If for the following data:
| Year \( (x_i) \) | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 |
| Sales (in Rs. crores) \( (y_i) \) | 6 | 8 | 9 | 11 | 13 | 17 | 20 |
If the equation of the straight line trend is \( y=a+b(x_i-2013) \), then the trend value for the year 2018 is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 23.45
Let u = x − 2013, so u = −3, −2, −1, 0, 1, 2, 3 and \( \sum u=0 \). The normal equations then give
\( \displaystyle a=\frac{\sum y}{n} \)
\( \displaystyle b=\frac{\sum uy}{\sum u^2} \)
From the table:
\( \sum y=6+8+9+11+13+17+20=84 \)
\( \sum uy=-18-16-9+0+13+34+60=64 \)
\( \sum u^2=9+4+1+0+1+4+9=28 \)
\( \displaystyle a=\frac{84}{7}=12 \)
\( \displaystyle b=\frac{64}{28}\approx 2.29 \)
For 2018, u = 2018 − 2013 = 5:
\( y=12+2.29\times 5 \)
\( =12+11.45 \)
\( =23.45 \)
Here b is taken to two decimal places, as is usual. The unrounded value \( b=\frac{16}{7} \) gives 23.43, which also points to Option (3).