In a certain demographic group the average height for men is

In a certain demographic group, the average height for men is 68 inches, with standard deviation 4 inches. The average weight is 185 lbs., with standard deviation 25 lbs. The correlation between height and weight is =0.4.
a. How heavy would you expect a 6-foot-tall (i.e. 72 inches) man to be?
b. How tall do you expect a man to be, who weighs 200 lbs.?
c. If a man is one standard deviation lighter than average, how many standard deviations taller or shorter than average do you expect him to be?
d. What fraction of the variation in weight is associated with variation in height?

Solution

Let X denote height of the men ~ N(68,16)

Let Y denot weigth of the men ~ N(185, 125)

a) How heavy would you expect a 6-foot-tall (i.e. 72 inches) man to be = E(Y|X = 72)

= Y+(Y/X)*(xX) = 185 + 0.4 * (25/4)*(72 - 68)

= 185 + 10 = 195 lbs

b) How tall do you expect a man to be, who weighs 200 lbs = E(X |y = 200)

=  X+(X/Y)*(yY) = 68 + 0.4 * (4/25)*(200 - 185) = 68 + 0.96 = 68.96 inches

c) If a men is one standard deviation lighter than average then he will be shorter than average as there is positive correlation between weight and height

i.e as average weight is 185 and one standard deviation is 25 i.e if person is 160 lbs then height is

E(X |y = 160) =  X+(X/Y)*(yY) = 68 + 0.4 * (4/25)*(160 - 185) = 68 - 1.6 = 66.4 inches

d) 0.4 variation in weight is associated with variation in height