Question

The heights of men in the United States are approximately normally distributed with mean 69.1 inches and standard deviation 2.9 inches. The heights of women in the United States are approximately nor- mally distributed with mean 63.7 inches and standard deviation 2.7 inches. Additionally, suppose that the heights of husbands and wives have a correlation of 0.3. Let X and Y be the heights of a married couple chosen at random. What are the mean and standard deviation of the average height,

Answer 1

Explanation:

Let M be the heights of men in the United States and W be the heights of women in the United States

Given that M is N(69.1, 2.9) and W (63.7, 2.7)

For a husband and wife we have average height as

`(x+y)/(2)`=Z (say)

`Mean =E(z) = (1)/(2) [E(M)+E(W)] = 66.4 inches`

Var (Z) =
`Var ((x+y)/(2) )=(1)/(4)[Var(x)+ Var(y)+2 cov (x,y)]\\`

=
`0.25[2.9^2+2.7^2+2*r*sx*sy]\\= 0.25(20.398)\\=5.0995`

Mean = 66.4" and std dev = 2.258