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

Answer:

This probability is the p-value of Z given $Z = \frac{X - \mu}{\sigma}$, considering X as less than X seconds, $\mu$ as the mean and $\sigma$ as the standard deviation.

Step-by-step 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:

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

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 $Z = \frac{X - \mu}{\sigma}$, considering X as less than X seconds, $\mu$ as the mean and $\sigma$ as the standard deviation.