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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.

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JeanaShupp

Answer: 0.0228

Step-by-step explanation:

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

[tex]\mu=30\\\\\sigma=5.5[/tex]

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 : [tex]z=\dfrac{x-\mu}{\sigma}[/tex]

[tex]z=\dfrac{19-30}{5.5}=-2[/tex]

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

[tex]P(X<19)=P(z<-2)=0.0227501\approx0.0228[/tex]

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

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