| Year (t) | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 |
| Sales (in Rs. crores) (y) | 6 | 8 | 9 | 11 | 13 | 17 | 20 |
In reference to the above data, which of the following statements are correct? (where \(x = t - 2013\))
(A) If the equation of the straight line trend is \(y = 12 + 2.29x\), then the trend value for the year 2017 is 21.16.
(B) If the equation of the straight line trend is \(y = 12 + 2.29x\), then the trend value for the year 2013 is 11.
(C) If the equation of the straight line trend is \(y = 12 + 2.29x\), then the trend value for the year 2015 is 17.
(D) If \((t_1, y_1), (t_2, y_2), (t_3, y_3) \ldots\ldots (t_n, y_n)\) denote the time series and \(y_t\) are the trend values of the variable y, then \(\sum(y - y_t) = 0\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (D) only
With \(x = t - 2013\), the year 2013 is x = 0 and each later year adds 1 to x. A trend value is read from the trend line, not from the data table.
(A) If the equation of the straight line trend is \(y = 12 + 2.29x\), then the trend value for the year 2017 is 21.16. — Correct. For 2017, x = 4:
\(y = 12 + 2.29 \times 4 = 12 + 9.16 = 21.16\)
(B) If the equation of the straight line trend is \(y = 12 + 2.29x\), then the trend value for the year 2013 is 11. — Incorrect. For 2013, x = 0, so the trend value is \(y = 12\). The figure 11 is the actual sales of 2013.
(C) If the equation of the straight line trend is \(y = 12 + 2.29x\), then the trend value for the year 2015 is 17. — Incorrect. For 2015, x = 2:
\(y = 12 + 2.29 \times 2 = 16.58 \ne 17\)
The figure 17 is the actual sales of 2015.
(D) If \((t_1, y_1), (t_2, y_2), (t_3, y_3) \ldots\ldots (t_n, y_n)\) denote the time series and \(y_t\) are the trend values of the variable y, then \(\sum(y - y_t) = 0\). — Correct. For a trend line fitted by the method of least squares, the deviations of the actual values from the trend values add up to zero. Check with these data:
\(\sum y = 6 + 8 + 9 + 11 + 13 + 17 + 20 = 84\)
\(\sum x = 0\), so \(\sum y_t = 7 \times 12 + 2.29 \times 0 = 84\)
\(\sum(y - y_t) = 84 - 84 = 0\)