91影视

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 $渭=64.3$.

The population standard deviation is equal to $蟽=2.6$.

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:

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

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:

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

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

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

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

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

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

$$\begin{gathered} x=渭+z蟽 \\ =64.3+\left(-2.05\right)(2.6) \\ =58.97 \\ 鈮�59.0 \end{gathered}$$

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.