It's true β 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 π = 10 feet per year and π = 3.7 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 button hyperlink to the SALT program that reads: Use SALT.
(a) an escape dune will move a total distance of more than 90 feet in 9 years
(b) an escape dune will move a total distance of less than 80 feet in 9 years
(c) an escape dune will move a total distance of between 80 and 90 feet in 9 years
To calculate the probabilities in this scenario, we need to use the properties of the normal distribution. Specifically, we will use the mean (π = 10 feet per year) and the standard deviation (π = 3.7 feet per year) provided.
First, let's calculate the probability for part (a):
(a) an escape dune will move a total distance of more than 90 feet in 9 years
To find this probability, we need to calculate the area under the normal distribution curve to the right of 90 feet. We can use the cumulative distribution function (CDF) of the normal distribution.
Using the given mean and standard deviation, the Z-score for 90 feet can be calculated as follows:
Z = (90 - π) / π = (90 - 10) / 3.7 = 21.62
Next, we can use a standard normal distribution table or a calculator to find the probability associated with this Z-score. Looking up the Z-score of 21.62, we get a probability of 1.0000, indicating that the probability of an escape dune moving more than 90 feet in 9 years is 1.
Now let's move on to part (b):
(b) an escape dune will move a total distance of less than 80 feet in 9 years
Similar to part (a), we need to calculate the area under the normal distribution curve to the left of 80 feet.
Calculating the Z-score for 80 feet:
Z = (80 - π) / π = (80 - 10) / 3.7 = 19.19
Using the Z-score, we can look up the probability associated with it. From the table or calculator, we find the probability to be approximately 0.0000. This means that the probability of an escape dune moving less than 80 feet in 9 years is extremely low.
Finally, let's determine part (c):
(c) an escape dune will move a total distance of between 80 and 90 feet in 9 years
To find this probability, we can subtract the probability of moving less than 80 feet from the probability of moving less than 90 feet.
From part (b), we know that the probability of moving less than 80 feet is approximately 0.0000. And from part (a), we know that the probability of moving more than 90 feet is 1. So, the probability of moving between 80 and 90 feet is 1 - 0 = 1.
In summary:
(a) The probability of an escape dune moving a total distance of more than 90 feet in 9 years is 1.
(b) The probability of an escape dune moving a total distance of less than 80 feet in 9 years is extremely low, approaching 0.
(c) The probability of an escape dune moving a total distance of between 80 and 90 feet in 9 years is 1.
Remember to round the answers to four decimal places as requested.