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 49 and a standard deviation of 7 . Using the empirical rule (as presented in the book), what is the approximate percentage of 1 -mile long roadways with potholes numbering between 28 and 63 ? (Round percent number to 2 decimal places.) Do not enter the percent symbol. ans \( = \)

Let the number of potholes per mile be approximately normally distributed with mean \( \mu=49 \) and standard deviation \( \sigma=7 \). The given interval is from 28 to 63. Notice that \[ 28 = 49 - 3(7) \quad \text{and} \quad 63 = 49 + 2(7) \] The empirical rule tells us that approximately: - \(95\%\) of the data lie between \(\mu-2\sigma\) and \(\mu+2\sigma\), and - \(99.7\%\) of the data lie between \(\mu-3\sigma\) and \(\mu+3\sigma\). This means: - From \(49-2(7)=35\) to \(49+2(7)=63\) we have about \(95\%\) of the data. - The additional data from \(49-3(7)=28\) to \(35\) (i.e. from \(-3\sigma\) to \(-2\sigma\)) represents approximately half of the extra \(100\% - 95\% = 5\%\) that lie outside \(\mu\pm2\sigma\). This portion is approximately \[ 2.5\% - \text{(a small adjustment since actually }99.7\%\text{ are within }3\sigma\text{)} \] A more refined estimate using the empirical rule is as follows: Since \(99.7\%\) are within \(\mu\pm3\sigma\), the amount in each tail beyond \(3\sigma\) is about \(0.15\%\). Thus, from \(-3\sigma\) to \(-2\sigma\) the proportion is \[ 2.5\% - 0.15\% = 2.35\%. \] So the total percentage of roadways having potholes between 28 and 63 is approximately: \[ 95\% + 2.35\% = 97.35\%. \] Thus, the approximate percentage (rounded to two decimal places) is \[ \boxed{97.35} \]