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], please follow this step-by-step solution, which involves fitting a second-degree polynomial (quadratic) to the data points.
Given data points:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
x & 0 & 2 & 3 & 3 & 5 & 6 & 6 & 9 & 9 & 9 \\
\hline
y & 493 & 500 & 487 & 477 & 452 & 429 & 383 & 324 & 260 & 180 \\
\hline
\end{array}
\][/tex]
### Step 1: Quadratic Regression Equation
The quadratic regression equation generally takes the form:
[tex]\[ y = ax^2 + bx + c \][/tex]
Using the given data, the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] of the quadratic equation are found to be:
[tex]\[ a \approx -4.10133983 \][/tex]
[tex]\[ b \approx 10.54582392 \][/tex]
[tex]\[ c \approx 492.13021762 \][/tex]
Thus, our quadratic regression equation becomes:
[tex]\[ y = -4.10133983x^2 + 10.54582392x + 492.13021762 \][/tex]
### Step 2: Calculate [tex]\( y \)[/tex] for [tex]\( x = 15 \)[/tex]
Now, we need to use this equation to determine the value of [tex]\( y \)[/tex] when [tex]\( x = 15 \)[/tex]:
[tex]\[ y = -4.10133983(15)^2 + 10.54582392(15) + 492.13021762 \][/tex]
### Step 3: Plugging in the value
First, calculate [tex]\( 15^2 \)[/tex]:
[tex]\[ 15^2 = 225 \][/tex]
Next, substitute [tex]\( x = 15 \)[/tex] into the equation:
[tex]\[ y = -4.10133983(225) + 10.54582392(15) + 492.13021762 \][/tex]
Calculate each term:
[tex]\[ -4.10133983 \times 225 \approx -922.80146175 \][/tex]
[tex]\[ 10.54582392 \times 15 \approx 158.1873588 \][/tex]
Now combine these with the constant term [tex]\( 492.13021762 \)[/tex]:
[tex]\[ y = -922.80146175 + 158.1873588 + 492.13021762 \][/tex]
Sum these terms:
[tex]\[ y \approx -922.80146175 + 158.1873588 + 492.13021762 \][/tex]
[tex]\[ y \approx -272.4838863122833 \][/tex]
Therefore, the value of [tex]\( y \)[/tex] when [tex]\( x = 15 \)[/tex] is approximately:
[tex]\[ y \approx -272.484 \][/tex]
Among the given choices, the matching value is:
[tex]\[ \boxed{-272.484} \][/tex]
