Question

The annual 2-mile fun-run is a traditional fund-raising event to support local arts and sciences activities. It is known that the mean and the standard deviation of finish times for this event are respectively \mu μ = 30 and \sigma σ = 5.5 minutes. Suppose the distribution of finish times is approximately bell-shaped and symmetric. Find the approximate proportion of runners who finish in under 19 minutes.

Answer 1

Answer: 0.0228

Explanation:

Given : The mean and the standard deviation of finish times (in minutes) for this event are respectively as :-

`\mu=30\\\\\sigma=5.5`

If the distribution of finish times is approximately bell-shaped and symmetric, then it must be normally distributed.

Let X be the random variable that represents the finish times for this event.

z score :
`z=(x-\mu)/(\sigma)`

`z=(19-30)/(5.5)=-2`

Now, the probability of runners who finish in under 19 minutes by using standard normal distribution table :-

`P(X<19)=P(z<-2)=0.0227501\approx0.0228`

Hence, the approximate proportion of runners who finish in under 19 minutes = 0.0228