Question

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

PLEASEE HELP ME

Answer 1

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.