Given the following least-squares regression equation: ŷ = -13.586 +4.340x where x represents the age of an elementary school student and y represents the score on a standardized test. (a) Interpret the value of the slope in this equation in the context of this data. (b) Give the y-intercept, and explain why it does not make sense in this situation. (c) Use the regression equation to predict the score on this standardized test for a student who is 8 years old. (d) Use the regression equation to predict the age of a student who scores a 35 on the standardized test. The least-squares regression line is ý = -13.586 + 4.340x, where x represents the age of an elementary school student and y represents the score on a standardized test.
Given the following least-squares regression equation: ŷ = -13.586 +4.340x where x represents the age of an elementary school student and y represents the score on a standardized test. (a) Interpret the value of the slope in this equation in the context of this data. (b) Give the y-intercept, and explain why it does not make sense in this situation. (c) Use the regression equation to predict the score on this standardized test for a student who is 8 years old. (d) Use the regression equation to predict the age of a student who scores a 35 on the standardized test. The least-squares regression line is ý = -13.586 + 4.340x, where x represents the age of an elementary school student and y represents the score on a standardized test.
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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![### Least-Squares Regression Analysis
The given least-squares regression equation is:
\[
\hat{y} = -13.586 + 4.340x
\]
where \( x \) represents the age of an elementary school student and \( y \) represents the score on a standardized test.
#### Tasks:
(a) **Interpret the Slope:**
The slope of the equation is 4.340. This means that for each additional year of age, the score on the standardized test is predicted to increase by approximately 4.34 points.
(b) **Interpret the y-intercept:**
The y-intercept is -13.586. This does not make practical sense in this context because a negative score on a standardized test is not feasible, especially for very young students (e.g., age 0).
(c) **Predict the Score for an 8-Year-Old:**
Using the regression equation, the predicted score for an 8-year-old student is:
\[
\hat{y} = -13.586 + 4.340(8) = 21.134
\]
(d) **Predict the Age for a Score of 35:**
To find the age of a student who scores 35, we set:
\[
35 = -13.586 + 4.340x
\]
Solving for \( x \):
\[
x = \frac{35 + 13.586}{4.340} \approx 11.25 \text{ years}
\]
#### Graph Explanation:
The accompanying graph is a scatter plot with a fitted regression line. It displays:
- **X-axis:** Age in years (ranging approximately from 4 to 12 years)
- **Y-axis:** Score on a standardized test (ranging approximately from 0 to 40)
Blue dots represent data points of students' ages versus their scores. The regression line, drawn diagonally, illustrates the positive relationship between age and test scores, supporting the calculated slope.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb321bf60-2279-406a-8c47-b6f3a554770f%2F6272a37f-4e30-44d6-8a68-8c87bc2125d0%2Fss0jwrj_processed.png&w=3840&q=75)
Transcribed Image Text:### Least-Squares Regression Analysis
The given least-squares regression equation is:
\[
\hat{y} = -13.586 + 4.340x
\]
where \( x \) represents the age of an elementary school student and \( y \) represents the score on a standardized test.
#### Tasks:
(a) **Interpret the Slope:**
The slope of the equation is 4.340. This means that for each additional year of age, the score on the standardized test is predicted to increase by approximately 4.34 points.
(b) **Interpret the y-intercept:**
The y-intercept is -13.586. This does not make practical sense in this context because a negative score on a standardized test is not feasible, especially for very young students (e.g., age 0).
(c) **Predict the Score for an 8-Year-Old:**
Using the regression equation, the predicted score for an 8-year-old student is:
\[
\hat{y} = -13.586 + 4.340(8) = 21.134
\]
(d) **Predict the Age for a Score of 35:**
To find the age of a student who scores 35, we set:
\[
35 = -13.586 + 4.340x
\]
Solving for \( x \):
\[
x = \frac{35 + 13.586}{4.340} \approx 11.25 \text{ years}
\]
#### Graph Explanation:
The accompanying graph is a scatter plot with a fitted regression line. It displays:
- **X-axis:** Age in years (ranging approximately from 4 to 12 years)
- **Y-axis:** Score on a standardized test (ranging approximately from 0 to 40)
Blue dots represent data points of students' ages versus their scores. The regression line, drawn diagonally, illustrates the positive relationship between age and test scores, supporting the calculated slope.
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