Chapter 4.5, Problem 12E

(a)

To determine

To find the equation of the line of the best fit for the data -

  

Answer to Problem 12E

The equation of the best fit of the line is $2y=x+16$

Explanation of Solution

Given:

  

Formula Used:

The line of fit is the line that most closely goes through the points of data.

The equation of a line is written as $y=mx+b$ where m is the slope and $b$ is the $y$ -intercept.

  $m=\frac{y_2-y_1}{x_2-x_1}$

Calculation:

Given is −

  

Plotting the above points on the graph, we have:

  

Considering the below two points as they fall on the same straight line, we have:

  $\left(0,8\right)$ and $\left(-2,7\right)$

Thus, we have:

Slope $=$$m=\frac{y_2-y_1}{x_2-x_1}$

  $\begin{array}{l}m=\frac{7-8}{-2-0}\\ m=\frac{1}{2}\end{array}$

Thus, the equation of the line is

  $y-y_1=m\left(x-x_1\right)$

  $\begin{array}{l}y-8=\frac{1}{2}\left(x-0\right)\\ y-8=\frac{1}{2}x\\ 2y-16=x\\ 2y=x+16\end{array}$

Thus, the equation of the best fit of the line is $2y=x+16$ .

Conclusion:

The equation of the best fit of the line is $2y=x+16$

(b)

To determine

To identify and interpret the correlation co-efficient -

  

Answer to Problem 12E

The correlation co-efficient is $-0.8858$ .

Explanation of Solution

Given:

  

Formula Used:

The formula to calculate correlation co-efficient is -

  $r=\frac{n\left({\displaystyle \sum xy}\right)-\left({\displaystyle \sum x}\right)\left(y\right)}{\sqrt{\left[n{\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2\right]\left[n{\displaystyle \sum y^2}-{\left(y\right)}^2\right]}}$

Calculation:

Given is −

  

Using the above formula to calculate correlation co-efficient, we have -

  $r=\frac{n\left({\displaystyle \sum xy}\right)-\left({\displaystyle \sum x}\right)\left(y\right)}{\sqrt{\left[n{\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2\right]\left[n{\displaystyle \sum y^2}-{\left(y\right)}^2\right]}}$

$x$	$y$	$xy$	$x^2$	$y^2$
$-4$	$17$	$-68$	$16$	$289$
$-2$	$7$	$-14$	$4$	$49$
$0$	$8$	$0$	$0$	$64$
$2$	$1$	$2$	$4$	$1$
$4$	$5$	$20$	$16$	$25$
$6$	$-2$	$-12$	$36$	$4$
$8$	$2$	$16$	$64$	$4$
$10$	$-8$	$-80$	$100$	$64$

Thus, we have:

  $r=\frac{n\left({\displaystyle \sum xy}\right)-\left({\displaystyle \sum x}\right)\left(y\right)}{\sqrt{\left[n{\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2\right]\left[n{\displaystyle \sum y^2}-{\left(y\right)}^2\right]}}$

  $r=-0.8858$

Conclusion:

The correlation co-efficient is $-0.8858$ .