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]
