Question

It's true — sand dunes in Colorado rival sand dunes of the Great Sahara Desert! The highest dunes at Great Sand Dunes National Monument can exceed the highest dunes in the Great Sahara, extending over 700 feet in height. However, like all sand dunes, they tend to move around in the wind. This can cause a bit of trouble for temporary structures located near the "escaping" dunes. Roads, parking lots, campgrounds, small buildings, trees, and other vegetation are destroyed when a sand dune moves in and takes over. Such dunes are called "escape dunes" in the sense that they move out of the main body of sand dunes and, by the force of nature (prevailing winds), take over whatever space they choose to occupy. In most cases, dune movement does not occur quickly. An escape dune can take years to relocate itself. Just how fast does an escape dune move? Let x be a random variable representing movement (in feet per year) of such sand dunes (measured from the crest of the dune). Let us assume that x has a normal distribution with μ = 14 feet per year and σ = 4.4 feet per year.

Under the influence of prevailing wind patterns, what is the probability of each of the following? (Round your answers to four decimal places.)

(a) an escape dune will move a total distance of more than 90 feet in 8 years


(b) an escape dune will move a total distance of less than 80 feet in 8 years


(c) an escape dune will move a total distance of between 80 and 90 feet in 8 years

Answer 1

Before we go on to solve the problem let us know a bit about Normal Distribution.

Normal Distribution

A continuous random variable X is said to have a normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

Notation:

Standard Normal Distribution

A continuous random variable X is said to have a standard normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

Exact evaluation of ?(x) is not possible but numerical method can be applied. The values of ?(x) has been tabulated extensively in Biometrika Volume I.

Notation:

Note:

Coming back to our problem

Given that,

X=Movement (in feet per year) of such sand dunes (measured from the crest of the dune)

and it is assumed that,

Let us define another random variable Y such that,

Y=Movement (in feet in 8 years) of such sand dunes (measured from the crest of the dune)

(a) Here we need to find the probability that an escape dune will move a total distance of more than 90 feet in 8 years.

Hence the probability that an escape dune will move a total distance of more than 90 feet in 8 years is 0.7357

(b) Here we need to find the probability that an escape dune will move a total distance of less than 80 feet in 8 years.

Hence the probability that an escape dune will move a total distance of less than 80 feet in 8 years is 0.1814

(c) Here we need to find the probability that an escape dune will move a total distance of between 80 and 90 feet in 8 years.

Hence the probability that an escape dune will move a total distance of between 80 and 90 feet in 8 years is 0.0829