If \(y = a + b(x - 2022)\) is a straight line trend using the least square method for the following data
| Year \((x)\) | 2020 | 2021 | 2022 | 2023 | 2024 |
| Profit (Rs. '000) \((y)\) | 2 | 3 | 4 | 5 | 2 |
Then the value of \(\displaystyle\frac{a}{b}\) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 16
Let \(t = x - 2022\), so the years 2020 to 2024 give t = −2, −1, 0, 1, 2 and \(\sum t = 0\). With the origin at the middle year, the normal equations of the least squares method reduce to:
\( \displaystyle a = \frac{\sum y}{n},\quad b = \frac{\sum ty}{\sum t^2} \)
The sums:
\(\sum y = 2 + 3 + 4 + 5 + 2 = 16\)
\(\sum ty = (-2)(2) + (-1)(3) + 0(4) + 1(5) + 2(2)\)
\(= -4 - 3 + 0 + 5 + 4 = 2\)
\(\sum t^2 = 4 + 1 + 0 + 1 + 4 = 10\)
Hence:
\( \displaystyle a = \frac{16}{5},\quad b = \frac{2}{10} = \frac{1}{5} \)
\( \displaystyle \frac{a}{b} = \frac{16/5}{1/5} = 16 \)