Question

A survey found that​ women's heights are normally distributed with mean 62.3 in. and standard deviation 3.6 in. The survey also found that​ men's heights are normally distributed with mean 67.7 in. and standard deviation 3.3 in. Consider an executive jet that seats six with a doorway height of 55.7 in. Complete parts​ (a) through​ (c) below.

a. What percentage of adult men can fit through the door without​ bending?

The percentage of men who can fit without bending is

___.

​(Round to two decimal places as​ needed.)

b. Does the door design with a height of

55.755.7

in. appear to be​ adequate? Why​ didn't the engineers design a larger​ door?

A.

The door design is​ inadequate, but because the jet is relatively small and seats only six​ people, a much higher door would require major changes in the design and cost of the​ jet, making a larger height not practical.

B.

The door design is​ adequate, because the majority of people will be able to fit without bending.​ Thus, a larger door is not needed.

C.

The door design is​ adequate, because although many men will not be able to fit without​ bending, most women will be able to fit without bending.​ Thus, a larger door is not needed.

D.

The door design is​ inadequate, because every person needs to be able to get into the aircraft without bending. There is no reason why this should not be implemented.

c. What doorway height would allow​ 40% of men to fit without​ bending?

The doorway height that would allow​ 40% of men to fit without bending is

__

in.

​(Round to one decimal place as​ needed.)

Answer 1

Solution:

Given:

Women's heights are normally distributed with mean 62.3 in. and standard deviation 3.6 in.

Men's heights are normally distributed with mean = 67.7 in. and standard deviation 3.3 in.

an executive jet that seats six with a doorway height of 55.7 in.

Part a) What percentage of adult men can fit through the door without​ bending?

That is find:

P( X < 55.7 ) = .........?

Find z score for x = 55.7

Thus we get:

P( X < 55.7 ) = P( Z< -3.64)

Look in z table for z = -3.6 and 0.04 and find corresponding area.

P( Z<-3.64 ) = 0.00014

Thus

P( X < 55.7 ) = P( Z< -3.64)

P( X < 55.7 ) = 0.00014

P( X < 55.7 ) = 0.014%

P( X < 55.7 ) = 0.01%

( Rounded to two decimal places)

Part b. Does the door design with a height of 55.7 in. appear to be​ adequate? Why​ didn't the engineers design a larger​ door?

Correct option is:

A.  The door design is​ inadequate, but because the jet is relatively small and seats only six​ people, a much higher door would require major changes in the design and cost of the​ jet, making a larger height not practical.

Part c) What doorway height would allow​ 40% of men to fit without​bending?

That is find x value such that:

P( X < x ) = 40%

P( X < x ) = 0.40

thus find z value such that:

P( Z < z ) = 0.4000

Look in z  table for Area = 0.4000 or its closest area and find corresponding z value.

Area 0.4013 is closest to 0.4000 and it corresponds to -0.2 and 0.05

thus z = -0.25

Now use following formula to find x value:

in.

The doorway height that would allow​ 40% of men to fit without bending is 66.9 in.