The times for completing one circuit of a bicycle course are

The times for completing one circuit of a bicycle course are normally distributed with a mean of 81.7 minutes and a standard deviation of 8.6 minutes. An association wants to sponsor a race and provide prizes for the best (that is, fastest) 15% of riders. Where should they expect to set the cutoff time to earn a prize?

Solution

P(X<x)=0.15

--> P((X-mean)/s <(x-81.7)/8.6) = 0.15

--> P(Z<(x-81.7)/8.6) =0.15

--> (x-81.7)/8.6 = -1.04 (from standard normal table)

So x= 81.7-1.04*8.6=72.756