The straight-line trend \(y = 30 + 2.5X\), where X = year − 2021, is fitted by the method of least squares to the yearly sales \(y\) (in lakh Rs.) of a firm for the years 2018 to 2024. By how many lakh Rs. does the trend value of the sales for 2024 exceed the trend value for 2018?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 15
In \(y = 30 + 2.5X\) the trend value rises by b = 2.5 lakh Rs. for each year. The X values of the two years are
\(2018: X = 2018 - 2021 = -3\)
\(2024: X = 2024 - 2021 = 3\)
The trend values are
\(2018: y = 30 + 2.5 \times (-3) = 22.5\)
\(2024: y = 30 + 2.5 \times 3 = 37.5\)
So the difference is
\(37.5 - 22.5 = 15\)
From 2018 to 2024 there are 6 yearly steps, not 7, so the rise is \(6 \times 2.5 = 15\).
Why the other options are incorrect
Option (2) 17.5 — Incorrect. This counts 7 steps (the number of years) instead of the 6 steps between 2018 and 2024.
Option (3) 2.5 — Incorrect. This is the rise for one year only.
Option (4) 7.5 — Incorrect. This is the rise for 3 years, from 2021 to 2024 only.