Question

A survey found that​ women's heights are normally distributed with mean 63.2 in and standard deviation

2.3 in The survey also found that​ men's heights are normally distributed with mean 69.1 in and standard deviation

3.3 in Consider an executive jet that seats six with a doorway height of 55.9 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 __%?

B) Does the door design with a height of 55.9 in. appear to be adequate? Why didn't the engineers design a larger door?

Options

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

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Finally C) The doorway height that would allow 40% of men to fit without bending ___ in.

(Round to one decimal place)

Answer 1

Solution:

Given:

i) The women's heights are normally distributed with mean 63.2 in and standard deviation 2.3 in

ii) The men's heights are normally distributed with mean = 69.1 in and standard deviation 3.3 in

iii) An executive jet that seats six with a doorway height of 55.9 in.

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

that is find:

P( X < 55.9 ) = ...............?

Find z score for x = 55.9

Thus we get:

P( X < 55.9 ) = P(Z < -4.00)

Look in z table for z = -4.0 and 0.00 and find corresponding area.

P( Z< -4.00) = 0.00003

Thus

P( X < 55.9 ) = P(Z < -4.00)

P( X < 55.9 ) = 0.00003

P( X < 55.9 ) = 0.003%

P( X < 55.9 ) = 0.00%

The percentage of men who can fit without bending is 0.00%

(Rounded to 2 decimal places)

Part B) Does the door design with a height of 55.9 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.

Part C) The doorway height that would allow 40% of men to fit without bending ___ in.

That is find x value such that:

P(X < x) = 40%

Thus find z value 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:

Thus

The doorway height that would allow 40% of men to fit without bending 68.3 in