Question

The following table shows the percent increase of donations made on behalf of a non-profit organization for the period of 1984 to 2003. Use a graphing calculator to make a scatter plot of the data. Find an equation for and graph the line of regression. Then use the equation to predict the percent donated in the year 2015.

Answer 1

Giving the table below which shows the percent increase of donations made on behalf of a non-profit organization for the period of 1984 to 2003.

Year:        1984     1989     1993     1997     2001     2003

Percent:    7.8       16.3       26.2      38.9     49.2      62.1

The scatter plot of the data is attached with the x-axis representing the number of years after 1980 and the y-axis representing the percent increase of donations made on behalf of a non-profit organization.

To find the equation for the line of regression where the x-axis representing the number of years after 1980 and the y-axis representing the percent increase of donations made on behalf of a non-profit organization.
$\begin{center} \begin{tabular}{ c| c| c| c| } x & y & x^2 & xy \\ [1ex] 4 & 7.8 & 16 & 31.2 \\ 9 & 16.3 & 81 & 146.7 \\ 13 & 26.2 & 169 & 340.6 \\ 17 & 38.9 & 289 & 661.3 \\ 21 & 49.2 & 441 & 1,033.2 \\ 23 & 62.1 & 529 & 1,428.3 \\ [1ex] \Sigma x=87 & \Sigma y=200.5 & \Sigma x^2=1,525 & \Sigma xy=3,641.3 \end{tabular} \end{center}$

Recall that the equation of the regression line is given by
$y=a+bx$
where
$a= \frac{(\Sigma y)(\Sigma x^2)-(\Sigma x)(\Sigma xy)}{n(\Sigma x^2)-(\Sigma x)^2} = \frac{200.5(1,525)-87(3,641.3)}{6(1,525)-(87)^2} \\ \\ = \frac{305,762.5-316793.1}{9,150-7,569} = \frac{-11,030.6}{1,581} =-6.977$
and
$b= \frac{n(\Sigma xy)-(\Sigma x)(\Sigma y)}{n(\Sigma x^2)-(\Sigma x)^2} = \frac{6(3,641.3)-(87)(200.5)}{6(1,525)-(87)^2} \\ \\ = \frac{21,847.8-17,443.5}{9,150-7,569} = \frac{4,404.3}{1,581} =2.7858$

Thus, the equation of the regresson line is given by
$y=-6.977+2.7858x$

The graph of the regression line is attached.

Using the equation, we can predict the percent donated in the year 2015. Recall that 2015 is 35 years after 1980. Thus x = 35.

The percent donated in the year 2015 is given by
$-6.977+2.7858(35)=-6.977+97.503=90.526$

Therefore, the percent donated in the year 2015 is predicted to be 90.5