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What is the quadratic regression equation for the data set? ​ ​ ​

PLEASEE HELP ME

What is the quadratic regression equation for the data set? ​ ​ ​ PLEASEE HELP ME-example-1
User Ueeieiie
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1 Answer

4 votes
You're trying to find constants
a_0,a_1,a_2 such that
\hat y=a_0+a_1\hat x+a_2{\hat x}^2. Equivalently, you're looking for the least-square solution to the following matrix equation.



\underbrace{\begin{bmatrix}1&6&6^2\\1&3&3^2\\\vdots&\vdots&\vdots\\1&9&9^2\end{bmatrix}}_(\mathbf A)\underbrace{\begin{bmatrix}a_0\\a_1\\a_2\end{bmatrix}}_(\mathbf x)=\underbrace{\begin{bmatrix}100\\110\\\vdots\\70\end{bmatrix}}_(\mathbf b)

To solve
\mathbf{Ax}=\mathbf b, multiply both sides by the transpose of
\mathbf A, which introduces an invertible square matrix on the LHS.


\mathbf{Ax}=\mathbf b\implies\mathbf A^\top\mathbf{Ax}=\mathbf A^\top\mathbf b\implies\mathbf x=(\mathbf A^\top\mathbf A)^(-1)\mathbf A^\top\mathbf b

Computing this, you'd find that


\mathbf x=\begin{bmatrix}a_0\\a_1\\a_2\end{bmatrix}\approx\begin{bmatrix}121.119\\-3.786\\-0.175\end{bmatrix}

which means the first choice is correct.
User Ajeet Lakhani
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