According to the empirical rule (also known as the 68-95-99.7 rule), for a bell-shaped distribution (or normal distribution), approximately:
- 68% of the data falls within one standard deviation of the mean
- 95% of the data falls within two standard deviations of the mean
- 99.7% of the data falls within three standard deviations of the mean
In this case, we can use the empirical rule to estimate the percentage of 1-mile long roadways with potholes numbering between 51 and 84.
To do this, we need to first calculate the z-scores for the values 51 and 84, using the formula:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
For x = 51:
z = (51 - 62) / 11 = -1
For x = 84:
z = (84 - 62) / 11 = 2
These z-scores tell us how many standard deviations away from the mean each value is. A z-score of -1 means that the value is 1 standard deviation below the mean, and a z-score of 2 means that the value is 2 standard deviations above the mean.
Now, we can use the empirical rule to estimate the percentage of 1-mile long roadways with potholes numbering between 51 and 84:
- The percentage of data within one standard deviation of the mean is approximately 68%. Since the mean is 62 and the standard deviation is 11, one standard deviation below the mean is 51, and one standard deviation above the mean is 73 (62 - 11 = 51, 62 + 11 = 73). Therefore, approximately 68% of the 1-mile long roadways have potholes numbering between 51 and 73.
- The percentage of data within two standard deviations of the mean is approximately 95%. Since two standard deviations below the mean is 40, and two standard deviations above the mean is 84 (62 - 2(11) = 40, 62 + 2(11) = 84), approximately 95% of the 1-mile long roadways have potholes numbering between 40 and 84.
Therefore, the approximate percentage of 1-mile long roadways with potholes numbering between 51 and 84 is approximately 95%.