Question

4.56 The data points in Exercise 4.44

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 $\hat{\mathrm{y}}=1.75+0.25x$,

(b) The graph for the regression equation is

Step-by-step solution

Part (a) Step 1: Given information

The data points in Exercise 4.44 as:

$x$	$1$	$1$	$5$	$5$
$y$	$1$	$3$	$2$	$4$

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	$(x_i-\overline{x})$	${(x_i-\overline{x})}^2$	$(y_i-\overline{y})$	$(x_i-\overline{x})(y_i-\overline{y})$
$1$	$1$	$-2$	$4$	$-1.5$	$3$
$1$	$3$	$-2$	$4$	$0.5$	$-1$
$5$	$2$	$2$	$4$	$-0.5$	$-1$
$5$	$4$	$2$	$4$	$1.5$	$3$
$\overline{x}=3$	$\overline{y}=2.5$		$\sum {(x_i-\overline{x})}^2=16$		$\sum (x_i-\overline{x})(y_i-\overline{y})=4$

Hence, $S_{xx}=16$and $S_{xy}=4$.

Part (a) Step 3: Calculation

Since,

$$\begin{gathered} b_1=\frac{S_{xy}}{S_{xx}} \\ {b}_1=\frac{4}{16} \\ {b}_1=0.25 \end{gathered}$$

Then,

$$\begin{gathered} b_0=\overline{y}-b_1\overline{x} \\ {b}_0=2.5-(0.25)3 \\ {b}_0=2.5-0.75 \\ {b}_0=1.75 \end{gathered}$$

As a result, the regression equation will be as follows:
$$\begin{gathered} \hat{y}=b_0+b_{1}x \\ \hat{y}=1.75+(0.25)x \\ \hat{\mathrm{y}}=1.75+0.25x \end{gathered}$$

Part (b) Step 1: Given information

Given in the question that

Part (b) Step 2: Graphical representation

The graph for the regression equation $\widehat{y}=1.75+0.25x$is