Throughout the semester we have seen the importance of standard deviation and how it is needed to calculate probabilities of certain outcomes. In chapter 11, you will find that a majority of the formulas are long. If you investigate the formulas, what you will find is that the formulas in chapter 11 are applications of the Pythagorean Theorem of Statistics. Let us use this concept to answer a seemingly impossible question. What is the probability that a randomly chosen woman will be taller than a randomly chosen man? And is that outcome usual or unusual? The mean height of men is 69 inches with a standard deviation of 3 inches. The mean height of women is 63.6 inches with a standard deviation of 2.5 inches. Let’s create a mean and standard deviation by combining the data set of men and women. This is technically impossible because how is it possible to collect an entire data set of men, let alone women. Let’s consider the set X – Y, where X represent the data set of women and Y represent the data set of men. The mean of the newly created data set is X-Y = 5.4, use the Pythagorean Theorem of Statistics to calculate σ X-Y Remember standard deviations are never negative. Based on the mean and standard deviation, determine the P(X – Y > 0) the probability that a randomly chosen woman is taller than a randomly chosen man. It is reasonable to assume a normal distribution.
