Question

According to Boeing data, the $757$airliner carries $200$passengers and has doors with a height of $72$inches. Assume for a certain population of men we have a mean height of $69.0$inches and a standard deviation of $2.8$ inches.
a. What doorway height would allow $95\%$of men to enter the aircraft without bending?
b. Assume that half of the $200$passengers are men. What mean doorway height satisfies the condition that there is a$0.95$probability that this height is greater than the mean height of $100$men?

c. For engineers designing the $757$ , which result is more relevant: the height from part a or part b? Why?

Short answer

(a) The mean doorway height to enter the aircraft without bending is$73.60$inches.
(b) The mean doorway height for the men is $69.46$inches.

(c) The result for the height from part (a) is more relevant.

Step-by-step solution

Part (a) Step 1: Given information 

Given in the question that, the 757 airliner carries 200 passengers and has doors with a height of 72 inches.

The mean height is $69.0$inches.

The standard deviation is$2.8$inches.

Part (a) Step 2: Explanation 

According to the information, We observed that:

Men with height $\mu =69$

Standard deviation $\sigma =2.8$

We can use Ti-83 calculator to find the ${95}^{th}$percentile:

Therefore,

The 95th percentile $=invNorm(0.95,69,2.8)=73.6$

Hence, mean doorway height is$73.60$inches

Part (b) Step 1: Given information

Given in the question that, the 757 airliner carries 200 passengers and has doors with a height of 72 inches.

The mean height is $69.0$inches.

The standard deviation is $2.8$inches.

Part (b) Step 2: Explanation 

The standard deviation is $2.8$ based on the information provided.

If half of the passengers are men, the standard deviation will be:

${\sigma }_{\mathrm{X}}=\frac{\sigma }{\sqrt{n}}$

$=\frac{2.8}{10}$

$=0.28$

Hence, the ${95}^{th}$percentile$=invNorm(0.95,69,0.28)=69.46$

Part (c) Step 1: Given information 

The $757$airliner carries $200$passengers and has doors with a height of $72$ inches.

Part (c) Step 2: Explanation

The engineers designing the$757$, the result from part (a) will be more relevant comparing to part (b), because that is calculated for all passengers' height.

So the result from part (a) is more relevant.