Answer:
The approximate proportion of 1-mile long roadways with potholes numbering between 20 and 70 is 0.9735.
Step-by-step explanation:
The Empirical Rule states that, for a normally distributed random variable:
Approximately 68% of the measures are within 1 standard deviation of the mean.
Approximately 95% of the measures are within 2 standard deviations of the mean.
Approximately 99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we have that:
Mean = 50, standard deviation = 10.
Between 20 and 70.
The normal distribution is symmetric, which means that 50% of the measures are below the mean and 50% are above.
20
20 = 50 - 3*10
So 20 is 3 standard deviations below the mean. Of the 50% of the measures below the mean, 99.7% are within 3 standard deviations of the mean, that is, above 20.
70
70 = 50 + 2*10
So 70 is 2 standard deviations above the mean. Of the 50% of the measures above the mean, 95% are within 2 standard deviations of the mean, that is, below 70.
Percentage:
0.997*50% + 0.95*50% = 97.35%
As a proportion, 97.35%/100 = 0.9735.
The approximate proportion of 1-mile long roadways with potholes numbering between 20 and 70 is 0.9735.
