If $y = a + b(x - 2022)$ is a straight line trend using the least square method for the following data

Then the value of $\frac{a}{b}$ is:

26

The correct answer is Option (3) → 26

Given trend equation

$y=a+b(x-2022)$

Let $t=x-2022$

Then for years $2020,2021,2022,2023,2024$

$t=-2,-1,0,1,2$

Corresponding $y$ values are $2,3,14,5,2$

Using least squares properties

$a=\frac{\sum y}{n}$ since $\sum t=0$

$\sum y=2+3+14+5+2=26$

$n=5$

$a=\frac{26}{5}$

$b=\frac{\sum ty}{\sum t^2}$

$\sum ty=(-2)(2)+(-1)(3)+(0)(14)+(1)(5)+(2)(2)$

$=-4-3+0+5+4=2$

$\sum t^2=4+1+0+1+4=10$

$b=\frac{2}{10}=\frac{1}{5}$

$\frac{a}{b}=\frac{\frac{26}{5}}{\frac{1}{5}}=26$

The value of $\frac{a}{b}$ is $26$.