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 minutes. What is the probability that a randomly selected runner has a time less than or equal to 3 hours and 20 minutes? Use the portion of the standard normal table below to help answer the question. Z Probability 0. 00 0. 5000 0. 50 0. 6915 1. 00 0. 8413 2. 00 0. 9772 3. 00 0. 9987 16% 32% 34% 84%.

Answer 1

The probability that a randomly selected runner has a timeless than or equal to 3 hours and 20 minutes will be 16%

What will be the probability?

It is given that

Standard deviaton = 30 minutes

Mean = 3 hours and 50 minutes

$\mu=3\times 60+50=230\ minutes$

The randomly selected runner has a time less than or equal to 3 hours and 20 minutes

$X=3\times60+20=200\ minutes$

Now the Z probability is given as

$Z=\dfrac{X-\mu}{\sigma}$

$Z=\dfrac{200-230}{30} =-1$

Thus the value of z at 1 in the given data is 0.8413

So the probability will be

$P=1-Z$

$P=1-0.84=16$

Thus the probability that a randomly selected runner has a timeless than or equal to 3 hours and 20 minutes will be 16%

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