Question

Simple linear regression: Explained and unaplained variation Renonia Bivanate data obtained for the paired variables \( x \) and \( y \) are shown below, in the table labeled "Sample data." These data are plotted in the scatter plot in Figure 1, which also displays the least-squares regression line for the data. The equation for this line is \( \hat{y}=141.14+0.48 x \). In the "Calculations" table are calculations involving the observed \( y \)-values, the mean \( \bar{y} \) of these values, and the values \( \hat{y} \) predicted from the regression equation. \begin{tabular}{|c|c|} \hline Sample & \\ \hline\( x \) & \( y \) \\ \hline 218.0 & 2502 \\ \hline 234.3 & 252.2 \\ \hline 254.4 & 2473 \\ \hline 275.4 & 291.3 \\ \hline 304.1 & 282.2 \\ \hline \end{tabular} Send data to Excel Calculations \begin{tabular}{|c|c|c|} \hline\( (\hat{y}-\overline{\boldsymbol{y}})^{\mathbf{2}} \) & \( (\boldsymbol{y}-\hat{\boldsymbol{y}})^{\mathbf{2}} \) & \( (\boldsymbol{y}-\overline{\boldsymbol{y}})^{\mathbf{2}} \) \\ \hline 355.6996 & 19.5364 & 208.5136 \\ \hline 121.7933 & 1.9712 & 154.7536 \\ \hline 1.9265 & 254.4663 & 300.6756 \\ \hline 75.5509 & 322.8490 & 710.7556 \\ \hline 504.8110 & 24.0885 & 308.3536 \\ \hline Column sum: & \begin{tabular}{c} Column sum: \\ 1059.7813 \end{tabular} & \begin{tabular}{c} Column sum: \\ 1622.9114 \end{tabular} \\ \hline \end{tabular} Figure 1 Answer the following. (a) The total variation in the sample \( y \)-values ispiven by the (Choose one) \( \square \) which for these data is (Choose one) \( \nabla \) (b) The proportion of the total variation in the sample \( y \)-values that can be explained by the estimated linear relationship between \( x \) and \( y \) is \( \square \). (Round your answer to at least 2 decimal places.) (c) The least-squares regression line given above is said to be a line that "best fits" the sample data. The term "best fits" is used because the line has an equation that minimizes the (Choose one) data is (Choose one) \( \mathbf{v} \). which for these data is (Choose one) \( \nabla \). Explanation Check

          Simple linear regression: Explained and unaplained variation
Renonia
Bivanate data obtained for the paired variables \( x \) and \( y \) are shown below, in the table labeled "Sample data." These data are plotted in the scatter plot in Figure 1, which also displays the least-squares regression line for the data. The equation for this line is \( \hat{y}=141.14+0.48 x \).

In the "Calculations" table are calculations involving the observed \( y \)-values, the mean \( \bar{y} \) of these values, and the values \( \hat{y} \) predicted from the regression equation.
\begin{tabular}{|c|c|}
\hline Sample & \\
\hline\( x \) & \( y \) \\
\hline 218.0 & 2502 \\
\hline 234.3 & 252.2 \\
\hline 254.4 & 2473 \\
\hline 275.4 & 291.3 \\
\hline 304.1 & 282.2 \\
\hline
\end{tabular}
Send data to Excel
Calculations
\begin{tabular}{|c|c|c|}
\hline\( (\hat{y}-\overline{\boldsymbol{y}})^{\mathbf{2}} \) & \( (\boldsymbol{y}-\hat{\boldsymbol{y}})^{\mathbf{2}} \) & \( (\boldsymbol{y}-\overline{\boldsymbol{y}})^{\mathbf{2}} \) \\
\hline 355.6996 & 19.5364 & 208.5136 \\
\hline 121.7933 & 1.9712 & 154.7536 \\
\hline 1.9265 & 254.4663 & 300.6756 \\
\hline 75.5509 & 322.8490 & 710.7556 \\
\hline 504.8110 & 24.0885 & 308.3536 \\
\hline Column sum: & \begin{tabular}{c} 
Column sum: \\
1059.7813
\end{tabular} & \begin{tabular}{c} 
Column sum: \\
1622.9114
\end{tabular} \\
\hline
\end{tabular}

Figure 1
Answer the following.
(a) The total variation in the sample \( y \)-values ispiven by the (Choose one)
\( \square \) which for these data is (Choose one) \( \nabla \)
(b) The proportion of the total variation in the sample \( y \)-values that can be explained by the estimated linear relationship between \( x \) and \( y \) is \( \square \). (Round your answer to at least 2 decimal places.)
(c) The least-squares regression line given above is said to be a line that "best fits" the sample data. The term "best fits" is used because the line has an equation that minimizes the (Choose one) data is (Choose one) \( \mathbf{v} \). which for these data is (Choose one) \( \nabla \).
Explanation
Check

Simple linear regression: Explained and unaplained variation
Renonia
Bivanate data obtained for the paired variables x and y are shown below, in the table labeled "Sample data." These data are plotted in the scatter plot in Figure 1, which also displays the least-squares regression line for the data. The equation for this line is ŷ=141.14+0.48 x.

In the "Calculations" table are calculations involving the observed y-values, the mean y̅ of these values, and the values ŷ predicted from the regression equation.

Sample

x y

218.0 2502

234.3 252.2

254.4 2473

275.4 291.3

304.1 282.2

Send data to Excel
Calculations

(ŷ-y)^2 (y-ŷ)^2 (y-y)^2

355.6996 19.5364 208.5136

121.7933 1.9712 154.7536

1.9265 254.4663 300.6756

75.5509 322.8490 710.7556

504.8110 24.0885 308.3536

Column sum:
Column sum:

1059.7813

Column sum:

1622.9114

Figure 1
Answer the following.
(a) The total variation in the sample y-values ispiven by the (Choose one)
□ which for these data is (Choose one) ∇
(b) The proportion of the total variation in the sample y-values that can be explained by the estimated linear relationship between x and y is □. (Round your answer to at least 2 decimal places.)
(c) The least-squares regression line given above is said to be a line that "best fits" the sample data. The term "best fits" is used because the line has an equation that minimizes the (Choose one) data is (Choose one) 𝐯. which for these data is (Choose one) ∇.
Explanation
Check

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