Doorway Height The Boeing 757-200 ER airliner carries 200 passengers and has doors with a height of 72 in. Heights of men are normally distributed with a mean of 68.6 in. and a standard deviation of 2.8 in. (based on Data Set 1 “Body Data” Appendix B).

a. If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending.

b. If half of the 200 passengers are men, find the probability that the mean height of the 100 men is less than 72 in.

c. When considering the comfort and safety of passengers, which result is more relevant: the probability from part (a) or the probability from part (b)?Why?

d. When considering the comfort and safety of passengers, why are women ignored in this case?

The height of the men is normally distributed with a mean $\left(\mu \right)$equal to 68.6 inches and a standard deviation $\left(\sigma \right)$equal to 2.8 inches.

a.

The height of doors in an airliner is equal to inches.

Let X denote the height of the men.

The probability of selecting a man who can fit through the doorway is computed using the standard normal table, as shown below.

$$\begin{gathered} P\left(x<72\right)=P\left(\frac{x-\mu }{\sigma }<\frac{72-\mu }{\sigma }\right) \\ =P\left(z<\frac{72-68.6}{2.8}\right) \\ =P\left(z<1.21\right) \\ =0.8869 \end{gathered}$$

Therefore, the probability that a randomly selected man can fit through the doorway is equal to 0.8869.

b.

Let$\overline{x}$ denote the sample mean height of the men.

The sample mean height of the men follows a normal distribution with a mean equal to${\mu }_{\overline{x}}=\mu$ and a standard deviation equal to ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$.

The sample size is equal to n=100.

The probability that the sample mean height of the sample of 100 men is less than 72 inches is computed using the standard normal table, as shown below.

$$\begin{gathered} P\left(\overline{x}<72\right)=P\left(\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{72-\mu }{\frac{\sigma }{\sqrt{n}}}\right) \\ =P\left(z<\frac{72-68.6}{\frac{2.8}{\sqrt{100}}}\right) \\ =P\left(z<12.14\right) \\ =1.0000 \end{gathered}$$

Therefore, the probability that the mean height of the sample of 100 men is less than 72 inches is equal to 1.0000.

c.

The outcome of part (b) tells us about the average height for a group of 100 men, but it doesn't tell us anything about the comfort and safety of the individual male passengers.

For the comfort of all the passengers on board, the individual height of the passengers should be kept in mind.

Therefore, the probability in part (a) that represents that the probability of a single man with a height less than 72 inches is equal to 88.69% is more relevant.

d.

Here, the comfort of the passengers depends on whether a passenger can fit through the doorway without bending.

In general, men are taller than women.

Thus, to examine the comfort and safety of the passengers, it is sufficient to consider the heights of men and ignore the heights of women.