An escape dune can take years to relocate itself. Just how fast does an escape d
ID: 3151522 • Letter: A
Question
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 xhas a normal distribution with = 12 feet per year and = 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
Explanation / Answer
This boils down to the last sentences:
Given x is normally distributed, = 12 for one year, and = 4.1,
a. Find the probability that 8x > 90
The nice thing about the normal distribution is that if x is normally distributed and multiplied by a constant, the mean and standard deviation of the new variable are also just multiplied by the same constant. So your new mean is 120 and your new standard deviation is 41. So you get your Z scores as usual with these new parameters:
Z = (90 - 120)/41 = -10/22 = -0.73. So you just need to look in your Z tables to find the probability that z is GREATER than -0.73. Probably the table will give you the probability that it is less than -0.73, so you may have to subtract from 1. (or just look up the probability that Z is LESS than 0.73, which will be the same.it is 0.9984
b. Same way:
Z = (80-120)/41 = -20/22 = -0.975
But this tome you want the probability that Z is LESS than -0.975, so you just read it directly off the table.
=0.0016
c. Just subtract the probability you found in b from the probability you found in a. =0.998-0.0016=0.9964
These are a lot easier to inderstand and follow if you draw a sketch of the normal curve and label the sections with the values for x and for z.