a. What is a residual? b. In what sense is the regression line the straight line that "best" fits the points in a scatterplot? a. What is a residual? OA. A residual is a value of y-y, which is the difference between an observed value of y and a predicted value of y. OB. A residual is the amount that one variable changes when the other variable changes by exactly one unit. OC. A residual is a point that has a strong effect on the regression equation. OD. A residual is a value that is determined exactly, without any error. b. In what sense is the regression line the straight line that "best" fits the points in a scatterplot? The regression line has the property that the sum ... of the residuals is sum of squares possible sum.

MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
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Chapter1: Starting With Matlab
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### Understanding Residuals and Regression Lines

**a. What is a residual?**

1. **Definition:**
   - A residual is defined as the value of \( y - \hat{y} \), which represents the difference between an observed value of \( y \) and a predicted value of \( y \).

2. **Options Explained:**
   - **A.** Correct. A residual is a value of \( y - \hat{y} \), which is the difference between an observed value of \( y \) and a predicted value of \( y \).
   - **B.** Incorrect. This option describes a change in variables, not specifically a residual.
   - **C.** Incorrect. This option suggests residuals have a strong effect on the regression equation, which is not accurate.
   - **D.** Incorrect. This option implies residuals are determined exactly without error, which is incorrect since residuals represent the error.

**b. In what sense is the regression line the straight line that "best" fits the points in a scatterplot?**

1. **Understanding the Regression Line:**
   - The regression line is considered the "best" fit line for a scatterplot in the sense that it minimizes the differences (residuals) between the observed values and the predicted values.

2. **Property of the Regression Line:**
   - The regression line has the property that the sum of the squared residuals is the smallest possible sum.

3. **Dropdown and Options:**
   - The dropdown menu provides options such as "sum" and "sum of squares." The correct choice is "sum of squares."

**Complete Statement:**
   - The regression line has the property that the **sum of squares** of the residuals is the **smallest** possible sum.
Transcribed Image Text:### Understanding Residuals and Regression Lines **a. What is a residual?** 1. **Definition:** - A residual is defined as the value of \( y - \hat{y} \), which represents the difference between an observed value of \( y \) and a predicted value of \( y \). 2. **Options Explained:** - **A.** Correct. A residual is a value of \( y - \hat{y} \), which is the difference between an observed value of \( y \) and a predicted value of \( y \). - **B.** Incorrect. This option describes a change in variables, not specifically a residual. - **C.** Incorrect. This option suggests residuals have a strong effect on the regression equation, which is not accurate. - **D.** Incorrect. This option implies residuals are determined exactly without error, which is incorrect since residuals represent the error. **b. In what sense is the regression line the straight line that "best" fits the points in a scatterplot?** 1. **Understanding the Regression Line:** - The regression line is considered the "best" fit line for a scatterplot in the sense that it minimizes the differences (residuals) between the observed values and the predicted values. 2. **Property of the Regression Line:** - The regression line has the property that the sum of the squared residuals is the smallest possible sum. 3. **Dropdown and Options:** - The dropdown menu provides options such as "sum" and "sum of squares." The correct choice is "sum of squares." **Complete Statement:** - The regression line has the property that the **sum of squares** of the residuals is the **smallest** possible sum.
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