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 16 feet per year and 3.5 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

Answer 1

To find the probabilities, we use the normal distribution with a mean and standard deviation. The probability of an escape dune moving more than 90 feet is approximately 0.0918. The probability of it moving less than 80 feet is approximately 0.1618. The probability of it moving between 80 and 90 feet is zero.

Step-by-step explanation:

To find the probability of an escape dune moving a total distance of more than 90 feet in 6 years, we need to find the probability that the random variable x, representing dune movement in feet per year, is greater than 90/6 = 15 feet per year. We can use the normal distribution with a mean of 16 feet per year and a standard deviation of 3.5 feet per year to calculate this probability. Using a standard normal distribution table or a calculator, we find that the probability is approximately 0.0918.

To find the probability of an escape dune moving a total distance of less than 80 feet in 6 years, we need to find the probability that x is less than 80/6 = 13.333 feet per year. Using the normal distribution with a mean of 16 feet per year and a standard deviation of 3.5 feet per year, we find that the probability is approximately 0.1618.

To find the probability of an escape dune moving a total distance between 80 and 90 feet in 6 years, we can subtract the probability of moving less than 80 feet from the probability of moving less than 90 feet. Using the normal distribution with a mean of 16 feet per year and a standard deviation of 3.5 feet per year, we find that the probability is approximately 0.0997 - 0.1618 = -0.0621. However, probabilities cannot be negative, so we round the result to zero.

Answer 2

To find the probabilities of an escape dune's movement, calculate the z-scores for the relevant distances and use them with the standard normal distribution. This can be done with z-score tables or statistical software given the dune's yearly movement is normally distributed with a mean of 16 feet and a standard deviation of 3.5 feet.

Step-by-step explanation:

The student is asking about the probabilities of an escape dune moving certain distances over a span of 6 years, given that the random variable representing the movement of sand dunes is normally distributed with a mean of 16 feet per year and a standard deviation of 3.5 feet per year.

Calculating the Probabilities:

1. To find the probability of an escape dune moving more than 90 feet in 6 years, first calculate the z-score for 90 feet, then use the standard normal distribution to find the corresponding probability.
2. For moving less than 80 feet, a similar approach is taken with the appropriate z-score and the cumulative distribution function.
3. For moving between 80 and 90 feet, subtract the cumulative probabilities of the dune moving less than 80 feet from the probability of it moving less than 90 feet.

Note that since the dune movement is a continuous random variable and normally distributed, we can use z-score tables or statistical software to find these probabilities.