Question

according to a physical fitness organization, the amount of time, in seconds, it takes high school students to run a mile is normally distributed, with a standard deviation of 56 seconds. the accompanying figure shows this distribution and marks where the mean of the distribution is. based on what you know about the empirical rule and what you observe in the accompanying figure, which one of the following statements is most accurate?

about 68% of students have run times between 336 seconds and 560 seconds.
about 95% of students take between 392 seconds and 504 seconds to run a mile.
about 16% of students take less than 392 seconds to run a mile.
about 5% of students take longer than 560 seconds to run a mile.
about 34% of students take between 336 seconds and 392 seconds to run a mile

Explanation:

Step1: Recall Empirical Rule

For a normal - distribution, about 68% of data is within 1 standard deviation of the mean ($\mu\pm\sigma$), about 95% is within 2 standard deviations of the mean ($\mu\pm2\sigma$), and about 99.7% is within 3 standard deviations of the mean ($\mu\pm3\sigma$). Given $\sigma = 56$ and assume $\mu=448$.

Step2: Calculate 1 - standard - deviation interval

$\mu-\sigma=448 - 56=392$ and $\mu+\sigma=448 + 56=504$. About 68% of students have run - times between 392 and 504 seconds.

Step3: Calculate 2 - standard - deviation interval

$\mu - 2\sigma=448-2\times56=448 - 112 = 336$ and $\mu + 2\sigma=448+2\times56=448 + 112 = 560$. About 95% of students have run - times between 336 and 560 seconds.

Step4: Analyze left - tail probabilities

Since the normal distribution is symmetric, the area to the left of $\mu-\sigma$ is $\frac{100 - 68}{2}=16\%$. The value of $\mu-\sigma = 392$. So about 16% of students take less than 392 seconds to run a mile.

Answer:

About 16% of students take less than 392 seconds to run a mile.