Question

The times of the runners in a marathon are normally distributed, with a mean of 3 hours and 50 minutes and a standard deviation of 30 minute. What is the probability that a randomly selected runner has a time less than or equal to 3 hours and 20 minutes?

Answer 1

`3\text{ hr }50\text{ min }=230\text{ min }`

`3\text{ hr }20\text{ min }=200\text{ min }`


`\mathbb P(X<200)=\mathbb P\left((X-230)/(30)<(200-230)/(30)\right)=\mathbb P(Z<-1)\approx0.1587`

(Same answer using the empirical rule: recalling that approximately 68% of a normal distribution lies within one standard deviation of the mean, so that 32% lies without, and due to symmetry of the distribution you know that approximately 16% of the distribution lies to the left of one standard deviation from the mean.)