Fit a straight-line trend by the method of least squares to the following data on a firm's profit (Rs '000) for 2018-2022, and estimate the trend value for each year.

| Year | 2018 | 2019 | 2020 | 2021 | 2022 |
|---|---|---|---|---|---|
| Profit | 10 | 12 | 18 | 22 | 28 |

Question

Fit a straight-line trend by the method of least squares to the following data on a firm's profit (Rs '000) for 2018-2022, and estimate the trend value for each year.

| Year | 2018 | 2019 | 2020 | 2021 | 2022 |
|---|---|---|---|---|---|
| Profit | 10 | 12 | 18 | 22 | 28 |

Accepted Answer

**Step 1 — set up the coding table** (origin at the middle year, 2020, since $n=5$ is odd):

| Year | Profit ($Y$) | $X$ | $XY$ | $X^2$ |
|---|---|---|---|---|
| 2018 | 10 | -2 | -20 | 4 |
| 2019 | 12 | -1 | -12 | 1 |
| 2020 | 18 | 0 | 0 | 0 |
| 2021 | 22 | 1 | 22 | 1 |
| 2022 | 28 | 2 | 56 | 4 |
| **Total** | $\Sigma Y = 90$ | $\Sigma X = 0$ | $\Sigma XY = 46$ | $\Sigma X^2 = 10$ |

**Step 2 — compute $a$ and $b$:**
$$a = \frac{\Sigma Y}{n} = \frac{90}{5} = 18 \qquad b = \frac{\Sigma XY}{\Sigma X^2} = \frac{46}{10} = 4.6$$

So the trend equation is $Y_c = 18 + 4.6X$, with origin 2020 and $X$ measured in years.

**Step 3 — trend values:**

| Year | $X$ | $Y_c = 18+4.6X$ |
|---|---|---|
| 2018 | -2 | $18-9.2=8.8$ |
| 2019 | -1 | $18-4.6=13.4$ |
| 2020 | 0 | $18.0$ |
| 2021 | 1 | $18+4.6=22.6$ |
| 2022 | 2 | $18+9.2=27.2$ | …

Year	Profit ( $Y$ )	$X$	$XY$	$X^2$
2018	10	-2	-20	4
2019	12	-1	-12	1
2020	18	0	0	0
2021	22	1	22	1
2022	28	2	56	4
Total	$\Sigma Y = 90$	$\Sigma X = 0$	$\Sigma XY = 46$	$\Sigma X^2 = 10$

Year	$X$	$Y_c = 18+4.6X$
2018	-2	$18-9.2=8.8$
2019	-1	$18-4.6=13.4$
2020	0	$18.0$
2021	1	$18+4.6=22.6$
2022	2	$18+9.2=27.2$

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