Question

use the empirical rule. the mean speed of a sample of vehicles along a stretch of highway is 68 miles per hour, with a standard deviation of 3 miles per hour. estimate the percent of vehicles whose speeds are between 65 miles per hour and 71 miles per hour. (assume the data set has a bell - shaped distribution.)
approximately \\% of vehicles travel between 65 miles per hour and 71 miles per hour.

Explanation:

Step1: Calculate the number of standard deviations from the mean

The mean $\mu = 68$ and the standard deviation $\sigma=3$.
For $x = 65$, $z=\frac{65 - 68}{3}=\frac{- 3}{3}=-1$.
For $x = 71$, $z=\frac{71 - 68}{3}=\frac{3}{3}=1$.

Step2: Apply the Empirical Rule

The Empirical Rule states that for a bell - shaped (normal) distribution:

• Approximately $68\%$ of the data lies within $z=-1$ and $z = 1$ (i.e., $\mu\pm\sigma$), $95\%$ within $z=-2$ and $z = 2$ ($\mu\pm2\sigma$), and $99.7\%$ within $z=-3$ and $z = 3$ ($\mu\pm3\sigma$).

Answer:

Approximately $68\%$ of vehicles travel between 65 miles per hour and 71 miles per hour.