Question

7. Property taxes for an Ontario resident over the years are listed.

Year: 2007 2008 2009 2010 2011 2012

Taxes: $2,900 $3,000 $3,150 $3,300 $3,400 $3,600

b) Calculate the least squares equation. (4 marks)
c) Predict the property tax for the year 2020. (2 marks)

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c} X & Y \\ \hline 0 & 2900\\ 1 & 3000 \\ 2 & 3150 \\ 3 & 3300 \\ 4 & 3400 \\ 5 & 3600 \end{array}$$

$$\bar{x}={\textstyle\sum_{i=1}^nx_i \over n} ={0+1+2+3+4+5\over 6}=2.5$$

$$\bar{y}={\textstyle\sum_{i=1}^ny_i \over n} ={2900+3000+3150+3300+3400+3600\over 6}=3225$$

$$\textstyle\sum_{i=1}^nx_i =15$$

$$\textstyle\sum_{i=1}^nx_i ^2 =55$$

$$\textstyle\sum_{i=1}^ny_i =19350$$

$$\textstyle\sum_{i=1}^ny_i ^2=62742500$$

$$\textstyle\sum_{i=1}^nx_iy_i =50800$$

$$S_{xx}=\textstyle\sum_{i=1}^n(x_i -\bar{x})^2=\textstyle\sum_{i=1}^nx_i ^2-n\bar{x}^2=17.5$$

$$S_{yy}=\textstyle\sum_{i=1}^n(y_i -\bar{y})^2=\textstyle\sum_{i=1}^ny_i ^2-n\bar{y}^2=338750$$

$$S_{xy}=\textstyle\sum_{i=1}^n(x_i-\bar{x})(y_i -\bar{y})=$$

$$=\textstyle\sum_{i=1}^nx_iy_i -n\bar{x}\bar{y}=2425$$

b. Trend line: $y=A+Bx$

$$B={S_{xy} \over S_{xx}}={2425 \over 17.5}\approx138.57142857$$

$$A=\bar{y}-B\bar{x}=3225-{2425 \over 17.5}\cdot2.5\approx2878.57142857$$

$$y=2878.5714+138.5714x$$

$x$ is the number of years from 2007.

c. $2020-2007=13$

$$y=2878.5714+138.5714(13)\approx4680$$

The property tax for the year 2020 will be $4680.