MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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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.
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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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