Question

4.54 The data points in Exercise$4.42$

a. find the regression equotion for the data points. Use the defining formulas in Definition $4.4$to obtain $S_{xx}$and $S_{xy}$.

b. graph the regression equation and the data points.

Short answer

(a) The regression equation is$\begin{array}{r}\widehat{y}=9-2x\end{array}$.

(b) the graph for the regression equation is

Step-by-step solution

Part (a) Step 1: Given information

The data points in Exercise $4.42$is:

$x$	$1$	$3$	$4$	$4$
$y$	$8$	$0$	$3$	$1$

Part (a) Step 2: Explanation

The terms $b_0$and$b_1$ are used to generate the regression equation.
For a set of$n$ data points, the regression equation is:
$\hat{y}=b_0+b_{1}x$
Note that,
$$\begin{gathered} b_1=\frac{S_{xy}}{S_{xx}} \\ {b}_0=\frac{1}{n}\left(\sum y_i-b_1\sum x_i\right) \\ {b}_0=\overline{y}-b_1\overline{x} \end{gathered}$$

The table is available in the following formats:

$x$	$y$	$\left(x_i-\overline{x}\right)$	${\left(x_i-\overline{x}\right)}^2$	$\left(y_i-\overline{y}\right)$	localid="1650867898682" $\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)$
$1$	$8$	$-2$	$4$	$5$	$-10$
$3$	$0$	$0$	$0$	$-3$	$0$
$4$	$3$	$1$	$1$	$0$	$0$
$4$	$1$	$1$	$1$	$-2$	$-2$
$\overline{x}=3$	$\overline{y}=3$		localid="1650867836278" $\sum {\left(x_i-\overline{x}\right)}^2=6$		localid="1650867920723" $\sum \left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)=-12$

Hence, $S_{xx}=6$and $S_{xy}=-12$.

Part (a) Step 3: Calculation

Since,
$$\begin{gathered} b_1=\frac{S_{xy}}{S_{xx}} \\ {b}_1=\frac{-12}{6} \\ {b}_1=-2 \end{gathered}$$

Then,

$$\begin{gathered} b_0=\overline{y}-b_1\overline{x} \\ {b}_0=3-(-2)3 \\ {b}_0=3+6 \\ {b}_0=9 \end{gathered}$$

As a result, the regression equation will be as follows:

$$\begin{gathered} \begin{array}{r}\widehat{y}=b_0+b_{1}x\end{array} \\ \begin{array}{r}\widehat{y}=9+(-2)x\end{array} \\ \begin{array}{r}\widehat{y}=9-2x\end{array} \end{gathered}$$

Part (b) Step 1: Given information

Given in the question that

Part (b) Step 2: Graphical representation

The graph for the regression equation $\widehat{y}=9-2x$is