A multiple regression analysis produced the following tables. $\begin{array} { | c | c | c | c | c | } \hline & \text { Coefficients } & \text { Stardard Error } & t \text { Statistic } & p \text {-value } \\\hline \text { Irtercept } & 1411.876 & 7.621533 & & \\\hline \boldsymbol { x } _ { 1 } & 35.18215 & 96.8433 & & \\\hline \boldsymbol { x } _ { 1 } ^ { 2 } & - 7.721648 & \mathbf { 3 . 0 0 7 9 4 3 } & & \\\hline\end{array}$ $\begin{array} { | c | c | c | c | c | } \hline & \mathrm { df } & \mathrm { SS } & \mathrm { MS } & F \\\hline \text { Regression } & 2 & 58567032 & 29283516 & 57.34861 \\\hline \text { Residual } & 25 & 12765573 & 510622.9 & \\\hline \text { Total } & 27 & 71332605 & & \\\hline\end{array}$ The minimum value of the predicted value of the dependent variable is reached when
X₁ = ______.

Question

A multiple regression analysis produced the following tables.

\begin{array} { | c | c | c | c | c | } \hline & \text { Coefficients } & \text { Stardard Error } & t \text { Statistic } & p \text {-value } \\\hline \text { Irtercept } & 1411.876 & 7.621533 & & \\\hline \boldsymbol { x } _ { 1 } & 35.18215 & 96.8433 & & \\\hline \boldsymbol { x } _ { 1 } ^ { 2 } & - 7.721648 & \mathbf { 3 . 0 0 7 9 4 3 } & & \\\hline\end{array}

\begin{array} { | c | c | c | c | c | } \hline & \mathrm { df } & \mathrm { SS } & \mathrm { MS } & F \\\hline \text { Regression } & 2 & 58567032 & 29283516 & 57.34861 \\\hline \text { Residual } & 25 & 12765573 & 510622.9 & \\\hline \text { Total } & 27 & 71332605 & & \\\hline\end{array}

The minimum value of the predicted value of the dependent variable is reached when
X₁ = ______.

Quizplus · Accepted Answer

To find the minimum value of the predicted dependent variable, we need to find the vertex of the parabola represented by the quadratic equation in the regression model. The general form of a quadratic equation is $y = ax^2 + bx + c$, where $a$ and $b$ are the coefficients of $x^2$ and $x$, respectively, and $c$ is the constant. In this case, $a = -7.721648$ and $b = 35.18215$. The x-coordinate of the vertex (which gives the minimum or maximum of the parabola depending on the sign of $a$) is given by $-\frac{b}{2a}$. Substituting the given values:$$x = -\frac{35.18215}{2 \times (-7.721648)} = \frac{35.18215}{15.443296} \approx 2.277$$Since this value is not listed among the options, the correct choice is E, a value not listed here.

A Multiple Regression Analysis Produced the Following Tables The Minimum Value of the Predicted Value of the Dependent