Question

Running times for 400 meters are Normally distributed for young men between 18 and 30 years of age with a mean of 93 seconds and a standard deviation of 16 seconds. How fast does a man have to run to be in the top 5% of runners (quickest runner)?

Answer 1

Answer: The running time should at least 119.32 seconds to be in the top 5% of runners.

Explanation:

Let X= random variable that represents the running time of men between 18 and 30 years of age.

As per given, X is normally distrusted with mean
`\mu=93\text{ seconds}` and standard deviation
`\sigma=16\text{ seconds}`.

To find: x in top 5% i.e. we need to find x such that P(X<x)=95% or 0.95.

i.e.
`P((X-\mu)/(\sigma)<(x-93)/(16))=0.95`

`P(Z<(x-93)/(16))=0.95\ \ \ \ \ [Z=(X-\mu)/(\sigma)]`

Since, z-value for 0.95 p-value ( one-tailed) =1.645

So,

Hence, the running time should at least 119.32 seconds to be in the top 5% of runners.