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Chapter 6: Problem 3
What is a population parameter? Give three examples.
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Find the \(z\) value described and sketch the area described.Find \(z\) such that \(55 \%\) of the standard normal curve lies to the left of \(z\).
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 \(\mu=17\) feet per year and \(\sigma=3.3\) feet per year. (For more information, see Hydrologic, Geologic, and Biologic Research at Great Sand Dunes National Monument and Vicinity, Colorado, proceedings of the National Park Service Research Symposium.) Under the influence of prevailing wind patterns, what is the probability that (a) an escape dune will move a total distance of more than 90 feet in 5 years? (b) an escape dune will move a total distance of less than 80 feet in 5 years? (c) an escape dune will move a total distance of between 80 and 90 feet in 5 years?
Crime: Murder What are the chances that a person who is murdered actually knew the murderer? The answer to this question explains why a lot of police detective work begins with relatives and friends of the victim! About \(64 \%\) of people who are murdered actually knew the person who committed the murder (Source: Chances: Risk and Odds in Everyday Life, by James Burke). Suppose that a detective file in New Orleans has 63 current unsolved murders. What is the probability that (a) at least 35 of the victims knew their murderers? (b) at most 48 of the victims knew their murderers? (c) fewer than 30 victims did not know their murderers? (d) more than 20 victims did not know their murderers?
(a) If we have a distribution of \(x\) values that is more or less mound-shaped and somewhat symmetric, what is the sample size needed to claim that the distribution of sample means \(\bar{x}\) from random samples of that size is approximately normal? (b) If the original distribution of \(x\) values is known to be normal, do we need to make any restriction about sample size in order to claim that the distribution of sample means \(\bar{x}\) taken from random samples of a given size is normal?
Consider two \(\bar{x}\) distributions corresponding to the same \(x\) distribution. The first \(\bar{x}\) distribution is based on samples of size \(n=100\) and the second is based on samples of size \(n=225 .\) Which \(\bar{x}\) distribution has the smaller standard error? Explain.
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