Chapter 4.5, Problem 20E

a.

To determine

To find: Equation of line of best fit using graphing calculator.

Answer to Problem 20E

The equation of line of best fit is $y=4.87x-37.68$ .

Explanation of Solution

Given Information: Table with data values of length and cost.

Calculation:

Inputting the given data in a graphing calculator, results are,

  $\begin{array}{l}\text{LinReg}\\ \text{ y=ax+b}\\ \text{ a= 4}\text{.87218045112782}\\ \text{ b= -37}\text{.676691729323316}\\ {\text{ r}}^2\text{= 0}\text{.8755332591045001}\\ \text{ r= 0}\text{.9356993422593072}\end{array}$

As $\text{y=ax+b}$ ,

Rounding (if needed) and substituting the values,

  $y=4.87x-37.68$

Thus, the equation of line of best fit is $y=4.87x-37.68$ .

b.

To determine

To interpret: The correlation coefficient.

Answer to Problem 20E

Model is a good fit for the data. As the length of sailboat increases cost increases.

Explanation of Solution

Given Information: Table with data values of length and cost, graphing calculator results from previous part.

Interpretation:

In the graphing calculator coefficient correlation is shown by the symbol $r$ . As can be seen, its value is $0.935$ , this implies that the points lie around a positive slope and there is a strong positive correlation. In this case, as the length of sailboat increases cost increases. As the magnitude of $r$ is near to $1$ . Model is a good fit for the data.

c.

To determine

To interpret: The slope and $y-\mathrm{int}ercept$ of the line of best fit.

Answer to Problem 20E

Slope signifies an increase of $\$4870$ in cost per every feet.

  $\text{y-intercept}$ has no significance or meaning in this particular instance.

Explanation of Solution

Given Information: Table with data values of length and cost, graphing calculator results from one of the previous parts.

Calculation:

Interpretation:

As can be seen in graphing calculator screen, slope $(a)$

  $=4.87$ , this means that marked points make an uphill from left to right i.e. they tend to head higher and higher as we move along the horizontal axis. In the concerned case, it signifies an increase of $\$4870$ in cost per every feet.

Putting $x=0$ in $y=4.87x-37.68$ to calculate $y-\mathrm{int}ercept$ ,

  $y=-37.68$

In our case, length of sailboat cannot be $0$ , it results in negative cost. Clearly, $\text{y-intercept}$ has no significance or meaning in this particular instance.

d.

To determine

Cost of a sail boat with length of $20feet$ .

Answer to Problem 20E

The cost is $\$59720$ .

Explanation of Solution

Given Information: Table with data values of length and cost, graphing calculator results from one of the previous parts.

Calculation:

Equation is $y=4.87x-37.68$ ,

Putting $x=20$ in above stated equation,

  $y=4.87(20)-37.68$

  $\begin{array}{l}y=4.87(20)-37.68\\ y=97.4-37.68\\ y=59.72\end{array}$

Thus, the cost is $\$59720$ .

e.

To determine

To predict: The length of a sailboat that costs $\$147,000$ .

Answer to Problem 20E

The length would be $37.92feet$ .

Explanation of Solution

Given Information: Table with data values of length and cost, graphing calculator results from one of the previous parts.

Calculation:

Equation is $y=4.87x-37.68$ ,

Putting $y=147$ in above stated equation,

  $\begin{array}{l}147=4.87x-37.68\\ 4.87x=147+37.68\\ 4.87x=184.68\\ x=\frac{184.68}{4.87}\\ x\approx 37.92\end{array}$

Thus, the length would be $37.92feet$ .