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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?

Step by step solution

01

Understanding an SRS

An SRS, or Simple Random Sample, involves randomly selecting a certain number of pine trees from the entire population, ensuring that every tree has an equal chance of being included in the sample. This approach, while unbiased in theory, can be impractical in a large, forested area like Rocky Mountain National Park due to the sheer number of trees and challenges in accessibility and visibility.

02

Identify Bias in Roadside Sampling

Using only the pine trees along the park's main road introduces selection bias, as these trees may not represent the entire population. Trees along the road might have different exposure to beetles due to environmental factors or human activity that affects tree health, leading to non-representative data.

03

Interpret Sample Results

Although a sampling plan with some randomization might report 35% infestation, this is only an estimate and might not reflect the true proportion for the entire park. Due to sampling variability and potential biases even in a sophisticated method, scientists can't definitively conclude that exactly 35% of all trees are infested.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Random Sample

A Simple Random Sample (SRS) is a method where each member of the population has an equal chance of being chosen. Imagine every tree in the Rocky Mountain National Park has its name in a hat, and you randomly select a few names. That's SRS in action! But, it's not always practical. In a vast, forested park, trees are scattered and some are hidden from paths. Accessing every tree to give it an equal chance can be quite tough. Not to mention, it would take a lot of time and resources. That's why, even though SRS is unbiased because of its random nature, it's not always an option in complex environments like this park.

Bias in Sampling

Sampling bias occurs when the method used to select the sample causes it to be unrepresentative of the population as a whole. If only trees near the main road are studied, you're likely missing out on the trees deeper in the forest, which might have different exposure levels to beetles. These roadside trees could be more affected due to human interference, like car emissions or more visitors potentially disrupting natural defenses. This skews results because it doesn't give you a true picture of what's happening with all those pine trees in the park. Think of it like tasting only the top layer of a cake — you might miss all the layers and flavors beneath.

Sampling Variability

Each time you take a sample, you might get a slightly different result. This is known as sampling variability. Suppose you grab different groups of trees to study each day. Some samples might tell you 35% are infested, others might say only 32%. This happens because every sample is just a small peek into the whole park. It’s important to remember that our 35% figure might just be one side of the story. So, don't take one specific result as the definitive answer when there’s so much variation possible.

Proportion Estimation

Proportion estimation uses sample data to make guesses about the whole population. Let's say your sample shows 35% of the trees are infested. The goal is to estimate the infestation rate for all the park’s trees. But given we've learned about sampling variability, 35% doesn’t mean every pocket in the park has the same rate. There could be areas with more or less infestation. It's like guessing the whole movie's plot with just a short trailer. Estimates give us a starting point, but don't forget the broader picture could differ significantly.