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Chapter 13: Problem 8

Explain the least squares method and least squares regression line. Why are they called by these names?

Short Answer

Expert verified
The 'Least Squares Method' is a mathematical procedure in regression analysis that minimizes the sum of the squares of the residuals to find the best fit line for data points. The 'Least Squares Regression Line' is the line that minimizes the sum of the squares of the residuals. They are called so because they signify that the sum of squares of the residuals is at its least.

Step by step solution

01

Definition and Explanation

The 'Least Squares Method' is a mathematical procedure used in regression analysis to find the best fit line for data points by minimizing the sum of the squares of the differences or the residuals between the observed and estimated values of the dependent variable configured as a straight line. The 'Least Squares Regression Line' is the line which minimizes the sum of the squares of the residuals. The residuals being the difference between observed outcome and the outcome predicted by the line.
02

Reason Behind the Name

These methods are named as 'Least Squares and Least Squares Regression Line' because they involve the principle of minimizing the sum of the squares of the residuals. The term 'least squares' signifies that the overall solution minimizes the sum of the squares of the residuals. The residuals are the discrepancy between observed and predicted outcomes which we aim to minimize to achieve the best fit line.
03

Quick Recap

The 'Least Squares Method' is a well-known mathematical procedure used in regression analysis to approximate the best fit line for data points by minimizing the sum of the squares of the differences between observed and estimated values. This principle highlights the minimization of the sum of squared residuals, is the factor that gives these methods their names - 'Least Squares' and 'Least Squares Regression Line'.

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Most popular questions from this chapter

A researcher took a sample of 10 years and found the following relationship between \(x\) and \(y\), where \(x\) is the number of major natural calamities (such as tornadoes, hurricanes, earthquakes, floods, etc.) that occurred during a year and \(y\) represents the average annual total profits (in millions of dollars) of a sample of insurance companies in the United States. $$ \hat{y}=342.6-2.10 x $$ a. A randomly selected year had 24 major calamities. What are the expected average profits of U.S. insurance companies for that year? b. Suppose the number of major calamities was the same for each of 3 years. Do you expect the average profits for all U.S. insurance companies to be the same for each of these 3 years? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact?

The following information is obtained from a sample data set. $$ n=10, \quad \Sigma x=100, \quad \Sigma y=220, \quad \Sigma x y=3680, \quad \Sigma x^{2}=1140 $$ Find the estimated regression line.

The following table gives information on the number of megapixels and the prices of nine randomly selected point-and-shoot digital cameras that were available on BestBuy.com on July 22, 2009 . $$ \begin{array}{l|rrrrrrrrr} \hline \text { Megapixels } & 10.3 & 10.2 & 7.0 & 9.1 & 10.0 & 12.1 & 8.0 & 5.0 & 14.7 \\ \hline \text { Price (\$) } & 130 & 150 & 62 & 160 & 200 & 280 & 125 & 60 & 400 \\ \hline \end{array} $$ Compute the following. a. \(\mathrm{SS}_{\mathrm{xr}} \mathrm{SS}_{y,}\) and \(\mathrm{SS}_{x y}\) b. Standard deviation of errors c. SST, SSE, and SSR d. Coefficient of determination

Explain the difference between linear and nonlinear relationships between two variables.

For a sample data set, the linear correlation coefficient \(r\) has a positive value. Which of the following is true about the slope \(b\) of the regression line estimated for the same sample data? a. The value of \(b\) will be positive. b. The value of \(b\) will be negative. c. The value of \(b\) can be positive or negative.
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