Question

3. (a) The sales of a small factory since 2008 are as follows: (6)

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

$$\bar{x}=\dfrac{1}{n}\displaystyle\sum_{i=1}^nx_i=\dfrac{15}{6}$$

$$\bar{y}=\dfrac{1}{n}\displaystyle\sum_{i=1}^ny_i=\dfrac{61}{6}$$

$$SS_{xx}=\displaystyle\sum_{i=1}^nx_i^2-\dfrac{1}{n}(\displaystyle\sum_{i=1}^nx_i)^2=$$

$$=55-\dfrac{15^2}{6}=\dfrac{105}{6}$$

$$SS_{yy}=\displaystyle\sum_{i=1}^ny_i^2-\dfrac{1}{n}(\displaystyle\sum_{i=1}^ny_i)^2=$$

$$=631-\dfrac{61^2}{6}=\dfrac{65}{6}$$

$$SS_{xy}=\displaystyle\sum_{i=1}^nx_iy_i-\dfrac{1}{n}(\displaystyle\sum_{i=1}^nx_i)(\displaystyle\sum_{i=1}^ny_i)=$$

$$=165-\dfrac{15(61)}{6}=\dfrac{75}{6}$$

$$m=\dfrac{SS_{xy}}{SS_{xx}}=\dfrac{\dfrac{75}{6}}{\dfrac{105}{6}}=\dfrac{5}{7}$$

$$n=\bar{y}-m\bar{x}=\dfrac{61}{6}-\dfrac{5}{7}(\dfrac{15}{6})=\dfrac{176}{21}$$

Therefore, we find that the regression equation is:

$$Y=\dfrac{176}{21}+\dfrac{5}{7}X$$

$$Y=8.3810+0.7143X$$