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 62 and a standard deviation of 11 . Using the empirical rule (as presented in the book), what is the approximate percentage of 1 -mile long roadways with potholes numbering between 29 and 73 ? (Round percent number to 2 decimal places.). Do not enter the percent symbol. ans \( = \)

To solve this problem, we will use the empirical rule, which states that for a bell-shaped (normal) distribution: 1. Approximately 68% of the data falls within one standard deviation of the mean. 2. Approximately 95% of the data falls within two standard deviations of the mean. 3. Approximately 99.7% of the data falls within three standard deviations of the mean. ### Step 1: Identify the Mean and Standard Deviation - Mean (\( \mu \)) = 62 - Standard Deviation (\( \sigma \)) = 11 ### Step 2: Calculate the Range for 1 Standard Deviation - Lower limit: \( \mu - \sigma = 62 - 11 = 51 \) - Upper limit: \( \mu + \sigma = 62 + 11 = 73 \) ### Step 3: Calculate the Range for 2 Standard Deviations - Lower limit: \( \mu - 2\sigma = 62 - 2(11) = 62 - 22 = 40 \) - Upper limit: \( \mu + 2\sigma = 62 + 2(11) = 62 + 22 = 84 \) ### Step 4: Determine the Range of Interest We are interested in the range between 29 and 73. - 29 is below the range of 2 standard deviations (40). - 73 is at the upper limit of 1 standard deviation. ### Step 5: Apply the Empirical Rule Since 29 is below the 2 standard deviations, we can conclude that: - The percentage of roadways with potholes between 29 and 73 includes all values from 29 to 40 (which is less than 2 standard deviations) and all values from 40 to 73 (which is within 1 standard deviation). From the empirical rule: - Approximately 95% of the data falls between 40 and 84 (2 standard deviations). - Since 29 is below 40, we can assume that the percentage of roadways with potholes between 29 and 73 is slightly more than 95%. ### Step 6: Estimate the Percentage Since we know that 95% of the data falls between 40 and 84, and we are including values below 40 (down to 29), we can estimate that the percentage of roadways with potholes between 29 and 73 is approximately 95%. ### Final Answer Thus, the approximate percentage of 1-mile long roadways with potholes numbering between 29 and 73 is: \[ \text{ans} = 95.00 \]