Chapter 4.6, Problem 7PPS

a.

To determine

To graph: An equation for the median line fit.

Answer to Problem 7PPS

The equation of line of median fit is $y=601.44x+2395$.

Explanation of Solution

Given Information: A table consisting of the number of entrants in the Boston Marathon every five years since 1975.

Calculation:

We have a dataset of 8 points. Divide the dataset into three groups with first and third group having same number of values that is 3 in each group and second group having 2 values in it.

Let the first group be $\left(0,2395\right),\left(5,5417\right),\left(10,5594\right)$ .

The second group be $\left(15,9412\right),\left(20,9415\right)$

And the third group be $\left(25,17813\right),\left(30,20453\right),\left(35,26735\right)$

• Here, 0 is for the year 1975.

Now, find the median of each group,

  $\begin{array}{c}{G}_1=\left(5,5417\right)\\ G_2=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{15+20}{2},\frac{9412+9415}{2}\right)\\ =\left(\frac{35}{2},\frac{18827}{2}\right)\\ =\left(17.5,9413.5\right)\\ G_3=\left(30,20453\right)\end{array}$

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

The slope of the median-line is,

  $\begin{array}{c}m=\frac{\left(20453-5417\right)}{\left(30-5\right)}\\ =\frac{15036}{25}\\ =601.44\end{array}$

The equation of line is,

  $y=mx+c$

• Here, x is the number of years from 1975

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

  $\begin{array}{c}y=mx+c\\ 2395=601.44\left(0\right)+c\\ c=2395\end{array}$

Therefore, the equation of line of median fit is,

  $y=601.44x+2395$

The graph of the median-fit line is,

  

b.

To determine

To calculate: The number of entrants in 2003.

Answer to Problem 7PPS

The number of entrants in 2003 were $19235.32$.

Explanation of Solution

Given Information: The equation of line of median fit is $y=601.44x+2395$ .

Calculation:

Substitute $x=28$ in the equation of line of median fit $y=601.44x+2395$ , since 2003 is 28 years after 1975.

  $\begin{array}{c}y=601.44x+2395\\ =601.44\left(28\right)+2395\\ =19235.32\end{array}$