Question

4.57 The data points in Exercise 4.45

a. find the regression equation 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 $\hat{y}=2.875-0.625x$,

(b) The graph for the regression equation will be,

Step-by-step solution

Part (a) Step 1: Given information

The data points in Exercise $4.45$as:

$x$	$0$	$2$	$2$	$5$	$6$
$y$	$4$	$2$	$0$	$-2$	$1$

Part (a) Step 2: Explanation

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

$x$	$y$	$\sum (x_i-\overline{x)}$	$\sum {(x_i-\overline{x})}^2$	$(y_i-\overline{y})$	$(x_i-\overline{x})(y_i-\overline{y})$
$0$	$4$	$-3$	$9$	$3$	$-9$
$2$	$2$	$-1$	$1$	$1$	$-1$
$2$	$0$	$-1$	$-1$	$-1$	$1$
$5$	$-2$	$2$	$4$	$-3$	$-6$
$6$	$1$	$3$	$9$	$0$	$0$
$\overline{x}=3$	$\overline{y}=1$		$\sum {(x_i-\overline{x})}^2=24$		$\sum (x_i-\overline{x})(y_i-\overline{y})=-15$

Hence, $S_{xx}=24$and $S_{xy}=-15$.

Part (a) Step 3: Calculation

Since,

$$\begin{gathered} b_1=\frac{S_{xy}}{S_{xx}} \\ {b}_1=\frac{-15}{24} \\ {b}_1=-0.625 \end{gathered}$$

Then,

$$\begin{gathered} b_0=\overline{y}-b_1\overline{x} \\ {b}_0=1-(-0.625)3 \\ {b}_0=1+1.875 \\ {b}_0=2.875 \end{gathered}$$

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

$$\begin{gathered} \hat{y}=b_0+b_{1}x \\ \hat{y}=2.875+(-0.625)x \\ \hat{y}=2.875-0.625x \end{gathered}$$

Part (b) Step 1: Given information

Given in the question that

Part (b) Step 2: Graphical representation

The graph for the regression equation $\widehat{y}=2.875-0.625x$is