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 = 17 feet per year and σ =
3.9 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 6 years
(b) an escape dune will move a total distance of less than 80 feet
in 6 years
(c) an escape dune will move a total distance of between 80 and 90
feet in 6 years
expected distance in 6 years =17*6=102
standard deviation =3.9*sqrt(6)=9.553
a)
for normal distribution z score =(X-μ)/σx | |
here mean= μ= | 102 |
std deviation =σ= | 9.553 |
probability =P(X>90)=P(Z>(90-102)/9.553)=P(Z>-1.26)=1-P(Z<-1.26)=1-0.1038=0.8962 |
b)
probability =P(X<80)=(Z<(80-102)/9.553)=P(Z<-2.3)=0.0107 |
c)
probability =P(80<X<90)=P((80-102)/9.553)<Z<(90-102)/9.553)=P(-2.3<Z<-1.26)=0.1038-0.0107=0.0931 |
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