91Ó°ÊÓ

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 exactly \(35 \%\) of \(a l l\) the pine trees on the west side of the park are infested? Why or why not?

Step by step solution

01

Understand Sampling Challenges

In part (a), the scientists cannot obtain a Simple Random Sample (SRS) because the terrain and vast area of Rocky Mountain National Park make it impractical to randomly select trees and survey them. The geographic barriers and the sheer number of trees impose logistical and resource constraints.

02

Identify Biased Sampling Method

In part (b), using only trees along the main road introduces bias because these trees are not representative of the entire park area. Trees along the road might have different exposure to environmental factors, like vehicle emissions or different degrees of human intervention, compared to trees deeper in the forest.

03

Analyze Sample Proportion

In part (c), even though 35% of the sampled trees are infested, scientists cannot conclusively state that exactly 35% of all trees in the park are infested. This sample proportion is an estimate with inherent variability, and the true proportion of infested trees could be higher or lower than 35%. Confidence intervals and additional study findings are needed to make such inferences.

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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 foundational concept in statistics, ideally used when each member of a population has an equal chance of being selected. However, in realistic scenarios like the Rocky Mountain National Park, obtaining an SRS can be quite the challenge. Why? Because the vast terrain creates significant logistical hurdles. Imagine trying to randomly pick trees across a sprawling and rugged landscape. This is impractical, as you’d need substantial resources to reach and study each randomly-selected tree. In such cases, scientists often have to consider alternative sampling methods that balance practicality with the goal of minimizing bias.

Bias in Sampling

Bias in sampling occurs when certain members of a population are more likely to be included than others, leading to skewed results. In the context of sampling in Rocky Mountain National Park, selecting every tree along the main road, as suggested in the exercise, introduces bias. Here's why:

• These trees may experience different environmental conditions compared to trees further from the road.
• They might be more affected by human activity, such as pollution and road maintenance, which isn't present deeper in the forest.

Such factors can result in data that does not accurately represent the entire population of trees. To reduce bias, sampling methods should strive to capture diversity within the entire park's ecosystem.

Proportion Estimation

Estimating the proportion of a trait within a population, such as the infestation rate of pine trees, involves some level of uncertainty. In the given exercise, a 35% infestation rate was observed in a sample of trees. It’s crucial to understand that this figure is merely an estimate of the true proportion. Why is it an estimate?

• The number reflects only the sampled trees and carries inherent variability.
• Due to sampling error, the actual infestation rate across the entire park could be higher or lower.

Further statistical methods, such as calculating confidence intervals, help contextualize this variability and offer a more reliable range for the true infestation proportion.