Chapter 29.8, Problem 8SRE

To determine

Obtain the regression equation for predicting y from x.

Answer to Problem 8SRE

The regression equation for predicting y from x is $\underset{\_}{\widehat{y}=1.6667+0.5333x}$.

Explanation of Solution

The general equation of the regression line is given below:

$\widehat{y}=\text{Intercept}+\text{Slope}\times x$

The value of Intercept is given below:

$y\text{-intercept}=\overline{y}-\text{Slope}\times \overline{x}$

Where,

$\begin{array}{l}\overline{x}-\text{sample average of }x\\ \overline{y}-\text{sample average of }y\\ \text{Slope}=r\times \frac{s_y}{s_x}\\ r=\frac{{\displaystyle \sum \left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}}{\left(n-1\right)s_x{s}_y}\\ s_x=\sqrt{\frac{{\displaystyle \sum {\left(x_i-\overline{x}\right)}^2}}{n_x-1}}\\ s_y=\sqrt{\frac{{\displaystyle \sum {\left(y_i-\overline{y}\right)}^2}}{n_y-1}}\end{array}$

The sample average of x is shown below:

$\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum x_i}}{n}\\ =\frac{1+8+10+10+14+17}{6}\\ =\frac{60}{6}\\ =10\end{array}$

The sample average of y is shown below:

$\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum x_i}}{n}\\ =\frac{1+4+6+12+12+7}{6}\\ =\frac{42}{6}\\ =7\end{array}$

The table below represents the values that help to find the regression equation:

x	y	$\left(x_i-\overline{x}\right)$	$\left(y_i-\overline{y}\right)$	$\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)$	${\left(x_i-\overline{x}\right)}^2$	${\left(y_i-\overline{y}\right)}^2$
1	1	-9	-6	54	81	36
8	4	-2	-3	6	4	9
10	6	0	-1	0	0	1
10	12	0	5	0	0	25
14	12	4	5	20	16	25
17	7	7	0	0	49	0
Total				80	150	96

The value of $s_x$ is given below:

$\begin{array}{c}{s}_x=\sqrt{\frac{{\displaystyle \sum {\left(x_i-\overline{x}\right)}^2}}{n_x-1}}\\ =\sqrt{\frac{150}{6-1}}\\ =\sqrt{\frac{150}{5}}\\ =5.4772\end{array}$

The value of $s_y$ is given below:

$\begin{array}{c}{s}_x=\sqrt{\frac{{\displaystyle \sum {\left(y_i-\overline{y}\right)}^2}}{n_y-1}}\\ =\sqrt{\frac{96}{6-1}}\\ =\sqrt{\frac{96}{5}}\\ =4.3818\end{array}$

The value of r is given below:

$\begin{array}{c}r=\frac{80}{\left(6-1\right)\left(5.4772\right)\left(4.3818\right)}\\ =\frac{80}{119.9999}\\ =0.6667\end{array}$

The slope of the regression line is given below:

$\begin{array}{c}\text{Slope}=r\times \frac{s_y}{s_x}\\ =0.6667\times \frac{4.3818}{5.4772}\\ =0.5333\end{array}$

The value of Intercept is given below:

$\begin{array}{c}y\text{-intercept}=\overline{y}-\text{Slope}\times \overline{x}\\ =7-0.5333\times 10\\ =1.6667\end{array}$

Substitute the corresponding values in the regression equation.

The regression equation for predicting y from x is $\underset{\_}{\widehat{y}=1.6667+0.5333x}$.