Chapter 4.1, Problem 18E

(a)

To determine

To explain: The scientists for obtaining the SRS in the setting.

Answer to Problem 18E

They had information about the entire population while the process being less time-consuming and less costly.

Explanation of Solution

Given information:

To estimate the proportion of all pine trees in the area that bar been infected

Simple random sampling uses a sample of size n in which every sample of size n has an equal chance of being chosen.

To determine a simple random sample, then first have to determine how many trees are in the National Park (which will most likely be thousands of trees).

To assign a unique integer to each tree, which will be very time-consuming as the trees all have to be marked in some way (such that know which tree has which label) and it will be very hard to male sure that you do not miss one of the trees (especially the very small trees or the areas where the forest is very dense).

Finally, on drawing a simple random sample from the trees. To search those trees in the sample again and check if they have been infected or not.

Note that a simple random sample will NOT be practical for scientists, because it will be very time-consuming and costly to first have to find every single tree, then mark them and then look up the trees again that ended up in the random sample. Moreover, since the scientists already had to check each tree in person, they could have easier determined if each tree was infected (which would have been less time-consuming and costly) and then they had information about the entire population while the process being less time-consuming and less costly.

(b)

To determine

To explain: The sampling method being biased.

Answer to Problem 18E

It will consistently underestimate the true proportion of infected pine trees.

Explanation of Solution

DEFINITIONS

A voluntary response sample is a sample for which the subjects can decide if they want to be in the sample or not.

Convenience sampling uses a subgroup from the population that is conveniently chosen.

Bias: consistently overestimating or consistently underestimating the value that you're interested in.

SOLUTION

The sample in the exercise prompt is a convenience sample, because the trees that were conveniently at the park‘s main road were selected to be in the sample (which will most likely also be the trees that are the most easily accessible by people).

A convenience sample is very often biased, because the trees at the park‘s main road will be less likely to be infected than the other trees in the National Park (as beetles tend to avoid people and prefer to live in areas where not much human activity is present) and thus will consistently underestimate the true proportion of infected pine trees.

(c)

To determine

To explain: The scientists can conclude 35% of the pine trees on the west side are infested.

Answer to Problem 18E

On expectation about 35% of all pine trees to be infested, but not exactly 35%.

Explanation of Solution

A random sample resulted in 35% of the pine trees in the sample to be infested.

The population will then most likely NOT contain exactly 35% infected pine trees (although it is possible), since we expect the population to deviate slightly from the sample (as the sample was only a part of the population).

Thus on expectation about 35% of all pine trees to be infested, but not exactly 35%.