Question

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 62 miles per hour and 78 miles per hour. (assume the data set has a bell - shaped distribution.) approximately % of vehicles travel between 62 miles per hour and 78 miles per hour.

Explanation:

Step1: Recall the Empirical Rule

The Empirical Rule for a normal - distribution states that approximately 68% of the data lies within 1 standard deviation of the mean, 95% lies within 2 standard deviations of the mean, and 99.7% lies within 3 standard deviations of the mean. The mean speed is $\mu = 70$ miles per hour and the standard deviation is $\sigma=4$ miles per hour.

Step2: Calculate the number of standard - deviations

We want to find the percentage of vehicles with speeds between 62 and 78 miles per hour. First, calculate the number of standard - deviations for the lower and upper bounds. For the lower bound $x_1 = 62$, the z - score is $z_1=\frac{x_1-\mu}{\sigma}=\frac{62 - 70}{4}=\frac{- 8}{4}=-2$. For the upper bound $x_2 = 78$, the z - score is $z_2=\frac{x_2-\mu}{\sigma}=\frac{78 - 70}{4}=\frac{8}{4}=2$.

Step3: Apply the Empirical Rule

Since the values 62 and 78 are 2 standard deviations below and above the mean respectively, and approximately 95% of the data in a normal distribution lies within $z=-2$ and $z = 2$.

Answer:

95%