With a trend line equation of y = -0.669x + 101.09, modify the table below showing the values as well as the residuals. Show your work and explain the steps you used to solve. X y ŷ e 106 28 83 51 DELL 61 39 56 76
With a trend line equation of y = -0.669x + 101.09, modify the table below showing the values as well as the residuals. Show your work and explain the steps you used to solve. X y ŷ e 106 28 83 51 DELL 61 39 56 76
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![### Introduction
In statistical analysis, a trend line (or line of best fit) is a straight line that best represents the data on a scatter plot. This line can be used to predict future points. Given a trend line equation, we can modify the existing dataset to include the predicted values (\(\hat{y}\)) and the residuals (\(e\)), which represent the differences between the observed values and the predicted values.
### Problem Statement
Given the trend line equation:
\[ y = -0.669x + 101.09 \]
Modify the table below to show the \(\hat{y}\) values as well as the residuals (\(e\)).
### Original Table
\[
\begin{array}{c|c|c|c}
x & y & \hat{y} & e \\
\hline
106 & 28 & & \\
83 & 51 & & \\
61 & 56 & & \\
39 & 76 & & \\
\end{array}
\]
The columns in the table are defined as follows:
- \(x\): Independent variable
- \(y\): Actual dependent variable
- \(\hat{y}\): Predicted value from the trend line equation
- \(e\): Residual (difference between \(y\) and \(\hat{y}\))
### Steps to Solve
1. **Calculate \(\hat{y}\)**: Use the trend line equation \( y = -0.669x + 101.09 \) to find the predicted value for each \(x\).
2. **Calculate Residuals (\(e\))**: Subtract the predicted value (\(\hat{y}\)) from the actual value (\(y\)).
Let's go through each calculation step-by-step:
#### For \(x = 106\):
\[ \hat{y} = -0.669(106) + 101.09 = -70.914 + 101.09 = 30.176 \]
\[ e = y - \hat{y} = 28 - 30.176 = -2.176 \]
#### For \(x = 83\):
\[ \hat{y} = -0.669(83) + 101.09 = -55.527 + 101.09 = 45.563 \]
\[ e = y](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc4056cef-3867-489f-a260-f13d42f9bf0e%2F86cf1fb1-f9c9-426a-8c01-255fb1244b67%2Fi47e5i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Introduction
In statistical analysis, a trend line (or line of best fit) is a straight line that best represents the data on a scatter plot. This line can be used to predict future points. Given a trend line equation, we can modify the existing dataset to include the predicted values (\(\hat{y}\)) and the residuals (\(e\)), which represent the differences between the observed values and the predicted values.
### Problem Statement
Given the trend line equation:
\[ y = -0.669x + 101.09 \]
Modify the table below to show the \(\hat{y}\) values as well as the residuals (\(e\)).
### Original Table
\[
\begin{array}{c|c|c|c}
x & y & \hat{y} & e \\
\hline
106 & 28 & & \\
83 & 51 & & \\
61 & 56 & & \\
39 & 76 & & \\
\end{array}
\]
The columns in the table are defined as follows:
- \(x\): Independent variable
- \(y\): Actual dependent variable
- \(\hat{y}\): Predicted value from the trend line equation
- \(e\): Residual (difference between \(y\) and \(\hat{y}\))
### Steps to Solve
1. **Calculate \(\hat{y}\)**: Use the trend line equation \( y = -0.669x + 101.09 \) to find the predicted value for each \(x\).
2. **Calculate Residuals (\(e\))**: Subtract the predicted value (\(\hat{y}\)) from the actual value (\(y\)).
Let's go through each calculation step-by-step:
#### For \(x = 106\):
\[ \hat{y} = -0.669(106) + 101.09 = -70.914 + 101.09 = 30.176 \]
\[ e = y - \hat{y} = 28 - 30.176 = -2.176 \]
#### For \(x = 83\):
\[ \hat{y} = -0.669(83) + 101.09 = -55.527 + 101.09 = 45.563 \]
\[ e = y
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman