Chapter 4.6, Problem 12PPS

a.

To determine

To find: An equation for the median line fit.

Answer to Problem 12PPS

The equation of line of median fit is $y=-0.84141x+59.0644$

Explanation of Solution

Given Information: A table showing the statistics from the U.S. Administration.

Calculation:

The dataset is of 6 points. Divide the dataset into three groups such that all the groups have 2 values each.

Let the first group be $\left(18,40.919\right),\left(25,245.516\right)$ .

The second group be $\left(35,330.516\right),\left(45,296.665\right)$

And the third group be $\left(55,167.087\right),\left(65,54.559\right)$

Now, find the median of each group,

  $\begin{array}{c}{G}_1=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{18+25}{2},\frac{40.919+245.516}{2}\right)\\ =\left(\frac{43}{2},\frac{286.435}{2}\right)\\ =\left(21.5,143.2175\right)\\ G_2=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{35+45}{2},\frac{330.516+296.665}{2}\right)\\ =\left(\frac{80}{2},\frac{627.181}{2}\right)\\ =\left(40,313.5905\right)\\ G_3=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{55+65}{2},\frac{167.087+54.559}{2}\right)\\ =\left(\frac{120}{2},\frac{221.646}{2}\right)\\ =\left(60,110.823\right)\end{array}$

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

The slope of the median-line is,

  $\begin{array}{c}m=\frac{\left(110.823-143.2175\right)}{\left(60-21.5\right)}\\ =\frac{-32.3945}{38.5}\\ =-0.84141\end{array}$

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,

  $\begin{array}{c}y=mx+c\\ 43.919=-0.84141\left(18\right)+c\\ c=59.0644\end{array}$

Therefore, the equation of line of median fit is,

  $y=-0.84141x+59.0644$

b.

To determine

To graph: The median fit line.

Answer to Problem 12PPS

The graph is shown.

Explanation of Solution

Given Information: The equation of line of median fit is $y=-0.84141x+59.0644$

Calculation:

The graph of the median-fit line is,

  

The dots represent the three centroids of the three groups. Red line represents the line of median-fit.

c.

To determine

To predict: Whether the median-fit line give an accurate model of the number of firefighters.

Answer to Problem 12PPS

The median-fit line does not give an accurate model of the number of firefighters.

Explanation of Solution

Given Information: The graph of the median-fit model.

Calculation:

From the graph it is evident that the median-fit line does not pass from any of the points and is at a considerable distance. Hence, the median-fit line gives to a lot of error values leading to not so accurate model of the number of the firefighters.