the times of all 15 year olds who run a certain race are approximately normally distributed with a given…

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?\n0.15%\n0.30%\n0.60%\n2.50%

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?\n0.15%\n0.30%\n0.60%\n2.50%

Answer

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 $\sigma = 1.2$. $z=\frac{14.4 - 18}{1.2}=\frac{- 3.6}{1.2}=-3$

Step2: Find the percentage using the standard normal distribution

The standard normal distribution table gives the area to the left of a z - score. For $z=-3$, the area to the left (the percentage of values less than the corresponding $x$ - value) is approximately $0.135%+0.015% = 0.15%$ (from the properties of the normal distribution: the area between $z=-3$ and $z = - 2$ is about $2.35% - 2.2%=0.15%$ and we know the cumulative area up to $z=-3$).

Answer:

A. $0.15%$