Chapter 4.5, Problem 13E

(a)

To determine

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

  

Answer to Problem 13E

The equation of the best fit of the line is $5y=6x+80$

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:

  $\left(0,16\right)$ and $\left(5,22\right)$

Thus, we have:

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

  $\begin{array}{l}m=\frac{22-16}{5-0}\\ m=\frac{6}{5}\end{array}$

Thus, the equation of the line is

  $y-y_1=m\left(x-x_1\right)$

  $\begin{array}{l}y-16=\frac{6}{5}\left(x-0\right)\\ 5y-80=6x\\ 5y=6x+80\end{array}$

Thus, the equation of the best fit of the line is $5y=6x+80$ .

Conclusion:

The equation of the best fit of the line is $5y=6x+80$

(b)

To determine

To identify and interpret the correlation co-efficient -

  

Answer to Problem 13E

The correlation co-efficient is $0.9987$ .

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 -

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

$x$	$y$	$xy$	$x^2$	$y^2$
$-15$	$-4$	$60$	$225$	$16$
$-10$	$2$	$-20$	$100$	$4$
$-5$	$7$	$-35$	$25$	$49$
$0$	$16$	$0$	$0$	$256$
$5$	$22$	$110$	$25$	$484$
$10$	$30$	$300$	$100$	$900$
$15$	$37$	$555$	$225$	$1369$
$20$	$43$	$860$	$400$	$1849$

Thus, we have:

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

  $r=0.9987$

Conclusion:

The correlation co-efficient is $0.9987$ .