Question

when samuel commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of 31 minutes and a standard deviation of 3 minutes. using the empirical rule, what percentage of his commutes will be between 28 and 34 minutes?

Explanation:

Step1: Calculate the number of standard deviations from the mean

The formula for the number of standard deviations \(z=\frac{x - \mu}{\sigma}\), where \(\mu = 31\) (mean), \(\sigma=3\) (standard deviation).
For \(x = 28\): \(z_1=\frac{28 - 31}{3}=\frac{- 3}{3}=-1\)
For \(x = 34\): \(z_2=\frac{34 - 31}{3}=\frac{3}{3}=1\)

Step2: Apply the empirical rule

The empirical rule (68 - 95 - 99.7 rule) states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(z=-1\) and \(z = 1\) (i.e., \(\mu\pm\sigma\)), \(95\%\) lies within \(\mu\pm2\sigma\), and \(99.7\%\) lies within \(\mu\pm3\sigma\)

Answer:

\(68\%\)