/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 32 Explain why it can be dangerous ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 32

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
Using the least squares line helps predict one variable based on another, but extrapolation, i.e., predicting for values outside the sample range, is dangerous. That's because the linearity assumption may not hold, all influencing factors may not be captured, and prediction errors are larger outside the sample range.

Step by step solution

01

Understanding the Role of Least Squares Line

The least squares line, also known as the line of best fit, is used in regression analysis to predict the value of one variable (dependent variable) based on the value of another variable (independent variable). It works under the assumption that there is a linear relationship between these variables.
02

Conceptualizing Out-of-Sample Predictions

An out-of-sample prediction refers to using the regression line (least squares line) to predict the dependent variable for values of the independent variable that are not within the range of the sample data. If \(x\) values are substantially larger or smaller than those of in the sample, it means the prediction is extrapolated far beyond the range of observed data.
03

Explaining the Dangers of Extrapolation

Predicting values for a variable outside the range of sample data is known as extrapolation. It can be dangerous for several reasons: 1) The assumption of linearity might not hold outside the sample range. Real-world relationships can be non-linear, and trends may change. 2) The sample data may not incorporate all factors affecting the relationship, which may be more prevalent outside the sample range. 3) Errors of estimates are larger and predictions are less reliable outside the sample range. Therefore, using the least squares line to obtain predictions for \(x\) values substantially larger or smaller than those in the sample can be risky and misleading.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The following data on \(x=\) soil depth (in centimeters) and \(y=\) percentage of montmorillonite in the soil were taken from a scatterplot in the paper "Ancient Maya Drained Field Agriculture: Its Possible Application Today in the New River Floodplain, Belize, C.A." (Agricultural Ecosystems and Environment \([1984]: 67-84)\) : $$ \begin{array}{lllllllr} x & 40 & 50 & 60 & 70 & 80 & 90 & 100 \\ y & 58 & 34 & 32 & 30 & 28 & 27 & 22 \end{array} $$ a. Draw a scatterplot of \(y\) versus \(x\). b. The equation of the least-squares line is \(\hat{y}=64.50-\) \(0.45 x\). Draw this line on your scatterplot. Do there appear to be any large residuals? c. Compute the residuals, and construct a residual plot. Are there any unusual features in the plot?

An auction house released a list of 25 recently sold paintings. Eight artists were represented in these sales. The sale price of each painting appears on the list. Would the correlation coefficient be an appropriate way to summarize the relationship between artist \((x)\) and sale price (y)? Why or why not?

In the article "Reproductive Biology of the Aquatic Salamander Amphiuma tridactylum in Louisiana" (Journal of Herpetology [1999]: \(100-105\) ), 14 female salamanders were studied. Using regression, the researchers predicted \(y=\) clutch size (number of salamander eggs) from \(x=\) snout-vent length (in centimeters) as follows: $$ \hat{y}=-147+6.175 x $$ For the salamanders in the study, the range of snout-vent lengths was approximately 30 to \(70 \mathrm{~cm}\). a. What is the value of the \(y\) intercept of the least-squares line? What is the value of the slope of the least-squares line? Interpret the slope in the context of this problem. b. Would you be reluctant to predict the clutch size when snout-vent length is \(22 \mathrm{~cm}\) ? Explain.

The paper "Root Dentine Transparency: Age Determination of Human Teeth Using Computerized Densitometric Analysis" (American Journal of Physical Anthropology \([1991]: 25-30\) ) reported on an investigation of methods for age determination based on tooth characteristics. With \(y=\) age (in years) and \(x=\) percentage of root with transparent dentine, a regression analysis for premolars gave \(n=36\), SSResid \(=5987.16\), and \(\mathrm{SSTo}=\) \(17,409.60 .\) Calculate and interpret the values of \(r^{2}\) and \(s_{e}\)

A sample of 548 ethnically diverse students from Massachusetts were followed over a 19 -month period from 1995 and 1997 in a study of the relationship between TV viewing and eating habits (Pediatrics [2003]: 1321- 1326). For each additional hour of television viewed per day, the number of fruit and vegetable servings per day was found to decrease on average by \(0.14\) serving. a. For this study, what is the dependent variable? What is the predictor variable? b. Would the least-squares line for predicting number of servings of fruits and vegetables using number of hours spent watching TV as a predictor have a positive or negative slope? Explain.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.