Assuming the same rate of change continues for the following data:
| Year (x) | 2019 | 2020 | 2021 | 2022 | 2023 |
| Profit (in Percentage) (y) | 38 | 40 | 65 | 72 | 69 |
The equation of the straight line trend using the least square method is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(y=9.4x-18940.6\)
The years are consecutive and the middle year is 2021, so shift the origin to 2021: let \(u = x - 2021\). Then u = −2, −1, 0, 1, 2 and \(\sum u = 0\).
\(\sum y = 38 + 40 + 65 + 72 + 69 = 284\)
\(\sum uy = -76 - 40 + 0 + 72 + 138 = 94\)
\(\sum u^2 = 4 + 1 + 0 + 1 + 4 = 10\)
For the trend \(y = a + bu\), the least squares normal equations with \(\sum u = 0\) give
\( \displaystyle a = \frac{\sum y}{n} = \frac{284}{5} = 56.8 \)
\( \displaystyle b = \frac{\sum uy}{\sum u^2} = \frac{94}{10} = 9.4 \)
So \(y = 56.8 + 9.4u\). Going back to the years, \(u = x - 2021\):
\(y = 56.8 + 9.4(x - 2021)\)
\(= 9.4x - 18997.4 + 56.8\)
\(= 9.4x - 18940.6\)
Why the other options are incorrect
Option (1) \(y=9.4x+ 18940.6\) — Incorrect. The constant is \(56.8 - 9.4 \times 2021 = -18940.6\), which is negative.
Option (3) \(y=9.4x+ 56.8\) — Incorrect. This is the trend \(y = 56.8 + 9.4u\) with the origin at 2021, written with x in place of u; for x in years it gives absurd values.
Option (4) \(y=9.4x- 56.8\) — Incorrect. At x = 2021 the trend value must be the mean 56.8, but this gives \(18997.4 - 56.8 = 18940.6\).