To determine the quadratic regression equation for the given data set, follow these steps:
1. Identify the data points:
[tex]\[
\begin{tabular}{|c|c|}
\hline $x$ & $y$ \\
\hline 3 & 3.45 \\
\hline 5 & 6.9 \\
\hline 6 & 8.79 \\
\hline 8 & 12.91 \\
\hline 10 & 17.48 \\
\hline 12 & 22.49 \\
\hline 15 & 30.85 \\
\hline
\end{tabular}
\][/tex]
2. Understand the form of the quadratic equation:
We aim to fit a quadratic equation of the form:
[tex]\[
\hat{y} = ax^2 + bx + c
\][/tex]
3. Coefficients from the quadratic regression:
The coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] obtained from the regression analysis are:
[tex]\[
\begin{align*}
a &= 0.056 \\
b &= 1.278 \\
c &= -0.886
\end{align*}
\][/tex]
4. Formulate the quadratic regression equation:
Substituting the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the equation:
[tex]\[
\hat{y} = 0.056x^2 + 1.278x - 0.886
\][/tex]
5. Identifying the correct equation:
Compare the formulated equation with the given options:
[tex]\[
\begin{align*}
\hat{y} &= 0.056 x^2 - 1.278 x - 0.886 \\
\hat{y} &= 0.056 x^2 + 1.278 x - 0.886 \\
\hat{y} &= 0.056 x^2 + 1.278 x \\
\hat{y} &= 0.056 x^2 + 1.278
\end{align*}
\][/tex]
The correct equation based on the coefficients is:
[tex]\[
\hat{y} = 0.056 x^2 + 1.278 x - 0.886
\][/tex]
Therefore, the quadratic regression equation for the given data set is:
[tex]\[
\hat{y} = 0.056 x^2 + 1.278 x - 0.886
\][/tex]
Thus, the correct option is:
[tex]\[
\boxed{\hat{y} = 0.056 x^2 + 1.278 x - 0.886}
\][/tex]
