The least-square regression line for the given data is y = -0.206x + 2.097. Determine the residual of a data point for which x= -3 and y=-6, rounding to three decimal places. X y -5 11 A. 2.715 B. -6.333 C. - 8.715 D. -3.285 - 3 -6 4 8 1 - 3 -1 -2 -2 1 0 5 25 -5 3 6 -40 7
The least-square regression line for the given data is y = -0.206x + 2.097. Determine the residual of a data point for which x= -3 and y=-6, rounding to three decimal places. X y -5 11 A. 2.715 B. -6.333 C. - 8.715 D. -3.285 - 3 -6 4 8 1 - 3 -1 -2 -2 1 0 5 25 -5 3 6 -40 7
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question
![**Understanding Residuals in Least-Square Regression**
The least-square regression line for the given data is defined by the equation:
\[
\hat{y} = -0.206x + 2.097
\]
**Objective:**
Determine the residual of a data point where \( x = -3 \) and \( y = -6 \), rounding to three decimal places.
**Given Data:**
\[
\begin{array}{c|cccccccccc}
x & -5 & -3 & 4 & 1 & -1 & -2 & 0 & 2 & 3 & -4 \\
\hline
y & 11 & -6 & 8 & -3 & -2 & 1 & 5 & -5 & 6 & 7 \\
\end{array}
\]
**Solution Steps:**
1. **Calculate Predicted \( \hat{y} \):**
Substitute \( x = -3 \) into the regression line equation:
\[
\hat{y} = -0.206(-3) + 2.097 = 0.618 + 2.097 = 2.715
\]
2. **Calculate the Residual:**
The residual is the difference between the observed \( y \) value and the predicted \( \hat{y} \) value:
\[
\text{Residual} = y - \hat{y} = -6 - 2.715 = -8.715
\]
**Answer Options:**
- A. 2.715
- B. -6.333
- C. -8.715
- D. -3.285
**Conclusion:**
The residual for the data point \( x = -3 \) and \( y = -6 \) is \(-8.715\), which corresponds to option C.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff46c6c1c-9cf3-4090-96fd-ea5e3513a088%2F96c2e481-ae0a-481a-9712-9358d40f3a62%2F8qhpsvl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Residuals in Least-Square Regression**
The least-square regression line for the given data is defined by the equation:
\[
\hat{y} = -0.206x + 2.097
\]
**Objective:**
Determine the residual of a data point where \( x = -3 \) and \( y = -6 \), rounding to three decimal places.
**Given Data:**
\[
\begin{array}{c|cccccccccc}
x & -5 & -3 & 4 & 1 & -1 & -2 & 0 & 2 & 3 & -4 \\
\hline
y & 11 & -6 & 8 & -3 & -2 & 1 & 5 & -5 & 6 & 7 \\
\end{array}
\]
**Solution Steps:**
1. **Calculate Predicted \( \hat{y} \):**
Substitute \( x = -3 \) into the regression line equation:
\[
\hat{y} = -0.206(-3) + 2.097 = 0.618 + 2.097 = 2.715
\]
2. **Calculate the Residual:**
The residual is the difference between the observed \( y \) value and the predicted \( \hat{y} \) value:
\[
\text{Residual} = y - \hat{y} = -6 - 2.715 = -8.715
\]
**Answer Options:**
- A. 2.715
- B. -6.333
- C. -8.715
- D. -3.285
**Conclusion:**
The residual for the data point \( x = -3 \) and \( y = -6 \) is \(-8.715\), which corresponds to option C.
![**Problem: Calculating Residuals for a Least-Squares Regression Line**
The least-square regression line for the given data is defined by the equation:
\[
\hat{y} = 2.097x - 0.552
\]
**Task:** Determine the residual for a data point where \( x = -2 \) and \( y = -6 \). Round your answer to three decimal places.
**Data Table:**
| x | -5 | -3 | 4 | 1 | -1 | -2 | 0 | 2 | 3 | -4 |
|----|----|----|---|---|----|----|---|---|---|----|
| y | -10 | -8 | 9 | 1 | -2 | -6 | -1 | 3 | 6 | -8 |
**Solution Options:**
- A. 11.134
- B. -1.254
- C. -10.746
- D. -4.746
**Correct Answer:** B. -1.254
To calculate the residual:
1. **Calculate the predicted value (\(\hat{y}\)):**
\[
\hat{y} = 2.097(-2) - 0.552
\]
2. **Find the residual by subtracting the actual y-value from the predicted y-value:**
\[
\text{Residual} = y - \hat{y}
\]
Conduct the arithmetic to find the solution to the problem set above.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff46c6c1c-9cf3-4090-96fd-ea5e3513a088%2F96c2e481-ae0a-481a-9712-9358d40f3a62%2Fnxn1f7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem: Calculating Residuals for a Least-Squares Regression Line**
The least-square regression line for the given data is defined by the equation:
\[
\hat{y} = 2.097x - 0.552
\]
**Task:** Determine the residual for a data point where \( x = -2 \) and \( y = -6 \). Round your answer to three decimal places.
**Data Table:**
| x | -5 | -3 | 4 | 1 | -1 | -2 | 0 | 2 | 3 | -4 |
|----|----|----|---|---|----|----|---|---|---|----|
| y | -10 | -8 | 9 | 1 | -2 | -6 | -1 | 3 | 6 | -8 |
**Solution Options:**
- A. 11.134
- B. -1.254
- C. -10.746
- D. -4.746
**Correct Answer:** B. -1.254
To calculate the residual:
1. **Calculate the predicted value (\(\hat{y}\)):**
\[
\hat{y} = 2.097(-2) - 0.552
\]
2. **Find the residual by subtracting the actual y-value from the predicted y-value:**
\[
\text{Residual} = y - \hat{y}
\]
Conduct the arithmetic to find the solution to the problem set above.
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