The yearly sales (in lakh Rs.) of a shop are given below.
| Year | 2020 | 2021 | 2022 | 2023 | 2024 |
| Sales (in lakh Rs.) | 10 | 14 | 15 | 19 | 22 |
The straight-line trend fitted by the method of least squares is \(y_t = 16 + 2.9X\), where X = year − 2022. Which of the following statements are correct?
(A) The trend value for 2020 is 10.2 lakh Rs.
(B) The deviation \(y - y_t\) for 2022 is 1.
(C) The sum of the deviations \(y - y_t\) for the five years is 0.
(D) The trend value for 2024 is 22 lakh Rs.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (C) only
The trend values, with X = −2, −1, 0, 1, 2, are
\(2020: 16 + 2.9 \times (-2) = 10.2\)
\(2021: 16 + 2.9 \times (-1) = 13.1\)
\(2022: 16 + 2.9 \times 0 = 16\)
\(2023: 16 + 2.9 \times 1 = 18.9\)
\(2024: 16 + 2.9 \times 2 = 21.8\)
The deviations \(y - y_t\) are
\(10 - 10.2 = -0.2,\quad 14 - 13.1 = 0.9,\quad 15 - 16 = -1\)
\(19 - 18.9 = 0.1,\quad 22 - 21.8 = 0.2\)
\(-0.2 + 0.9 - 1 + 0.1 + 0.2 = 0\)
For a least squares line the sum of the deviations is always 0.
So the trend values are 10.2, 13.1, 16, 18.9, 21.8, the deviation for 2022 is −1, and the deviations add up to 0.
Checking the statements
(A) The trend value for 2020 is 10.2 lakh Rs. — Correct. 16 − 5.8 = 10.2.
(B) The deviation \(y - y_t\) for 2022 is 1. — Incorrect. It is 15 − 16 = −1.
(C) The sum of the deviations \(y - y_t\) for the five years is 0. — Correct. −0.2 + 0.9 − 1 + 0.1 + 0.2 = 0.
(D) The trend value for 2024 is 22 lakh Rs. — Incorrect. It is 16 + 5.8 = 21.8; 22 is the actual sales.