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 49 and a standard deviation of 9. Using the Empirical Rule, what is the approximate percentage of 1-mile long roadways with potholes numbering between 22 and 58?

Answer 1

83.85% of 1-mile long roadways with potholes numbering between 22 and 58

Explanation:

The Empirical Rule states that, for a normally distributed 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 = 49

Standard deviation = 9

Using the Empirical Rule, what is the approximate percentage of 1-mile long roadways with potholes numbering between 22 and 58?

22 = 49 - 3*9

So 22 is three standard deviations below the mean.

Since the normal distribution is symmetric, 50% of the measures are below the mean and 50% are above the mean.

Of those 50% which are below the mean, 99.7% of those are within 3 standard deviations of the mean, that is, greater than 22.

58 = 49 + 9

So 58 is one standard deviation of the mean.

Of those which are above the mean, 68% are within 1 standard deviation of the mean, that is, lesser than 58.

Then

0.997*0.5 + 0.68*0.5 = 0.8385 = 83.85%

83.85% of 1-mile long roadways with potholes numbering between 22 and 58