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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 an optimization technique used to minimize the sum of the squares of residuals (differences between observed and estimated values). The least squares regression line is the result of applying that method to obtain a line that best fits the data. These terms are so named because 'least squares' signifies minimizing the square of these residuals, and the 'least squares regression line' specifies the line obtained from this method.

Step by step solution

01

Definition of Least Squares Method

The least squares method is an optimization technique used to minimize the sum of the squares of the residuals. The residuals are the differences between the observed and estimated data. So, least squares refer to minimizing the sum of squares of these differences. This is mathematically expressed as \( \min_{\beta_0, \beta_1} \sum_{i}^{N} (y_i - (\beta_0 + \beta_1 X_i))^2 \) where \( \beta_0, \beta_1 \) are parameters we are solving for.
02

Definition of Least Squares Regression Line

The least squares regression line is the line that minimizes the sum of the squares of the residuals. This line is used in regression analysis to predict the dependent variable based on the independent variable. In equation form, it's typically represented as \( y = \beta_0 + \beta_1 x \) where \( \beta_0 \) is y-intercept (constant term) and \( \beta_1 \) is the slope (coefficient term). This line gives the best mathematical approximation of a set of data points.
03

Explanation of the Names

The term 'least squares' comes from the method's approach to minimize the sum of squares of the residuals. 'Least' signifies minimization and 'squares' points to the squared differences. Furthermore, the 'least squares regression line' is so called because this line is obtained by applying the least squares method.

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Most popular questions from this chapter

Can the values of \(B\) and \(\rho\) calculated for the same population data have different signs? Explain.

A sample data set produced the following information. $$ \begin{aligned} &n=12, \quad \Sigma x=66, \quad \Sigma y=588, \quad \Sigma x y=2244, \\ &\Sigma x^{2}=396, \quad \text { and } \quad \Sigma y^{2}=58,734 \end{aligned} $$ a. Calculate the linear correlation coefficient \(r .\) b. Using a \(1 \%\) significance level, can you conclude that \(\rho\) is negative?

The following table gives information on the incomes (in thousands of dollars) and charitable contributions (in hundreds of dollars) for the last year for a random sample of 10 households. $$ \begin{array}{rc} \hline \text { Income } & \text { Charitable Contributions } \\ \hline 76 & 15 \\ 57 & 4 \\ 140 & 42 \\ 97 & 33 \\ 75 & 5 \\ 107 & 32 \\ 65 & 10 \\ 77 & 18 \\ 102 & 28 \\ 53 & 4 \\ \hline \end{array} $$ a. With income as an independent variable and charitable contributions as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y^{\circ}}\) b. Find the regression of charitable contributions on income. c. Briefly explain the meaning of the values of \(a\) and \(b\). d. Calculate \(r\) and \(r^{2}\) and briefly explain what they mean. e. Compute the standard deviation of errors. f. Construct a \(99 \%\) confidence interval for \(B\). g. Test at a \(1 \%\) significance level whether \(B\) is positive. h. Using a \(1 \%\) significance level, can you conclude that the linear correlation coefficient is different from zero?

The following data give the experience (in years) and monthly salaries (in hundreds of dollars) of nine randomly selected secretaries. $$ \begin{array}{l|rrrrrrrrr} \hline \text { Experience } & 14 & 3 & 5 & 6 & 4 & 9 & 18 & 5 & 16 \\ \hline \text { Monthly salary } & 62 & 29 & 37 & 43 & 35 & 60 & 67 & 32 & 60 \\\ \hline \end{array} $$ a. Find the least squares regression line with experience as an independent variable and monthly salary as a dependent variable. b. Construct a \(98 \%\) confidence interval for \(B\). c. Test at the \(2.5 \%\) significance level whether \(B\) is greater than zero.

Construct a \(95 \%\) confidence interval for the mean value of \(y\) and a \(95 \%\) prediction interval for the predicted value of \(y\) for the following. a. \(\hat{y}=13.40+2.58 x\) for \(x=8\) given \(s_{e}=1.29, \bar{x}=11.30, \mathrm{SS}_{x x}=210.45\), and \(n=12\) b. \(\hat{y}=-8.6+3.72 x\) for \(x=24\) given \(s_{e}=1.89, \bar{x}=19.70, \mathrm{SS}_{x x}=315.40\), and \(n=10\)
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