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Interpret the Slope and the Y-Intercept of the Least-Squares Regression $\mathrm { y } = \beta _ { 0 } + \beta _ { 1 } \mathrm { x }$

question 156

Multiple Choice

Interpret the Slope and the y-intercept of the Least-Squares Regression Line
-Civil engineers often use the straight-line equation, $\mathrm { y } = \beta _ { 0 } + \beta _ { 1 } \mathrm { x }$ , to model the relationship between the mean shear strength of masonry joints and precompression stress, $x$ . To test this theory, a series of stress tests were performed on solid bricks arranged in triplets and joined with mortar. The precompression stress was varied for each triplet and the ultimate shear load just before failure (called the shear strength) was recorded. The stress results for $\mathrm { n } = 7$ triplet tests is shown in the accompanying table followed by a SAS printout of the regression analysis.
$\begin{array}{l|lcccccc}\text { Triplet Test } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline \text { Shear Strength (tons) , } \mathrm{y} & 1.00 & 2.18 & 2.24 & 2.41 & 2.59 & 2.82 & 3.06 \\\hline \text { Precomp. Stress (tons) , } \mathrm{x} & 0 & 0.60 & 1.20 & 1.33 & 1.43 & 1.75 & 1.75\end{array}$

$Interpret the Slope and the y-intercept of the Least-Squares Regression Line -Civil engineers often use the straight-line equation, \mathrm { y } = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } , to model the relationship between the mean shear strength of masonry joints and precompression stress, x . To test this theory, a series of stress tests were performed on solid bricks arranged in triplets and joined with mortar. The precompression stress was varied for each triplet and the ultimate shear load just before failure (called the shear strength) was recorded. The stress results for \mathrm { n } = 7 triplet tests is shown in the accompanying table followed by a SAS printout of the regression analysis. \begin{array}{l|lcccccc} \text { Triplet Test } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text { Shear Strength (tons) , } \mathrm{y} & 1.00 & 2.18 & 2.24 & 2.41 & 2.59 & 2.82 & 3.06 \\ \hline \text { Precomp. Stress (tons) , } \mathrm{x} & 0 & 0.60 & 1.20 & 1.33 & 1.43 & 1.75 & 1.75 \end{array} Give a practical interpretation of the estimate of the slope of the least squares line. A) For every 1 ton increase in precompression stress, we estimate the shear strength of the joint to increase by 0.987 ton. B) For a triplet test with a precompression stress of 1 ton, we estimate the shear strength of the joint to be 0.987 ton. C) For every 0.987 ton increase in precompression stress, we estimate the shear strength of the joint to increase by 1 ton. D) For a triplet test with a precompression stress of 0 tons, we estimate the shear strength of the joint to be 1.19 tons.$

$Interpret the Slope and the y-intercept of the Least-Squares Regression Line -Civil engineers often use the straight-line equation, \mathrm { y } = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } , to model the relationship between the mean shear strength of masonry joints and precompression stress, x . To test this theory, a series of stress tests were performed on solid bricks arranged in triplets and joined with mortar. The precompression stress was varied for each triplet and the ultimate shear load just before failure (called the shear strength) was recorded. The stress results for \mathrm { n } = 7 triplet tests is shown in the accompanying table followed by a SAS printout of the regression analysis. \begin{array}{l|lcccccc} \text { Triplet Test } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text { Shear Strength (tons) , } \mathrm{y} & 1.00 & 2.18 & 2.24 & 2.41 & 2.59 & 2.82 & 3.06 \\ \hline \text { Precomp. Stress (tons) , } \mathrm{x} & 0 & 0.60 & 1.20 & 1.33 & 1.43 & 1.75 & 1.75 \end{array} Give a practical interpretation of the estimate of the slope of the least squares line. A) For every 1 ton increase in precompression stress, we estimate the shear strength of the joint to increase by 0.987 ton. B) For a triplet test with a precompression stress of 1 ton, we estimate the shear strength of the joint to be 0.987 ton. C) For every 0.987 ton increase in precompression stress, we estimate the shear strength of the joint to increase by 1 ton. D) For a triplet test with a precompression stress of 0 tons, we estimate the shear strength of the joint to be 1.19 tons.$

Give a practical interpretation of the estimate of the slope of the least squares line.

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General Intelligence

A theory suggesting that a single factor underlies various cognitive abilities, enabling individuals to perform a range of mental tasks.

Factor Analysis

A statistical method used to identify underlying variables, or factors, that explain the pattern of correlations within a set of observed variables.

Variance Assessment

The process of evaluating the extent of variation or dispersion from the mean or expected value in a set of data.

Standardization

The process of implementing and developing technical standards to ensure consistency and compatibility across different processes or products.

Interpret the Slope and the Y-Intercept of the Least-Squares Regression y=β0+β1x\mathrm { y } = \beta _ { 0 } + \beta _ { 1 } \mathrm { x }y=β0​+β1​x

Definitions:

Interpret the Slope and the Y-Intercept of the Least-Squares Regression $\mathrm { y } = \beta _ { 0 } + \beta _ { 1 } \mathrm { x }$