a) escape dune move a distance of more than 90 feet in 8 yrs. B) escape dune move a distance of less than 80 feet in 8 yrs. C) escape dune will move a total distance of between 80 and 90 feet in 8 yrs

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 = 12 feet per year and 𝜎 = 4.1 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

The problem deals with the distribution of the sum of normally distributed variables from the same population. So long as the samples are independent, i.e., the values of one don't depend on the values in another, then the sum will also have a normal distribution,

Specifically, where the population mean is μ, variance is σ² , and n is number of samples taken.

Σ(X) ~ N(μ = nμ, nσ²)

In this problem, you have a population mean of 12 with sigma = 4.1 and a sample of 8 years, n = 8.

The distribution of interest is now mean = (8)(12) = 96, sigma = sqrt(n)*sigma = (4.1)(sqrt(8)) = 11.60.

The remainder of the problem is straight manipulation of Z-scores. The NORM.S.DIST(Z, True) function in Excel will return the p(Z < z).

Find sum(X) Z p(Z < Z) p(Z > Z) Note

p(sum(X) > 90) 90 -0.517 0.302 0.698 p(Z > Z) = 1 - p(Z < Z)

p(sum(X) < 80) 80 -1.380 0.084

p(80 < sum(X) < 90) = 0.219, p(Z < -.517) - p(Z < -1.380)