Chapter 4.5, Problem 19E

(a)

To determine

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

  

Answer to Problem 19E

The equation of the best fit of the line is $6y=-x+116$ .

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:

  $\left(8,18\right)$ and $\left(14,17\right)$

Thus, we have:

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

  $\begin{array}{l}m=\frac{17-18}{14-8}\\ m=-\frac{1}{6}\end{array}$

Thus, the equation of the line is

  $y-y_1=m\left(x-x_1\right)$

  $\begin{array}{l}y-18=-\frac{1}{6}\left(x-8\right)\\ 6y-108=-x+8\\ 6y=-x+116\end{array}$

Thus, the equation of the best fit of the line is $6y=-x+116$ .

Conclusion:

The equation of the best fit of the line is $6y=-x+116$ .

(b)

To determine

To identify and interpret the correlation co-efficient -

  

Answer to Problem 19E

The correlation co-efficient is $-0.9685$ .

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 -

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

$x$	$y$	$xy$	$x^2$	$y^2$
$22$	$16$	$352$	$484$	$256$
$14$	$17$	$238$	$196$	$289$
$18$	$17$	$306$	$324$	$289$
$30$	$14$	$420$	$900$	$196$
$8$	$18$	$144$	$64$	$324$
$24$	$15$	$360$	$576$	$225$

Thus, we have:

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

  $r=-0.9685$

Conclusion:

The correlation co-efficient is $-0.9685$ .

(c)

To determine

To interpret the slope and y-intercept of the line of the best fit-

  

Answer to Problem 19E

The slope is $-\frac{1}{6}$ and $y$ -intercept is $\frac{58}{3}$ .

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 −

  

Equation of the best fit of the line is $6y=-x+116$ .

Equation can be written as:

  $\begin{array}{l}y=-\frac{1}{6}x+\frac{116}{6}\\ y=-\frac{1}{6}x+\frac{58}{3}\end{array}$

  $m=-\frac{1}{6}$

  $y$ -intercept $=$$\frac{58}{3}$

Thus, Slope is $-\frac{1}{6}$ and $y$ -intercept is $\frac{58}{3}$ .

Conclusion:

The slope is $-\frac{1}{6}$ and $y$ -intercept is $\frac{58}{3}$ .

(d)

To determine

To approximate the mileage of an automobile that costs $\$15,500$

Answer to Problem 19E

The mileage of an automobile is $23$ (thousands of mile).

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 −

  

Equation of the best fit of the line is $6y=-x+116$ .

Equation can be written as:

  $\begin{array}{l}y=-\frac{1}{6}x+\frac{116}{6}\\ y=-\frac{1}{6}x+\frac{58}{3}\end{array}$

Since, $y=15.5$

  $x=23$

Conclusion:

The mileage of an automobile is $23$ (thousands of mile).

(e)

To determine

To find the price of automobile with $6000$ miles.

  

Answer to Problem 19E

The price of automobile with $6000$ miles is $18.33$ (thousands of dollars).

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 −

  

Equation of the best fit of the line is $6y=-x+116$ .

Equation can be written as:

  $\begin{array}{l}y=-\frac{1}{6}x+\frac{116}{6}\\ y=-\frac{1}{6}x+\frac{58}{3}\end{array}$

Since, $x=6$

  $y=18.33$

Thus, the price of automobile with $6000$ miles is $18.33$ (thousands of dollars).

Conclusion:

The price of automobile with $6000$ miles is $18.33$ (thousands of dollars).