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 $\frac{a}{b}$ is:

Given the trend line equation $y = a + b(x - 2022)$ with the data:

Year ($x$)	2020	2021	2022	2023	2024
Profit ($y$)	2	3	4	5	2

Let $u = x - 2022$ to simplify calculations.

The equation becomes $y = a + bu$

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Converting $x$-values to $u$-values:

When $x = 2020$: $u = 2020 - 2022 = -2$

When $x = 2021$: $u = 2021 - 2022 = -1$

When $x = 2022$: $u = 2022 - 2022 = 0$

When $x = 2023$: $u = 2023 - 2022 = 1$

When $x = 2024$: $u = 2024 - 2022 = 2$

The transformed data:

$u$	-2	-1	0	1	2
$y$	2	3	4	5	2

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Number of data points: $n = 5$

$\sum u = -2 + (-1) + 0 + 1 + 2$

$\sum u = 0$

$\bar{u} = \frac{0}{5} = 0$

$\sum y = 2 + 3 + 4 + 5 + 2$

$\sum y = 16$

$\bar{y} = \frac{16}{5} = 3.2$

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$\sum u^2 = (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2$

$\sum u^2 = 4 + 1 + 0 + 1 + 4$

$\sum u^2 = 10$

$\sum uy = (-2)(2) + (-1)(3) + (0)(4) + (1)(5) + (2)(2)$

$\sum uy = -4 - 3 + 0 + 5 + 4$

$\sum uy = 2$

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Since $\bar{u} = 0$:

$a = \bar{y} = \frac{16}{5}$

Using the least squares formula:

$b = \frac{\sum uy}{\sum u^2}$

$b = \frac{2}{10}$

$b = \frac{1}{5}$

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$\frac{a}{b} = \frac{\frac{16}{5}}{\frac{1}{5}}$

$\frac{a}{b} = \frac{16}{5} \times \frac{5}{1}$

$\frac{a}{b} = 16$

Therefore, the value of $\frac{a}{b} = 16$