91Ó°ÊÓ

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.

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:

\(\begin{aligned}{c}z = \frac{{x - \mu }}{\sigma }\\ = \frac{{70 - 64.3}}{{2.6}}\\ = 2.19\end{aligned}\)

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

a.

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

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

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

\(\begin{aligned}{c}{\rm{Percentage}} = 0.0143 \times 100\% \\ = 1.43\% \end{aligned}\)

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

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 \(2\% \)of the men from the tallest \(98\% \)of men has the following expression:

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

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, \(P\left( {z < - 2.05} \right) = 0.02\).

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

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

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.