Disney Monorail Consider the same Mark VI monorail described in the preceding exercise. Again assume that heights of men are normally distributed with a mean ofin. and a standard deviation of in.

a. In determining the suitability of the monorail door height, why does it make sense to consider men while women are ignored?

b. Mark VI monorail cars have a capacity of 60 passengers. If a car is loaded with 60 randomly selected men, what is the probability that their mean height is less than 72 in.?

c. Why can't the result from part (b) be used to determine how well the doorway height accommodates men?

The heights of men are normally distributed with a mean value equal to 68.6 inches and a standard deviation equal to 2.8 inches.

a.

In order to determine the monorail door height, the heights of men are considered because, in general, men are taller than women. Thus,considering the heights of men for the door design, it is safe to assume that most women can also fit through the door.

b.

Let X denote the height of the passengers.

A sample of heights of 60 men is considered.

Thus, n=60.

Now, the sample mean value is given as follows:

$\overline{x}=72\text{inches}$

It is known that if the population is normally distributed, the sampling distribution of the sample mean is also normally distributed with mean ${\mu }_{\overline{x}}=\mu$ and standard deviation ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$.

The value of the population mean $\left(\mu \right)$ is equal to 68.6 inches.

The value of the population standard deviation $\left(\sigma \right)$ is equal to 2.8 inches.

The probability that the mean height of the 60 passengers is less than 72 inches is computed using the standard normal table as below:

$$\begin{gathered} P\left(\overline{x}<72\right)=P\left(\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{72-\mu }{\frac{\sigma }{\sqrt{n}}}\right) \\ =P\left(z<\frac{72-68.6}{\frac{2.8}{\sqrt{60}}}\right) \\ =P\left(z<9.41\right) \\ \approx 1.000 \end{gathered}$$

Therefore, the probability that the mean height of the 60 passengers is less than 72 inches is approximately equal to 1.000.

c.

The probability that the mean height is less than 72 inches (computed in part (b) is inefficient in determining the effectiveness of the doorway height in accommodating men as it does not describe the proportion of individual men who can fit through the door.

The adequacy of the doorway height can only be determined by computing the probability of the individual height of men and not the mean height of men.