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Calculate an estimate of the error standard deviation in the simple regression model

Calculate an estimate of the error standard deviation in the simple regression model

MATLAB: An Introduction with Applications
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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Calculate an estimate of the error standard deviation in the simple regression model

**Title: Analyzing the Relationship Between Cube Strength and Axial Strength in Asphalt Using Regression** **Introduction:** Over the past decade, rubber powder has been utilized in asphalt cement to enhance performance. A study was conducted to examine the relationship between cube strength (MPa) and axial strength (MPa) using the following sample data: - \( x \) (cube strength): 112.3, 97.0, 92.7, 86.0, 109.9, 95.8, 103.5, 89.0, 105.8, 89.7 - \( y \) (axial strength): 75.2, 70.7, 57.9, 48.5, 73.9, 57.9, 63.7, 59.8, 68.4, 55.7 **Regression Analysis:** (a) **Equation of the Least Squares Line:** The regression equation derived from the data is: \[ y = 0.9847x - 31.5869 \] This equation indicates the predicted relationship between cube strength and axial strength, where the slope (\(0.9847\)) represents the rate of change. **Interpret the Slope:** The slope of the equation suggests: - A one MPa increase in cube strength is associated with an increase in the predicted axial strength equal to the slope. (b) **Coefficient of Determination:** The coefficient of determination is calculated as \(0.6351\). **Interpret the Coefficient of Determination:** This value indicates that approximately 63.51% of the variation in axial strength can be explained by the variation in cube strength. The remaining variation is due to other factors not accounted for in this linear model. **Conclusion:** The study highlights a significant linear relationship between cube strength and axial strength in asphalt samples. Understanding this relationship can aid in optimizing the use of rubber powder in asphalt mixes to achieve desired structural properties.
Transcribed Image Text:**Title: Analyzing the Relationship Between Cube Strength and Axial Strength in Asphalt Using Regression** **Introduction:** Over the past decade, rubber powder has been utilized in asphalt cement to enhance performance. A study was conducted to examine the relationship between cube strength (MPa) and axial strength (MPa) using the following sample data: - \( x \) (cube strength): 112.3, 97.0, 92.7, 86.0, 109.9, 95.8, 103.5, 89.0, 105.8, 89.7 - \( y \) (axial strength): 75.2, 70.7, 57.9, 48.5, 73.9, 57.9, 63.7, 59.8, 68.4, 55.7 **Regression Analysis:** (a) **Equation of the Least Squares Line:** The regression equation derived from the data is: \[ y = 0.9847x - 31.5869 \] This equation indicates the predicted relationship between cube strength and axial strength, where the slope (\(0.9847\)) represents the rate of change. **Interpret the Slope:** The slope of the equation suggests: - A one MPa increase in cube strength is associated with an increase in the predicted axial strength equal to the slope. (b) **Coefficient of Determination:** The coefficient of determination is calculated as \(0.6351\). **Interpret the Coefficient of Determination:** This value indicates that approximately 63.51% of the variation in axial strength can be explained by the variation in cube strength. The remaining variation is due to other factors not accounted for in this linear model. **Conclusion:** The study highlights a significant linear relationship between cube strength and axial strength in asphalt samples. Understanding this relationship can aid in optimizing the use of rubber powder in asphalt mixes to achieve desired structural properties.
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