Choose the One Alternative That Best Completes the Statement or Answers

question 12

Multiple Choice

Choose the one alternative that best completes the statement or answers the question.
-Civil engineers often use the straight-line equation, $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x }$ , to model the relationship between the mean shear strength $\mathrm { E } ( \mathrm { y } )$ 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.
$\begin{array}{l|ccccccc}\text { Triplet Test } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline \text { Shear Strength (tons) , } \mathrm{y} & 1.00 & 2.18 & 2.24 & 2.41 & 2.59 & 2.82 & 3.06 \\\hline \text { Precomp. Stress (tons) , } \mathrm{x} & 0 & 0.60 & 1.20 & 1.33 & 1.43 & 1.75 & 1.75\end{array}$

$\text { Analysis of Variance }$

$\begin{array}{lccccc} & {\text { Sum of }} & \text { Mean } \\\text { Source } & \text { DF } & \text { Squares } & \text { Square } & \text { F Value } & \text { Prob > F } \\\text { Model } & 1 & 2.39555 & 2.39555 & 47.732 & 0.0010 \\\text { Error } & 5 & 0.25094 & 0.05019 & & \\\text { C Total } & 6 & 2.64649 & & &\end{array}$

$\begin{array}{llll}\text { Root MSE } & 0.22403 & \text { R-square } & 0.9052 \\\text { Dep Mean } & 2.32857 & \text { Adj R-sq } & 0.8862 \\\text { C.V. } & 9.62073 & &\end{array}$

$\begin{array}{l}\text { Parameter Estimates }\\\begin{array}{llllll} & & \text { Parameter } & \text { Standard } & \text { T for HO: } & \\\text { Variable } & \text { DF } & \text { Estimate } & \text { Error } & \text { Parameter }=0 & \text { Prob }>|\mathrm{T}| \\\text { INTERCEP } & 1 & 1.191930 & 0.18503093 & 6.442 & 0.0013 \\\mathrm{X} & 1 & 0.987157 & 0.14288331 & 6.909 & 0.0010\end{array}\end{array}$

Give a practical interpretation of $R ^ { 2 }$ , the coefficient of determination for the least squares model.

• A) About \(91 \%\) of the total variation in the sample of \(y\)-values can be explained by (or attributed to) the linear relationship between shear strength and precompression stress.
• B) In repeated sampling, approximately \(91 \%\) of all similarly constructed regression lines will accurately predict shear strength.
• C) We expect to predict the shear strength of a triplet test to within about 91 ton of its true value.
• D) We expect about \(91 \%\) of the observed shear strength values to lie on the least squares line.

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