Honors AP Calculus
Worksheet for Students not on NSA Field Trip
This will be graded. Submit paper in Room R at end of F period 5/17/00.
If you have an excused absence for 5/17, due date is 5/18.

Assume that the heights of American men are normally distributed with mean 70 inches and standard deviation 3 inches, and that the heights of American woman are normally distributed with mean 65 inches and standard deviation 2.5 inches. Let the random variable X denote the height of a randomly selected man, Y the height of a randomly selected woman, and Y – X the difference in heights between an independently selected woman and man.

(a) What does P(Y > X) mean in plain English?
(b) Use a Riemann sum to compute a good numerical estimate for part (a). Observe that for a man of any height (i.e., some specific value x₁ for X), the probability that Y > X is normalcdf(x₁, 99999, 65, 2.5). These probabilities must be summed over all possible values of X. There are, of course, infinitely many possible values for X, but the probability that X falls in the interval (x₁ , x₁ + D x) is normalcdf(x₁, x₁+D x, 70, 3). The probability that X falls in the interval (x₁ , x₁ + D x) and that Y > X merely requires multiplying by the expression we found earlier for the probability that Y > X.

Note: The formula P(A and B) = P(A Ç B) = P(A) · P(B) is not true in general, but it is true here since the heights of the randomly selected man and woman are independent.

Collaboration with other students, including statistics students, is permitted if you document it. However, copying is not allowed. Each student must submit an original writeup.