Question

World class marathon runners are known to run that distance (26.2 miles) in an average of 146 minutes with a standard deviation of 14 minutes.

If we sampled a group of world class runners from a particular race, find the probability of the following:

**(use 4 decimal places)**

a.) The probability that one runner chosen at random finishes the race in less than 140 minutes.

b.) The probability that 10 runners chosen at random have an average finish time of less than 140 minutes.

c.) The probability that 50 runners chosen at random have an average finish time of less than 140 minutes.

Answer 1

a)

X ~ N ( µ = 146 , σ = 14 )
We convert this to standard normal as
P ( X < x ) = P ( Z < ( X - µ ) / σ )
P ( ( X < 140 ) = P ( Z < 140 - 146 ) / 14 )
= P ( Z < -0.43 )
P ( X < 140 ) = 0.3336

b)

X ~ N ( µ = 146 , σ = 14 )
P ( X < 140 )
Standardizing the value
Z = ( X - µ ) / (σ/√(n)
Z = ( 140 - 146 ) / ( 14 / √10 )
Z = -1.36
P ( ( X - µ ) / ( σ/√(n)) = ( 140 - 146 ) / ( 14 / √(10) )
P ( X < 140 ) = P ( Z < -1.36 )
= 0.0869

c)

X ~ N ( µ = 146 , σ = 14 )
P ( X < 140 )
Standardizing the value
Z = ( X - µ ) / (σ/√(n)
Z = ( 140 - 146 ) / ( 14 / √50 )
Z = -3.03
P ( ( X - µ ) / ( σ/√(n)) = ( 140 - 146 ) / ( 14 / √(50) )
P ( X < 140 ) = P ( Z < -3.03 )
= 0.0012