Question

The number of potholes in any given 1 mile stretch of freeway pavement in Pennsylvania has an approximate normal distribution with mean 50 and a standard deviation 10. Use the Empirical Rule to determine the approximate proportion of 1-mile long roadways with potholes numbering between 20 and 70.

Answer 1

The approximate proportion of 1-mile long roadways with potholes numbering between 20 and 70 is 0.9735.

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.