Statistics: how often can something “better” be expected to perform better?

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.

Solution:

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.

Source:

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.

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