The number of potholes in any given 1 mile stretch of freeway pavement in Pennsylvania has a Normal distribution.

Question

The number of potholes in any given 1 mile stretch of freeway pavement in Pennsylvania has a Normal distribution. This distribution has a mean of 59 and a standard deviation of 5. Using the Standard Deviation Rule, aka the Empirical Rule, what is the approximate percentage of 1-mile long roadways with potholes numbering between 49 and 74?

Answer 1

59 - 49 = 10, so 49 is 2 standard deviation below the mean

74 - 59 = 15, so 74 is 3 standard deviations above the mean

Based on the Standard Deviation Rule,

1. 2 standard deviation below mean will count for probability 95.45% / 2 = 47.725%
2. 3 standard deviation above mean will count for probability of 99.73% / 2 = 49.865%

So the final answer is 47.725% + 49.865% = 97.59%

Answer 2

54 - 49 = 5, so 49 is one standard deviation below the mean

74 - 59 = 15, so 74 is 15/5 = 3 three standard deviations above the mean

Per the empirical rule 34% of the data is between 1 standard deviation below the mean and the mean and 49.9% of the data is between the mean and 3 standard deviations above the mean, therefore 34 + 49.9 = 83.9% of the data is 49 and 74.