Question

The sales of a company from 1993-1998 are given below:

Year 1993 1994 1995 1996 1997 1998
Sales (in lakhs of rupees) 40 45 50 55 60 65
Fit a linear curve using the least squares method. Hence find out the company’s sales in 1999.

Answer 1

Let $x=$ the number of years since 1993, $y=$ sales (in lakhs of rupees)

$$\def\arraystretch{1.5} \begin{array}{c:c:c} & x & y & xy & x^2 & y^2 \\ \hline & 0 & 40 & 0 & 0 & 1600 \\ \hdashline & 1 & 45 & 45 & 1 & 2025 \\ & 2 & 50 & 100 & 4 & 2500\\ \hdashline & 3 & 55 & 165 & 9 & 3025\\ & 4 & 60 & 240 & 16 & 3600 \\ & 5 & 65 & 325 & 25 & 4225\\ \hdashline Sum= & 15 & 315 & 875 & 55 & 16975 \end{array}$$

$$\bar{x}={1\over n}\displaystyle\sum_{i=1}^nx_i=2.5, \bar{y}={1\over n}\displaystyle\sum_{i=1}^ny_i=52.5$$

$$S_{xx}=\displaystyle\sum_{i=1}^nx_i^2 -{1\over n}(\displaystyle\sum_{i=1}^nx_i)^2=17.5$$

$$S_{yy}=\displaystyle\sum_{i=1}^ny_i^2 -{1\over n}(\displaystyle\sum_{i=1}^ny_i)^2=437.5$$

$$S_{xy}=\displaystyle\sum_{i=1}^nx_iy_i -{1\over n}(\displaystyle\sum_{i=1}^nx_i)(\displaystyle\sum_{i=1}^ny_i)=87.5$$

$$m={S_{xy}\over S_{xx}}={87.5\over 17.5}=5$$

$$n=\bar{y}-m\cdot\bar{x}=52.5-5\times2.5=40$$

Therefore, we find that the regression equation is:

$$y=40+5x$$

Find out the company’s sales in 1999

$$y=40+5\times6=70(lakhs\ of\ rupees)$$

The company’s sales in 1999 will be 70 lakhs of rupees.