Question

In the accompanying table, we list the height h, the gender g, and the weightwof some young adults.

Fit a function of the form

$\mathit{w}\mathbf{=}{\mathit{c}}_{\mathbf{0}}\mathbf{+}{\mathit{c}}_{\mathbf{1}}\mathit{h}\mathbf{+}{\mathit{c}}_{\mathbf{2}}\mathit{g}$

To these data, using least squares. Before you do the computations, think about the signs of ${\mathit{c}}_{\mathit{1}}$ and ${\mathit{c}}_2$. What signs would you expect if these data were representative of the general population? Why? What is the sign of ${\mathit{c}}_{\mathbf{0}}$? What is the practical significance of ${\mathit{c}}_{\mathbf{0}}$?

Short answer

$\stackrel{r*}{c}=\left[\begin{array}{c}125\\ 5\\ -25\end{array}\right]$and the function that best fits the data points is $w^{*}=125+5h-25g$.

Step-by-step solution

Function of the data point.

From the given data it gets the terms,

$$\begin{gathered} c_0+2c_1+0c_2=110....\left(1\right) \\ {c}_0+12c_1+0c_2=120....\left(2\right) \\ {c}_0+5c_1+c_2=120....\left(3\right) \\ {c}_0+11c_1+c_2=160....\left(4\right) \\ {c}_0+6c_1+0c_2=160....\left(5\right) \end{gathered}$$

Thus, it has the system $\underset{A}{\underbrace{\left[\begin{array}{ccc}1& 2& 1\\ 1& 12& 0\\ 1& 5& 1\\ 1& 11& 1\\ 1& 6& 0\end{array}\right]}}\underset{\stackrel{\to *}{c}}{\underbrace{\left[\begin{array}{cc}{c}_0& ⬚\\ c_1& ⬚\\ c_2& ⬚\end{array}\right]}}=\left[\begin{array}{c}110\\ 180\\ 120\\ 160\\ 160\end{array}\right]$.

Recall that if Ker{A}={localid="1659697152741" $\stackrel{\perp }{0}$}. Then, $\stackrel{r*}{x}={\left(A^{T}A\right)}^{-1}{A}^T\stackrel{r}{b}*.$.

Since, the kernel of the matrix A is {$\stackrel{\perp }{0}$}.

So, it is found as:

$$\begin{gathered} A^{T}A=\left[\begin{array}{ccc}5& 36& 3\\ 36& 330& 18\\ 3& 18& 3\end{array}\right] \\ {\left(A^{T}A\right)}^{-1}=\left[\begin{array}{ccc}1.8500& -0.1500& -0.9500\\ -0.1500& 0.0167& 0.0500\\ -0.9500& 0.0500& 0.9833\end{array}\right] \\ {\left(A^{T}A\right)}^{-1}{A}^T=\left[\begin{array}{ccccc}0.6000& 0.0500& 0.1500& -0.7500& 0.9500\\ -0.0667& 0.0500& -0.0167& 0.0833& -0.0500\\ 0.1333& -0.3500& -0.2833& 0.5833& -0.6500\end{array}\right] \end{gathered}$$

${\left(A^{T}A\right)}^{-1}{A}^T\stackrel{r}{b*}=\left[\begin{array}{c}125\\ 5\\ -25\end{array}\right]$

Hence, $\stackrel{r}{c}*=\left[\begin{array}{c}125\\ 5\\ -25\end{array}\right]$and the function that fits the data points is $w*=125+5h-25g$.