Chapter 4.5, Problem 20E

(a)

To determine

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

  

Answer to Problem 20E

The equation of the best fit of the line is $y=7x-95$ .

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:

  $\left(26,87\right)$ and $\left(27,94\right)$

Thus, we have:

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

  $\begin{array}{l}m=\frac{94-87}{27-26}\\ m=7\end{array}$

Thus, the equation of the line is

  $y-y_1=m\left(x-x_1\right)$

  $\begin{array}{l}y-87=7\left(x-26\right)\\ y-87=7x-182\\ y=7x-95\end{array}$

Thus, the equation of the best fit of the line is $y=7x-95$ .

Conclusion:

The equation of the best fit of the line is $y=7x-95$ .

(b)

To determine

To identify and interpret the correlation co-efficient -

  

Answer to Problem 20E

The correlation co-efficient is $0.9357$ .

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 -

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

$x$	$y$	$xy$	$x^2$	$y^2$
$27$	$94$	$2538$	$729$	$8836$
$18$	$56$	$1008$	$324$	$3136$
$25$	$58$	$1450$	$625$	$3364$
$32$	$123$	$3936$	$1024$	$15129$
$18$	$60$	$1080$	$324$	$3600$
$26$	$87$	$2262$	$676$	$7569$
$36$	$145$	$5220$	$1296$	$21025$

Thus, we have:

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

  $r=0.9357$

Conclusion:

The correlation co-efficient is $0.9357$ .

(c)

To determine

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

  

Answer to Problem 20E

The slope is $7$ and $y$ -intercept is $-95$ .

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 $y=7x-95$ .

  $m=7$

  $y$ -intercept $=$$-95$

Thus, Slope is $7$ and $y$ -intercept is $-95$ .

Conclusion:

The slope is $7$ and $y$ -intercept is $-95$ .

(d)

To determine

To find the cost of sailboat that is $20$ feet long.

  

Answer to Problem 20E

The cost of sailboat that is $20$ feet long is $45$ (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 $y=7x-95$ .

Since, $x=20$

  $y=45$

Conclusion:

The cost of sailboat that is $20$ feet long is $45$ (thousands of dollars).

(e)

To determine

To find the length of sailboat that costs $\$147000$ .

  

Answer to Problem 20E

The length of sailboat that costs $\$147000$ is $34.57$ feet.

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 $y=7x-95$ .

Since, $y=147$

  $x=34.57$

Thus, the length of sailboat that costs $\$147000$ is $34.57$ feet.

Conclusion:

The length of sailboat that costs $\$147000$ is $34.57$ feet.