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.
[tex]\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}[/tex]
Recall that the equation of the regression line is given by
[tex]y=a+bx[/tex]
where
[tex]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[/tex]
and
[tex]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[/tex]
Thus, the equation of the regresson line is given by
[tex]y=-6.977+2.7858x[/tex]
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
[tex]-6.977+2.7858(35)=-6.977+97.503=90.526[/tex]
Therefore, the percent donated in the year 2015 is predicted to be 90.5
