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Chapter 5: Problem 25

Explain why it can be dangerous to use the leastsquares line to obtain predictions for \(x\) values that are substantially larger or smaller than those contained in the sample.

Short Answer

Expert verified
Extrapolating the least squares line can be dangerous as it assumes the relationship observed within the sample extends indefinitely, which may not hold true. As such, predictions for \(x\) values substantially larger or smaller than those in the sample may result in significant inaccuracies.

Step by step solution

01

Definition of Least Squares Line

The least squares line is a statistical tool used to find the best fit line for a set of data points. It is calculated by minimizing the sum of the squares of the differences between the observed and predicted values.
02

Application of Least Squares Line

In cases where the range of \(x\) values in the sample is utilized, the least squares line does a satisfactory job at predicting \(y\) values. This is because the line is designed based on existing data points.
03

Extrapolation with Least Squares Line

The trouble arises when we attempt to use this line to predict values outside the range of the sampled \(x\) values. This practice is called extrapolation. The further we depart from the original range, the less reliable our predictions become.
04

Explanation of Risks

The reason extrapolation can be risky is due to the key assumption that the relationship modeled by the least squares line remains the same beyond the sampled data. However, real world data often exhibits non-linear patterns or changes in behavior outside the sampled range.

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Most popular questions from this chapter

As part of a study of the effects of timber management strategies (Ecological Applications [2003]: IIIOII123) investigators used satellite imagery to study abundance of the lichen Lobaria oregano at different elevations. Abundance of a species was classified as "common" if there were more than 10 individuals in a plot of land. In the table below, approximate proportions of plots in which Lobaria oregano were common are given. Proportions of Plots Where Lobaria oregano Are Common \begin{tabular}{lrrrrrrr} \hline Elevation (m) & 400 & 600 & 800 & 1000 & 1200 & 1400 & 1600 \\ Prop. of plots & \(0.99\) & \(0.96\) & \(0.75\) & \(0.29\) & \(0.077\) & \(0.035\) & \(0.01\) \\ with lichen & & & & \end{tabular} with lichen \begin{tabular}{l} with lichen \\ common \\ \hline \end{tabular} a. As elevation increases, does the proportion of plots for which lichen is common become larger or smaller? What aspect(s) of the table support your answer? b. Using the techniques introduced in this section, calculate \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\) for each of the elevations and fit the line \(y^{\prime}=a+b(\) Elevation). What is the equation of the best-fit line? c. Using the best-fit line from Part (b), estimate the proportion of plots of land on which Lobaria oregano are classified as "common" at an elevation of \(900 \mathrm{~m} .\)

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