For the following data:
| Year \( (x_i) \) | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 |
| Sales (in Rs. ’000) \( (y_i) \) | 26 | 26 | 44 | 42 | 108 | 120 | 166 |
The equation of the straight line trend is \( y=a+b(x_i-2007) \), then the trend value for the year 2011 is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 172
Let \( u=x-2007 \), so u = −3, −2, −1, 0, 1, 2, 3 and \( \sum u=0 \). The normal equations then give
\( \displaystyle a=\frac{\sum y}{n},\qquad b=\frac{\sum uy}{\sum u^2} \)
\( \sum y=26+26+44+42+108+120+166=532 \)
\( \sum uy=-78-52-44+0+108+240+498=672 \)
\( \sum u^2=9+4+1+0+1+4+9=28 \)
\( \displaystyle a=\frac{532}{7}=76 \)
\( \displaystyle b=\frac{672}{28}=24 \)
For 2011, u = 2011 − 2007 = 4, so the trend value is
\( \displaystyle y=76+24\times 4=172 \)