Potholes Empirical F8004E

1. **State the problem:** We want to find the approximate percentage of 1-mile long roadways with potholes numbering between $46$ and $61$ given a normal distribution with mean $\mu=55$ and standard deviation $\sigma=3$. 2. **Recall the empirical rule:** For a bell-shaped (normal) distribution, about 68% of data lies within $1$ standard deviation of the mean, about 95% lies within $2$ standard deviations, and about 99.7% lies within $3$ standard deviations. 3. **Calculate the number of standard deviations from the mean for the bounds:** - Lower bound: $46$ - Upper bound: $61$ Calculate the z-scores: $$ z_{lower} = \frac{46 - 55}{3} = \frac{-9}{3} = -3 $$ $$ z_{upper} = \frac{61 - 55}{3} = \frac{6}{3} = 2 $$ 4. **Interpret the interval:** The interval $[46,61]$ corresponds to $[-3,2]$ standard deviations from the mean. 5. **Use the empirical rule:** - About 95% of data lies between $-2$ and $2$ standard deviations. - About 99.7% lies between $-3$ and $3$ standard deviations. Since our interval is from $-3$ to $2$, it covers slightly more than 95% but less than 99.7%. 6. **Estimate the percentage:** - From $-2$ to $2$ is about 95%. - From $-3$ to $-2$ is about 2.35% (half of the 0.3% outside $\pm 3$ SD is split evenly, so about 0.15% beyond $-3$ and 2.35% between $-3$ and $-2$). So the total percentage between $-3$ and $2$ is approximately $95\% + 2.35\% = 97.35\%$. **Final answer:** Approximately $97$ percent of 1-mile long roadways have potholes numbering between $46$ and $61$.