91Ó°ÊÓ

For the past decade, rubber powder has been used in asphalt cement to improve performance. The article "Experimental Study of Recycled Rubber-Filled High- Strength Concrete" (Magazine of Concrete Res., 2009: 549-556) includes a regression of \(y=\) axial strength (MPa) on \(x=c u b e\) strength (MPa) based on the following sample data: $$ \begin{array}{c|cccccccccc} x & 112.3 & 97.0 & 92.7 & 86.0 & 102.0 & 99.2 & 95.8 & 103.5 & 89.0 & 86.7 \\ \hline y & 75.0 & 71.0 & 57.7 & 48.7 & 74.3 & 73.3 & 68.0 & 59.3 & 57.8 & 48.5 \end{array} $$ a. Obtain the equation of the least squares line, and interpret its slope. b. Calculate and interpret the coefficient of determination. c. Calculate and interpret an estimate of the error standard deviation \(\sigma\) in the simple linear regression model.

Step by step solution

01

Calculate the Mean Values for x and y

First, we need to find the mean of the x-values (cube strength) and y-values (axial strength). Calculate \(\bar{x}\) and \(\bar{y}\) using:\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \] \[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i \] where \(n\) is the number of data points, which is 10.

02

Use the Least Squares Formula to Find the Slope and Y-Intercept

The slope \(b\) of the least squares line is calculated using the formula: \[ b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \] Once the slope \(b\) is found, calculate the y-intercept \(a\) using: \[ a = \bar{y} - b\bar{x} \] Thus, the least squares regression line is \( y = a + bx \).

03

Interpret the Slope

The slope \( b \) represents the change in the axial strength (MPa) for every one MPa increase in cube strength. If \( b > 0 \), the relationship is directly proportional; \( b < 0 \), the relationship is inversely proportional.

04

Calculate the Coefficient of Determination \(R^2\)

The coefficient of determination \(R^2\) is calculated as:\[ R^2 = 1 - \frac{SS_{res}}{SS_{tot}} \] where \( SS_{res} \) is the sum of the squares of the residuals, and \( SS_{tot} \) is the total sum of squares. This value indicates the proportion of the variance in the dependent variable that is predictable from the independent variable.

05

Interpret the Coefficient of Determination \(R^2\)

The value of \( R^2 \) indicates how well the regression line fits the data. If \( R^2 \) is close to 1, a large portion of the variance in \(y\) is explained by \(x\). If \( R^2 \) is close to 0, the regression line does not explain much of the variance in \(y\).

06

Estimate the Error Standard Deviation \(\sigma\)

The error standard deviation (\(\sigma\)) is calculated using:\[ \sigma = \sqrt{\frac{1}{n-2}SS_{res}} \]This provides a measure of the typical distance that the observed values fall from the regression line, indicating the spread of the residuals.

07

Interpret the Error Standard Deviation \(\sigma\)

The error standard deviation \(\sigma\) gives an estimate of the typical size of the prediction errors (residuals) when using the regression line. A smaller \(\sigma\) indicates the data points are closer to the fitted line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Least Squares Method

The least squares method is a fundamental approach in linear regression analysis. Its purpose is to find the best-fitting line through a set of data points by minimizing the sum of the squared differences between the observed values and the values predicted by the line. These differences are known as residuals.
Here's how you calculate it:

• Calculate the means: First, we find the average values of the x and y data sets, called \(\bar{x}\) and \(\bar{y}\).
• Determine the slope (\(b\)): This tells us how much \(y\) (axial strength) is expected to change with a one-unit change in \(x\) (cube strength). It's given by the formula: \[b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\]
• Calculate the y-intercept (\(a\)): Once we have the slope, the y-intercept can be found using: \[a = \bar{y} - b\bar{x}\]
• Form the equation: The equation of the least squares line is then \(y = a + bx\).

The resulting line is the one that best fits the data, where the sum of the squared residuals is at its minimum.
This is very useful for making predictions based on trends in the data.

Coefficient of Determination

The coefficient of determination, represented as \(R^2\), is a key metric that evaluates the goodness of fit of your regression line. It tells us what portion of the variance in the dependent variable can be predicted from the independent variable.
This is how it is calculated:

• Calculate \(SS_{res}\): This involves measuring the sum of squares of the residuals.
• Calculate \(SS_{tot}\): This is the total sum of squares, or the total variance in the data.
• Compute \(R^2\): The formula is \[R^2 = 1 - \frac{SS_{res}}{SS_{tot}}\].

Interpreting \(R^2\):

• Ranges from 0 to 1: An \(R^2\) closer to 1 implies that a significant extent of the variation in the output (\(y\)) is explained by the input (\(x\)).
• If \(R^2 = 0.8\), for instance, 80% of the variance in \(y\) can be accounted for by the line fit to \(x\). The remaining 20% is due to other factors or randomness.

Higher \(R^2\) values generally indicate a better fit, meaning the line closely follows the real data points.

Error Standard Deviation

The error standard deviation, denoted as \(\sigma\), is crucial in understanding the variability or spread of residuals. Residuals are essentially the differences between observed values and those predicted by the regression line.

• Calculation: It is determined through the formula: \[\sigma = \sqrt{\frac{1}{n-2}SS_{res}}\]where \(SS_{res}\) is the sum of the squared residuals, and \(n\) is the number of observations.
• Interpreting \(\sigma\): A small \(\sigma\) implies that the observed data points hover closely around the line of best fit, which means reliable predictions.

Understanding \(\sigma\) helps us comprehend how much prediction error exists on average when using the regression model.
By minimizing \(\sigma\), we aim for a tighter fit between the model and actual data, enhancing the predictability and reliability of regression analyses.