Chapter 8.9, Problem 1GP

(a)

To determine

Sketch a scatter plot with the line of fit.

Year	$1999$	$2000$	$2001$	$2002$	$2003$	$2004$	$2005$	$2006$
Percent	$3$	$7$	$10$	$14$	$17$	$19$	$21$	$23$

Answer to Problem 1GP

The scatter plot and the line of fits is shown by the following graph:

  

Explanation of Solution

Given:

Year	$1999$	$2000$	$2001$	$2002$	$2003$	$2004$	$2005$	$2006$
Percent	$3$	$7$	$10$	$14$	$17$	$19$	$21$	$23$

Concept Used:

The slope intercept form equation is

  $y=mx+b$

Where;

Slope: $m$

Y-intercept: $\left(0,b\right)$

Calculation:

Consider,

Year: $x$

Percent: $y$

There are seven points are given as:

  $\begin{array}{l}A=\left(x_1,y_1\right)=\left(1999,3\right)\\ B=\left(x_2,y_2\right)=\left(2000,7\right)\\ C=\left(x_3,y_3\right)=\left(2001,10\right)\\ D=\left(x_4,y_4\right)=\left(2002,14\right)\\ E=\left(x_5,y_5\right)=\left(2003,17\right)\\ F=\left(x_6,y_6\right)=\left(2004,19\right)\\ G=\left(x_7,y_7\right)=\left(2005,21\right)\\ H=\left(x_8,y_8\right)=\left(2006,23\right)\end{array}$

Now use any two points to calculate the slope intercept from equation:

  $m=\frac{y_3-y_1}{x_3-x_1}=\frac{10-3}{2001-1999}=\frac{7}{2}$

Now, the slope intercept form equation is written as:

  $\begin{array}{l}y=mx+b\\ y=\frac{7}{2}x+b\end{array}$

Now put the value of $\left(x_1,y_1\right)=\left(1999,3\right)$ in the above eq. to get the Y-intercept:

  $\begin{array}{l}y=mx+b\\ 3=\frac{7}{2}\left(1999\right)+b\\ b=-6996.5+3\\ b=-6993.5\end{array}$

So, the final slope intercept form equation is:

  $\begin{array}{l}y=mx+b\\ y=\frac{7}{2}x-6993.5\end{array}$

The scatter plot and the line of fits are shown by the following graph by using the above data points and line of fit eq. or slope-intercept form eq.

  

Conclusion:

Hence, with the help of slope-intercept form the line of fit is drawn as above.

(b)

To determine

The percent of households with the digital camera in the year $2015$ by using line of fit equation from the above subpart

Answer to Problem 1GP

There is $59$ percent of households with the digital camera in the year $2015$

Explanation of Solution

Given:

  $x=2015$

Concept Used:

The slope intercept form equation is

  $y=mx+b$

Where;

Slope: $m$

Y-intercept: $\left(0,b\right)$

Calculation:

From the first subpart:

The final slope intercept form equation is derived as:

  $\begin{array}{l}y=mx+b\\ y=\frac{7}{2}x-6993.5\end{array}$

Now, to calculate the number of Sunday newspapers $y$ , need to put $x=2015$ in above eq.

  $\begin{array}{l}y=mx+b\\ y=\frac{7}{2}x-6993.5\\ y=\frac{7}{2}\left(2015\right)-6993.5\\ y=7052.5-6993.5\\ y=59\end{array}$

Conclusion:

Hence, percent of households with the digital camera in the year $2015$ is calculated as $y=59$ by using the line fit.