Chapter 4.5, Problem 14E

(a)

To determine

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

  

Answer to Problem 14E

The equation of the best fit of the line is $y=-4x+35$

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:

  $\left(8,3\right)$ and $\left(9,-1\right)$

Thus, we have:

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

  $\begin{array}{l}m=\frac{-1-3}{9-8}\\ m=-4\end{array}$

Thus, the equation of the line is

  $y-y_1=m\left(x-x_1\right)$

  $\begin{array}{l}y-3=-4\left(x-8\right)\\ y-3=-4x+32\\ y=-4x+35\end{array}$

Thus, the equation of the best fit of the line is $y=-4x+35$ .

Conclusion:

The equation of the best fit of the line is $y=-4x+35$

(b)

To determine

To identify and interpret the correlation co-efficient -

  

Answer to Problem 14E

The correlation co-efficient is $-0.4435$ .

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 -

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

$x$	$y$	$xy$	$x^2$	$y^2$
$5$	$12$	$60$	$25$	$144$
$6$	$-2$	$-12$	$36$	$4$
$7$	$8$	$56$	$49$	$64$
$8$	$3$	$24$	$64$	$9$
$9$	$-1$	$-9$	$81$	$1$
$10$	$-4$	$-40$	$100$	$16$
$11$	$6$	$66$	$121$	$36$
$12$	$0$	$0$	$144$	$0$

Thus, we have:

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

  $r=-0.4435$

Conclusion:

The correlation co-efficient is $-0.4435$ .