Chapter 4.6, Problem 3CYU

a.

To determine

To graph: An equation for the median line fit.

Answer to Problem 3CYU

The equation of line of median fit is $y=-124.0740740x+362.222$.

Explanation of Solution

Given Information: A table consisting of the cost of the house per night based on how close it is to the water.

Calculation:

The dataset has 6 points. Divide the dataset into three groups with two values in each group.

Let the first group be $\left(0.3,325\right),\left(0.5,250\right)$ .

The second group be $\left(1,200\right),\left(1.25,150\right)$

And the third group be $\left(1.5,140\right),\left(2,100\right)$

Now, find the median of each group,

  $\begin{array}{c}{G}_1=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{0.3+0.5}{2},\frac{325+250}{2}\right)\\ =\left(\frac{0.8}{2},\frac{575}{2}\right)\\ =\left(0.4,287.5\right)\\ G_2=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{1+1.25}{2},\frac{200+150}{2}\right)\\ =\left(\frac{2.25}{2},\frac{350}{2}\right)\\ =\left(1.125,175\right)\\ G_3=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{1.5+2}{2},\frac{140+100}{2}\right)\\ =\left(\frac{3.5}{2},\frac{240}{2}\right)\\ =\left(1.75,120\right)\end{array}$

• Here, $G_1,G_2,G_3$ are the medians of group1, group2, group3 respectively.

The slope of the median-line is,

  $\begin{array}{c}m=\frac{\left(120-287.5\right)}{\left(1.75-0.4\right)}\\ =\frac{-167.5}{1.35}\\ =-124.0740740\end{array}$

The equation of line is,

  $y=mx+c$

Substitute the value of x, y and m in equation (1) to obtain the value of c,

  $\begin{array}{c}y=mx+c\\ 325=-124.0740740\left(0.3\right)+c\\ c=362.222\end{array}$

Therefore, the equation of line of median fit is,

  $y=-124.0740740x+362.222$

The graph of the median-fit line is,

  

b.

To determine

To calculate: The cost of a rental 1.75 miles from the lake.

Answer to Problem 3CYU

The cost of a rental 1.75 miles from the lake is 145.092

Explanation of Solution

Given Information: The equation of line of median fit is $y=-124.0740740x+362.222$ .

Calculation:

Substitute $x=1.75$ in the equation of line of median fit $y=-124.0740740x+362.222$ ,

  $\begin{array}{c}y=-124.0740740x+362.222\\ =-124.0740740\left(1.75\right)+362.222\\ =145.092\end{array}$