Question

In Grafton, a rural area in Vermont, the distance (in meters) between telephone poles has a uniform distribution between 40 and 65 meters. Suppose two consecutive telephone poles are selected at random. a) Any distance between poles greater than 50 meters is considered to be environment friendly. What is the probability that the distance is environment friendly? b) Find the mean of the distances between the telephone poles c) Find the standard deviation of the distance between the telephone poles.

Answer 1

a) 60% probability that the distance is environment friendly.

b) The mean of the distances between the telephone poles is 52.5 meters.

c) The standard deviation of the distance between the telephone poles is 7.22 meters.

Explanation:

An uniform probability is a case of probability in which each outcome is equally as likely.

For this situation, we have a lower limit of the distribution that we call a and an upper limit that we call b.

The probability that we find a value X lower than x is given by the following formula.

`P(X \leq x) = (x - a)/(b-a)`

Uniform distribution between 40 and 65 meters.

This means that
`a = 40, b = 65`

a) Any distance between poles greater than 50 meters is considered to be environment friendly. What is the probability that the distance is environment friendly?

This is
`P(X > 50)`

We have that

`P(X \leq 50) + P(X > 50) = 1`

So

`P(X > 50) =  1 - P(X \leq 50)`

`P(X \leq 50) = (50 - 40)/(65 - 40) = 0.4`

`P(X > 50) =  1 - P(X \leq 50) = 1 - 0.4 = 0.6`

60% probability that the distance is environment friendly.

b) Find the mean of the distances between the telephone poles

The mean of the uniform distribution is

`M = (a + b)/(2)`

So

`M = (a + b)/(2) = (40 + 65)/(2) = 52.5`

c) Find the standard deviation of the distance between the telephone poles.

The standard deviation of the uniform distribution is:

`S = \sqrt{((b-a)^(2))/(12)}`

So

`S = \sqrt{((65-40)^(2))/(12)} = 7.22`