Question

A survey found that​ women's heights are normally distributed with mean 62.4 in. and standard deviation 3.1 in. The survey also found that​ men's heights are normally distributed with mean 69.3 in. and standard deviation 3.9 in. Consider an executive jet that seats six with a doorway height of 55.8 in. a. What percentage of adult men can fit through the door without​ bending? The percentage of men who can fit without bending is nothing​%. ​(Round to two decimal places as​ needed.) b. Does the door design with a height of 55.8 in. appear to be​ adequate? Why​ didn't the engineers design a larger​ door? A. 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. B. 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. The door design is​ adequate, because the majority of people will be able to fit without bending.​ Thus, a larger door is not needed. D. 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. 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 nothing in. ​(Round to one decimal place as​ needed.)

Answer 1

Solution:

Given:

Women's heights are normally distributed with mean 62.4 in. and standard deviation 3.1 in.

Thus

Men's heights are normally distributed with mean 69.3 in. and standard deviation 3.9 in.

Thus

An executive jet that seats six with a doorway height of 55.8 in

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

That is we have to find:

P( X_m < 55.8) = ............?

Find z score for x_m = 55.8

z score formula:

P( X_m < 55.8) = P( Z < -3.46)

Look in z table for z= -3.4 and 0.06 and find area.

P( Z < -3.46) = 0.0003

P( X_m < 55.8) = P( Z < -3.46)

P( X_m < 55.8) = 0.0003

P( X_m < 55.8) = 0.03%

The percentage of men who can fit without bending is 0.03%

Part b) Does the door design with a height of 55.8 in. appear to be​ adequate? Why​ didn't the engineers design a larger​ door?

Correct option is:

D. 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 we have to find x value such that:

P( X_m < x_m ) = 0.40

Thus first find z such that P( Z < z ) =0.40

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.