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Chapter 4: Problem 28

What does it mean when we say that the regression line is the least squares line?

Short Answer

Expert verified
The regression line being the 'least squares line' means that it's the line that minimizes the sum of the squares of the residuals (the vertical distance of each data point from the line) among all possible lines.

Step by step solution

01

Understanding Regression Line

A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. We often use a regression line to predict the value of y for a given value of x.
02

Understanding Least Squares Method

The least squares method is a standard approach in regression analysis to approximate the solution of overdetermined systems by minimizing the sum of the squares of the residuals. The residuals are the difference between the actual value and the predicted value.
03

Connecting the Concepts

When we say that 'the regression line is the least squares line', we mean that out of all the possible lines that could be drawn, the regression line is the one that makes the sum of the squares of the vertical distances of the data points from the line as small as possible. It minimizes the sum of the squared residuals. This effectively minimizes the total 'error' of the model.

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Most popular questions from this chapter

An auction house released a list of 25 recently sold paintings. The artist's name and the sale price of each painting appear on the list. Would the correlation coefficient be an appropriate way to summarize the relationship between artist and sale price? Why or why not?

The article "Examined Life: What Stanley H. Kaplan Taught Us About the SAT" (The New Yorker [December 17, 2001]: \(86-92\) ) included a summary of findings regarding the use of SAT I scores, SAT II scores, and high school grade point average (GPA) to predict first-year college GPA. The article states that "among these, SAT II scores are the best predictor, explaining 16 percent of the variance in first-year college grades. GPA was second at 15.4 percent, and SAT I was last at 13.3 percent." a. If the data from this study were used to fit a least squares regression line with \(y=\) first-year college GPA and \(x=\) high school GPA, what would be the value of \(r^{2} ?\) b. The article stated that SAT II was the best predictor of first-year college grades. Do you think that predictions based on a least-squares line with \(y=\) first-year college GPA and \(x=\) SAT II score would be very accurate? Explain why or why not.

It may seem odd, but biologists can tell how old a lobster is by measuring the concentration of pigment in the lobster's eye. The authors of the paper "Neurolipofuscin Is a Measure of Age in Panulirus argus, the Caribbean Spiny Lobster, in Florida" (Biological Bulletin [2007]: 55-66) wondered if it was sufficient to measure the pigment in just one eye, which would be the case if there is a strong relationship between the concentration in the right eye and the concentration in the left eye. Pigment concentration (as a percentage of tissue sample) was measured in both eyes for 39 lobsters, resulting in the following summary quantities (based on data from a graph in the paper): $$ \begin{array}{cll} n=39 & \sum_{x}=88.8 & \sum y=86.1 \\ \sum x y=281.1 & \sum x^{2}=288.0 & \sum y^{2}=286.6 \end{array} $$ An alternative formula for calculating the correlation coefficient that doesn't involve calculating the z-scores is $$ r=\frac{\sum_{x y}-\frac{\left(\sum x\right)\left(\sum y\right)}{n}}{\sqrt{\sum x^{2}-\frac{\left(\sum x\right)^{2}}{n}} \sqrt{\sum y^{2}-\frac{\left(\sum y\right)^{2}}{n}}} $$ Use this formula to calculate the value of the correlation coefficient, and interpret this value.

For each of the following pairs of variables, indicate whether you would expect a positive correlation, a negative correlation, or a correlation close to \(0 .\) Explain your choice. a. Interest rate and number of loan applications b. Height and \(\mathrm{IQ}\) c. Height and shoe size d. Minimum daily temperature and cooling cost

Briefly explain why it is important to consider the value of \(s\) in addition to the value of \(r^{2}\) when evaluating the usefulness of the least squares regression line.
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