Consider the following data
| Year (x) | 2014 | 2016 | 2018 | 2020 | 2022 | 2024 |
| Profit (in Rs. Thousand) (y) | 7 | 9 | 10 | 12 | 14 | 14 |
Then for the above data the equation of straight line trend by method of least square is given by:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(35y = 26(x - 2019) + 385\)
There are 6 years (an even number), with middle point 2019. Take the origin there: let \(X = x - 2019\), so that \(\sum X = 0\).
\(X = -5, -3, -1, 1, 3, 5\)
\(y = 7, 9, 10, 12, 14, 14\)
For the trend line \(y = a + bX\), the normal equations of least squares with \(\sum X = 0\) give
\( \displaystyle a = \frac{\sum y}{n} \)
\( \displaystyle b = \frac{\sum Xy}{\sum X^2} \)
The sums:
\(\sum y = 7 + 9 + 10 + 12 + 14 + 14 = 66\)
\(\sum Xy = -35 - 27 - 10 + 12 + 42 + 70 = 52\)
\(\sum X^2 = 25 + 9 + 1 + 1 + 9 + 25 = 70\)
\( \displaystyle a = \frac{66}{6} = 11 \)
\( \displaystyle b = \frac{52}{70} = \frac{26}{35} \)
So the trend line is
\( \displaystyle y = 11 + \frac{26}{35}(x - 2019) \)
Multiplying by 35:
\(35y = 26(x - 2019) + 385\)