Question

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?

Answer 1

For this case we want to find this probability:

`P(57 <X<89)`

And we can calculate the number of deviations from the mean for the limits using the z score formula given by:

`z = (57-65)/(8)= -1`

`z = (89-65)/(8)= 3`

We know that within one deviation from the mean we have 68% of the values and on each tail we have (100-68)/2 % = 16%. And within 3 deviations we have 99.7% of the values and on each tail we have (100-99.7)/2% = 0.15%.

And the percentage desired would be:

(100%-0.15%) - 16% = 83.85%

Explanation:

Previous concepts

The empirical rule, also known as three-sigma rule or 68-95-99.7 rule, "is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ)".

Let X the random variable who representt the number of potholes

From the problem we have the mean and the standard deviation for the random variable X.
`E(X)=65, Sd(X)=8`

So we can assume
`\mu=65 , \sigma=8`

On this case in order to check if the random variable X follows a normal distribution we can use the empirical rule that states the following:

• The probability of obtain values within one deviation from the mean is 0.68

• The probability of obtain values within two deviation's from the mean is 0.95

• The probability of obtain values within three deviation's from the mean is 0.997

Solution to the problem

For this case we want to find this probability:

`P(57 <X<89)`

And we can calculate the number of deviations from the mean for the limits using the z score formula given by:

`z = (57-65)/(8)= -1`

`z = (89-65)/(8)= 3`

We know that within one deviation from the mean we have 68% of the values and on each tail we have (100-68)/2 % = 16%. And within 3 deviations we have 99.7% of the values and on each tail we have (100-99.7)/2% = 0.15%.

And the percentage desired would be:

(100%-0.15%) - 16% = 83.85%

Answer 2

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

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