Question

The heights of males in a population are approximately normally distributed with mean 69.2 inches and standard deviation 2.92. The heights of females in the same population are approximately normally distributed with mean 64.1 inches and standard deviation 2.75.

a. Suppose one male from this age group is selected at random and one female is independently selected at random and their heights added. Find the mean and standard error of the sampling distribution of this sum. Mean = Standard deviation = (round to three decimal places)

b. Find the probability that the sum of the heights is less than 125 inches. (round to four decimal places)

c. What total heights are reasonably likely? (, ) (round to two decimal places)

d. What is the probability that the male is at least 2 inches taller than the female? (round to four decimal places)

Answer 1

a) The key to these problems is that when you add (or subtract) two items from normal curves, you need to ADD the RMS of the standard deviation (even when subtracting). So:

Mean = mean1 + mean2 = 69.2 + 64.1 = 133.3
Std Dev = sqrt(2.92^2 + 2.75^2) ~= 4.01"

b) P(Sum < 125) = P(Z < -8.3 / 4.01)
                           = P(Z < -2.069)
                           =1?P ( Z<2.069 )
                           =1?0.9808
                           =0.0192

c) If we pick -3 < Z < 3, then you get sums between ~113" to 137".

d) This works the same as adding, except you subtract the means:

Mean diff = 69.2 - 64.1 = 5.1"
Std Dev = 4.01"

P(Diff > 2") = P(Z > -3.1 / 4) = P(Z > -0.775) =P ( Z<0.775 )=0.7823