Question

It is generally believed that the heights of adults males in the U.S. are approximately normally distributed with mean 70 inches (5 feet, 10 inches) and standard deviation 3 inches and that the heights of adult females in the U.S. are also approximately normally distributed with mean 64 inches (5 feet, 4 inches) and standard deviation 2.5 inches. A small university is considering custom ordering beds for their dorm rooms. Answer the following questions about the lengths of beds in dorm rooms at this university.

a) Should the university be concerned that females will not fit in the 75 inch beds? Numerically justify your answer.

b) The university decides it is too expensive to replace all the beds. Suppose the university has 2,150 beds all of which are 75 inches long. How many beds should they replace? You may assume that only those males taller than 75 inches will receive the longer beds and that females make up half of the population that will need a dorm room bed.

c) The university plans on ordering custom sized beds such that 99% of male students are expect to fit in them when lying perfectly straight. What length beds should they order? Round your answer to the nearest inch.

Answer 1

Let X be the height of adult males in US

then

Let Y be the height of adult females

then

(a) To find P( Y > 75)

=P(z > 4.4)

= 0.00001 ( from z table)

Since Probability of height of adult female being more than 75 inches is 0.00001 , which is very small.

So University should not be concerned about females not fitting in 75 inches bed .

b) Let us find P(X>75)

= P(z > 1.67)

= 0.0475 (from z table)

Probability of height of adult male being more than 75 inches is 0.0475

Males ( females) make up half of the population who need dorm bed

Thus number of males who need dorm bed is 2150/2= 1075

Number of males who need bed of more than 75 inches long = 1075* 0.0475=51 (rounding)

Thus number of beds to be replaced = 51

c) To find c such that

P( X <c ) =0.99

From z table

P( z < 2.33) =0.99

Thus

Length of bed should be ordered = 77 inches