Determine the quadratic regression equation for the following data set. Then, use it to determine [tex]\( y \)[/tex] if [tex]\( x = 15 \)[/tex].

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & 3 & 5 & 6 & 5 & 8 & 7 & 10 & 11 & 11 & 12 \\
\hline
[tex]$y$[/tex] & -126 & -150 & -172 & -229 & -273 & -335 & -420 & -506 & -598 & -708 \\
\hline
\end{tabular}

A. [tex]$-1124.46371$[/tex]
B. [tex]$-979.68417$[/tex]
C. [tex]$-1073.87175$[/tex]
D. [tex]$-997.01487$[/tex]



Answer :

To determine the quadratic regression equation for the given data set and use it to find the value of [tex]\( y \)[/tex] when [tex]\( x = 15 \)[/tex], follow these steps:

### Step 1: Organize the Data
The given data points are:

[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 3 & 5 & 6 & 5 & 8 & 7 & 10 & 11 & 11 & 12 \\ \hline y & -126 & -150 & -172 & -229 & -273 & -335 & -420 & -506 & -598 & -708 \\ \hline \end{array} \][/tex]

### Step 2: Calculate the Quadratic Regression Equation
The quadratic regression equation has the form:

[tex]\[ y = ax^2 + bx + c \][/tex]

To determine the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], use the method of least squares to fit a quadratic polynomial to the data points.

Given the coefficients from the calculations:

[tex]\[ a = -6.368968909878045 \][/tex]
[tex]\[ b = 37.338902007084506 \][/tex]
[tex]\[ c = -200.93699330972316 \][/tex]

Thus, the quadratic regression equation is:

[tex]\[ y = -6.368968909878045 x^2 + 37.338902007084506 x - 200.93699330972316 \][/tex]

### Step 3: Calculate [tex]\( y \)[/tex] for [tex]\( x = 15 \)[/tex]
Substitute [tex]\( x = 15 \)[/tex] into the quadratic regression equation:

[tex]\[ y = -6.368968909878045 (15)^2 + 37.338902007084506 (15) - 200.93699330972316 \][/tex]

Evaluate the expression:

[tex]\[ y = -6.368968909878045 \cdot 225 + 37.338902007084506 \cdot 15 - 200.93699330972316 \][/tex]

[tex]\[ y = -1433.5170057225601 + 560.0835301062676 - 200.93699330972316 \][/tex]

[tex]\[ y = -1073.8714679260156 \][/tex]

### Result
Therefore, when [tex]\( x = 15 \)[/tex], the value of [tex]\( y \)[/tex] is:

[tex]\[ y = -1073.8714679260156 \][/tex]

In conclusion, the quadratic regression equation for the given data set is:

[tex]\[ y = -6.368968909878045 x^2 + 37.338902007084506 x - 200.93699330972316 \][/tex]

And the value of [tex]\( y \)[/tex] when [tex]\( x = 15 \)[/tex] is:

[tex]\[ y = -1073.8714679260156 \][/tex]