.

The following formula is used to convert a given sample value (x=70) to a z-score:

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{70-64.3}{2.6} \\ =2.19 \end{gathered}$$

The required probability value can be computed using the value of z-score.

Required probability

a.

The probability of getting a standing height greater than 70 inches is computed using the standard normal table as:

$$\begin{gathered} P\left(x>70\right)=P\left(z>2.19\right) \\ =1-P\left(z<2.19\right) \\ =1-0.9857 \\ =0.0143 \end{gathered}$$

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

$$\begin{gathered} \text{Percentage}=0.0143\times 100\% \\ =1.43\% \end{gathered}$$

Therefore, the percentage of men who will find the height of the eye recognition system uncomfortable is equal to 1.43%.

Conversion of probability value to z-score

b.

Let X denote the standing height of men.

Now, it is given that the positioning of the eye recognition system suits the tallest 98% of the men.

Thus, the value that separates the bottom of the men from the tallest of men has the following expression:

$$\begin{gathered} P\left(Z>z\right)=0.98 \\ 1-P\left(Z<z\right)=0.98 \\ P\left(Z<z\right)=1-0.98 \\ =0.02 \end{gathered}$$

The corresponding z-score for the left-tailed probability value equal to 0.02 is seen from the standard normal table and is approximately equal to -2.05.

Thus, $P\left(z<-2.05\right)=0.02$.

Conversion of z-score to sample value

The value of the height corresponding to the z-score of -2.05 is computed below:

$$\begin{gathered} x=\mu +z\sigma \\ =64.3+\left(-2.05\right)(2.6) \\ =58.97 \\ \approx 59.0 \end{gathered}$$

Therefore, the standing height of men that separates the tallest 98% of standing eye heights from the lowest 2% is equal to 59.0 inches.