Question

A physical fitness association is including the mile run in its high school fitness test. The time for this event is known to possess a normal distribution with a mean of seconds and a standard deviation of seconds. Find the probability that a randomly selected high school student can run the mile in less than seconds. Round to four decimal places.

Answer 1

This probability is the p-value of Z given
`Z = (X - \mu)/(\sigma)`, considering X as less than X seconds,
`\mu` as the mean and
`\sigma` as the standard deviation.

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

In this question:

Mean
`\mu`, standard deviation
`\sigma`.

Find the probability that a randomly selected high school student can run the mile in less than X seconds.

This probability is the p-value of Z given
`Z = (X - \mu)/(\sigma)`, considering X as less than X seconds,
`\mu` as the mean and
`\sigma` as the standard deviation.