Questions: Use the Empirical Rule. The mean speed of a sample of vehicles along a stretch of highway is 70 miles per hour, with a standard deviation of 4 miles per hour. Estimate the percent of vehicles whose speeds are between 66 miles per hour and 74 miles per hour. (Assume the data set has a bell-shaped distribution.) Approximately % of vehicles travel between 66 miles per hour and 74 miles per hour.

Solution Steps

The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution:

• Approximately 68% of the data falls within one standard deviation of the mean.
• Approximately 95% of the data falls within two standard deviations of the mean.
• Approximately 99.7% of the data falls within three standard deviations of the mean.

Given:

• Mean speed (\(\mu\)) = 70 miles per hour
• Standard deviation (\(\sigma\)) = 4 miles per hour

The range for one standard deviation from the mean is: \[ \mu - \sigma = 70 - 4 = 66 \text{ miles per hour} \] \[ \mu + \sigma = 70 + 4 = 74 \text{ miles per hour} \]

According to the Empirical Rule, approximately 68% of the data falls within one standard deviation of the mean. Therefore, approximately 68% of vehicles travel between 66 miles per hour and 74 miles per hour.

Final Answer