mumto2kids
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given the data points (1,5), (3,13) and (5,21), which of the following equations describes the best-fit line?

y-hat=4+1x
y-hat=2+4x
y-hat=1+4x
y-hat=-1+4x

Respuesta :

LammettHash
We're looking for a line of best fit of the form [tex]\hat y=c_1x+c_0[/tex]. Set up a matrix equation:

[tex]\begin{bmatrix}1&1\\3&1\\5&1\end{bmatrix}\begin{bmatrix}c_1\\c_0\end{bmatrix}=\begin{bmatrix}5\\13\\21\end{bmatrix}[/tex]

Multiply both sides on the left by the transpose of the coefficient matrix:

[tex]\begin{bmatrix}1&3&5\\1&1&1\end{bmatrix}\begin{bmatrix}1&1\\3&1\\5&1\end{bmatrix}\begin{bmatrix}c_1\\c_0\end{bmatrix}=\begin{bmatrix}1&3&5\\1&1&1\end{bmatrix}\begin{bmatrix}5\\13\\21\end{bmatrix}[/tex]
[tex]\begin{bmatrix}35&9\\9&3\end{bmatrix}\begin{bmatrix}c_1\\c_0\end{bmatrix}=\begin{bmatrix}149\\39\end{bmatrix}[/tex]

Multiply both sides by the inverse of the new coefficient matrix:

[tex]\begin{bmatrix}35&9\\9&3\end{bmatrix}^{-1}\begin{bmatrix}35&9\\9&3\end{bmatrix}\begin{bmatrix}c_1\\c_0\end{bmatrix}=\begin{bmatrix}35&9\\9&3\end{bmatrix}^{-1}\begin{bmatrix}149\\39\end{bmatrix}[/tex]
[tex]\begin{bmatrix}c_1\\c_0\end{bmatrix}=\begin{bmatrix}4\\1\end{bmatrix}[/tex]

So we end up with a best-fit line of

[tex]\hat y=4x+1[/tex]

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