Question

the times of all 15 - year - olds who run a certain race are approximately normally distributed with a given mean $mu = 18$ sec and standard deviation $sigma=1.2$ sec. what percentage of the runners have times less than 14.4 sec?
0.15%
0.30%
0.60%
2.50%

Explanation:

Step1: Calculate the z - score

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x = 14.4$, $\mu=18$, and $\sigma = 1.2$. So, $z=\frac{14.4 - 18}{1.2}=\frac{- 3.6}{1.2}=-3$.

Step2: Use the standard normal distribution table

The standard - normal distribution table gives the cumulative probability $P(Z\lt z)$. Looking up $z=-3$ in the standard normal distribution table, we find that $P(Z\lt - 3)=0.00135\approx0.15\%$.

Answer:

0.15%