Question

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 this area that have been infested.

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 exactly $35\%$ of all the pine trees on the west side of the park are infested? Why or why not?

Short answer

a. Scientists would not be able to obtain an SRS in this situation, according to the definition of SRS.

b. It's likely that the first $200$trees aren't as infected as the rest of the trees, or that the infection pine beetles are more concentrated near the highway than in other trees.

c. Starting from a random point on a sample frame, every $n$th element in the frame is selected at equal intervals in systematic sampling, selection interval is$25$

Step-by-step solution

Part (a) Step 1 : Given Information

To explain why it would not be practical for scientists to obtain an SRS in this setting.

Part (a) Step 2 ; Simplification

Given that:

Total pine trees =$5000$

Population size = $5000$

Sample pine trees = $200$

This implies sample size = $200$

Now, in case of $5000$pine trees which is spread around highway.

Now we have to choose $200$ trees from that, but in practice, we choose trees that are closer to the highway than trees that are farther away, or vice versa. However, we do not select trees in such a way that each tree has an equal probability of being selected. As a result, the likelihood of selecting each tree is not the same. As a result, scientists would not be able to obtain an SRS in this situation, according to the definition of SRS.

Part (b) Step 1 : Given Information

We have to explain why is this sampling method biased.

Part (b) Step 2 ; Simplification

The first $200$pine trees along the highway as you enter the park might be used as an alternative.

Only the top $200$pine trees are chosen in this scenario. As a result, it's likely that the first $200$trees aren't as infected as the rest of the trees, or that the infection pine beetles are more concentrated near the highway than in other trees.

As a result, if we just investigate the first $200$trees, the results will be biased, and the study will not yield a perfect result.

Part (c) Step 1 : Given Information

We have to find out can scientist conclude that exactly $35\%$of all the pine trees on the west side of the park are infected and explain why or why not.

Part (c) Step 2 ; Simplification

Starting from a random point on a sample frame, every $n$th element in the frame is selected at equal intervals in systematic sampling.

The total number of trees in this study is = $5000$

Also, have a look at some pine trees =$200$

As a result, the selection interval,

$\frac{5000}{200}=25$

It indicates that we choose every $25$th tree starting with the first three and ending with the $500$th tree. So, from $5000$trees, we get $200$trees at intervals of $25$trees. This sampling covers the entire population. As a result, analysis of this sampling yields reliable results for tree infection and infection proportion.