Question

the number of potholes in any given 1 mile stretch of freeway pavement in pennsylvania has a bell - shaped distribution. this distribution has a mean of 56 and a standard deviation of 6. using the empirical rule (68 - 95 - 99.7% rule), what is the approximate percentage of 1 - mile long roadways with potholes numbering less than 38?

Explanation:

Step1: Calculate the number of standard deviations from the mean

The mean $\mu = 56$ and the standard deviation $\sigma=6$. We want to find the value $x = 38$. The number of standard deviations $z=\frac{x-\mu}{\sigma}=\frac{38 - 56}{6}=\frac{-18}{6}=- 3$

Step2: Use the Empirical Rule

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

• Approximately $68\%$ of the data lies within $1$ standard deviation of the mean ($\mu\pm\sigma$)
• Approximately $95\%$ of the data lies within $2$ standard deviations of the mean ($\mu\pm2\sigma$)
• Approximately $99.7\%$ of the data lies within $3$ standard deviations of the mean ($\mu\pm3\sigma$)

The total area under the normal curve is $100\%$. The percentage of data within $\mu\pm3\sigma$ is $99.7\%$. The percentage of data less than $\mu - 3\sigma$ is $\frac{100\% - 99.7\%}{2}=0.15\%$

Answer:

$0.15\%$