Tutoring statistics, you might imagine everyday situations. The tutor brings up one.
Let’s imagine we have two mile runners. Runner 1, called R1, has mean time 4:45, with standard deviation 10s; R2 has mean time 5:00 with standard deviation 12s.
In any given race, give the probability R1 will beat R2.
First, we convert the mile times to seconds: R1’s mean is 285s, while R2’s is 300.
The expected difference between R2 and R1’s time is 300-285=15.
We can’t add standard deviations, but rather variances: 10^2 + 12^2 = 244. The standard deviation of the difference is then 244^0.5 = 15.6.
The standardized statistic is z = (x-15)/15.6. We wonder p(x>0), which means p(z>-0.96). From the z-table, the answer is 0.8315.
So, R1 should beat R2 about 83% of the time.
Harnett, Donald L. and James L. Murphy. Statistical Analysis for Business and Economics. Don Mills: Addison-Wesley, 1993.
Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.