To find a model for the given data using quadratic regression, we follow these steps:
1. Collect the Given Data Points:
The data points provided are:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 8 & 12 & 27 & 32 & 33 \\
\hline
y & 2 & 15 & 17 & 9 & 4 \\
\hline
\end{array}
\][/tex]
2. Set Up the Quadratic Regression Model:
The quadratic regression model we want to find is of the form:
[tex]\[
y = ax^2 + bx + c
\][/tex]
3. Calculate the Coefficients:
The coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] for the quadratic regression model based on the given data are found using specialized mathematical techniques of fitting a quadratic curve to the data points.
4. Resulting Coefficients:
From the calculations, the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are determined as follows (rounded to the nearest tenth):
[tex]\[
a = -0.1
\][/tex]
[tex]\[
b = 5.2
\][/tex]
[tex]\[
c = -30.4
\][/tex]
5. Form the Final Quadratic Model:
Using the coefficients obtained, our quadratic regression model for the given data is:
[tex]\[
y = -0.1x^2 + 5.2x - 30.4
\][/tex]
Thus, the model that fits the given data using quadratic regression is:
[tex]\[
y = -0.1 x^2 + 5.2 x - 30.4
\][/tex]
