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 65 and a standard deviation of 8. Using the empirical (68-95-99.7) rule, what is the approximate percentage of 1-mile long roadways with potholes numbering between 57 and 89?

83.85% of 1-mile long roadways with potholes numbering between 57 and 89

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed(bell-shaped) random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 65

Standard deviation = 8

Using the empirical (68-95-99.7) rule, what is the approximate percentage of 1-mile long roadways with potholes numbering between 57 and 89?

It is important to remember that the normal distribution is symmetric, which means that 50% of the measures are below the mean and 50% are avobe.

57 = 65 - 8

So 57 is one standard deviation below the mean.

By the Empirical Rule, 68% of the 50% below the mean is within 1 standard deviation of the mean.

89 = 65 + 3*8

So 89 is three standard deviations above the mean.

By the Empirical Rule, 99.7% of the 50% above the mean is within 3 standard deviations of the mean.

0.68*0.5 + 0.997*0.5 = 0.8385

83.85% of 1-mile long roadways with potholes numbering between 57 and 89