Question

A survey found that women’s heights are normally distributed with mean 63.5 in. and standard deviation 2.9 in. The survey also found that men’s heights are normally distributed with mean 67.2 in. and standard deviation 3.8 in. Consider an executive jet that seats six with a doorway height of 56.5 in. Complete Parts (a) through (c) below.

A) What percentage of adult men can fit through the door without bending?

B) Does the door design with a height of 56.5 appear to be adequate? Why didn’t the engineers design a larger door?

C) What doorway height would allow 40% of men to fit without bending?

Can you please give me step by step ways to work this problem....not just answer!! Thank you!

Answer 1

Part A

Solution:

Here, we have to find the probability of adult men can fit through the door without bending.

That is, we have to find P(X<56.5)

We are given that heights of men are normally distributed with

Mean = 67.2 in.

SD = 3.8 in.

Z = (X – mean) / sd

Z = (56.5 – 67.2) / 3.8

Z = -2.81579

P(Z< -2.81579) = 0.002433 (by using z-table or excel)

About 0.2% of adult men can fit through the door without bending.

Part B

The door design with a height of 56.5 does not appear to be adequate, because very low proportion of men will fit with this height. Engineers didn’t design a large door for optimization purpose of overall jet design.

Part C

Here, we have to find height X for which 40% of men will fit without bending.

X = Mean + Z*SD

Z for lower 40% area is given as below:

Z = -0.25335 (by using z-table or excel)

Mean = 67.2 in.

SD = 3.8 in.

X = 67.2 + (-0.25335)*3.8

X = 66.23727

Required doorway height = 66.2 inch