Answer to Question #200451 in Statistics and Probability for Raghav

Question

The sales of a small factory since 2008 are as follows:

Year Sales (in ₹ lakhs)

• 2008 8
• 2009 10
• 2010 9
• 2011 11
• 2012 11
• 2013 12

Using 2008 as the zero year, find the least-square trend-line equation.

Answer 1

$$\bar{x}=\dfrac{\sum_ix_i}{n}=\dfrac{15}{6}=2.5$$

$$\bar{y}=\dfrac{\sum_iy_i}{n}=\dfrac{61}{6}\approx10.166667$$

$$SS_{xx}=\sum_i(x_i-\bar{x})^2=\sum_ix_i^2-n\cdot\bar{x}^2$$

$$=55-6\cdot(2.5)^2=17.5$$

$$SS_{yy}=\sum_i(y_i-\bar{y})^2=\sum_iy_i^2-n\cdot\bar{y}^2$$

$$=631-6\cdot(\dfrac{61}{6})^2=\dfrac{65}{6}\approx10.833333$$

$$SS_{xy}=\sum_i(x_i-\bar{x})(y_i-\bar{y})=\sum_ix_iy_i-n\cdot\bar{x}\bar{y}$$

$$=165-6\cdot(2.5)(\dfrac{61}{6})=12.5$$

$$b=\dfrac{SS_{xy}}{SS_{xx}}=\dfrac{12.5}{17.5}=\dfrac{5}{7}\approx0.714286$$

$$a=\bar{y}-b\bar{x}=\dfrac{61}{6}-\dfrac{5}{7}(2.5)=\dfrac{176}{21}\approx8.380952$$

$$y=8.380952+0.714286x$$

$$r=\dfrac{SS_{xy}}{\sqrt{SS_{xx}}\sqrt{SS_{yy}}}\approx0.9078413$$

$$r^2\approx0.8241758$$