Consider the following data Year (x) 2014 2016 2018 2020 2022 2024 Pr…

The correct answer is Option (3) → $35y = 26(x-2019) + 385$

Given

Years $x$: 2014, 2016, 2018, 2020, 2022, 2024

Profit $y$ (in Rs. thousand): 7, 9, 10, 12, 14, 14

Use least squares method for straight line trend: $y = a + b(x - \bar{x})$

Let $x' = x - 2019$ (midpoint of years), then $x' = -5, -3, -1, 1, 3, 5$

Corresponding $y = 7, 9, 10, 12, 14, 14$

Compute $b = \frac{\sum x'y}{\sum x'^2}$:

$\sum x'y = (-5*7) + (-3*9) + (-1*10) + (1*12) + (3*14) + (5*14) = -35 -27 -10 +12 +42 +70 = 52$

$\sum x'^2 = (-5)^2 + (-3)^2 + (-1)^2 + 1^2 +3^2 +5^2 = 25 +9 +1 +1 +9 +25 =70$

$b = \frac{52}{70} = \frac{26}{35}$

Compute $a = \bar{y} = \frac{7+9+10+12+14+14}{6} = \frac{66}{6} = 11$

Straight line trend equation: $y = 11 + \frac{26}{35}(x - 2019)$

Multiply both sides by 35: $35y = 35*11 + 26(x-2019) = 385 + 26(x-2019)$

Answer

$35y = 26(x - 2019) + 385$