Dead trees On the west side of Rocky Mountain National Park, many mature pine trees are dying due to infestation by pine beetles. Scientists would like to

use sampling to estimate the proportion of all pine trees in the area that have been infected.

(a) Explain why it wouldn’t be practical for scientists to obtain an SRS in this setting.

(b) A possible alternative would be to use every pine tree along the park’s main road as a sample. Why is this sampling method biased?

(c) Suppose that a more complicated random sampling plan is carried out, and that $35\%$ of the pine trees in the sample are infested by the pine beetle. Can

scientists conclude that 35% of all the pine trees on the west side of the park are infested? Why or why not?

Many mature pine trees on the west side of Rocky Mountain National Park are dying due to pine beetle infestation. Scientists want to utilise sampling to figure out what percentage of all pine trees in the area are diseased.

A simple random sample (SRS) of size $n$ is made up of n people chosen from the population with an equal chance of being the sample that is actually chosen.

To calculate the percentage of all pine trees in the region that have been affected A sample of size n is used in simple random sampling, and each sample has an equal probability 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). Assigning a unique integer to each tree will take a long time because the trees must all be recorded in some way (such as knowing which tree has which label) and it will be difficult to avoid missing any of the trees (especially the extremely small trees or regions where the forest is dense). Finally, a simple random sample was taken from the trees. To go over the sample again and see whether any of the trees have been affected. Note that a simple random sample will not be practicable for scientists because finding every single tree, marking them, and then looking up the trees that ended up in the random sample will be exceedingly time-consuming and costly. Furthermore, because the scientists had to inspect each tree in person, they could have more easily identified if each tree was sick (which would have taken less time and money) and then had knowledge about the entire population while saving time and money.

The sample in the exercise prompt is a convenience sample because the trees in the sample were chosen because they were conveniently located near the park's main road (which will most likely also be the trees that are the most easily accessible by people). A convenience sample is frequently skewed because trees near the park's main road are less likely to be infected than other trees in the park (because beetles avoid people and prefer to live in areas where there is little human activity), resulting in an underestimation of the true proportion of infected pine trees.

A random sample revealed that $35\%$of the pine trees were infected. Because we expect the population to differ slightly from the sample, the population is unlikely (but not impossible) to contain exactly $35$percent sick pine trees (because the sample comprised only a small portion of the whole population). As a result, approximately $35\%$of all pine trees should be damaged, but not exactly $35\%$