The yearly profit (in lakh Rs.) of a company is given below.
| Year | 2020 | 2021 | 2022 | 2023 | 2024 |
| Profit (in lakh Rs.) | 16 | 21 | 23 | 28 | 32 |
A straight-line trend \(y = a + bX\), where X = year − 2022, is fitted to these data by the method of least squares. Which of the following statements are correct?
(A) \(a = 24\)
(B) The trend value of the profit rises by 7.8 lakh Rs. every year.
(C) The trend value of the profit for the year 2024 is 31.8 lakh Rs.
(D) The 3-year moving average of the profit for the year 2022 is 23 lakh Rs.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
With the middle year 2022 as origin, X = −2, −1, 0, 1, 2 and \(\sum X = 0\), so
\( \displaystyle a = \frac{\sum y}{n}, \quad b = \frac{\sum Xy}{\sum X^2} \)
The sums are
\(\sum y = 16 + 21 + 23 + 28 + 32 = 120\)
\(\sum Xy = -32 - 21 + 0 + 28 + 64 = 39\)
\(\sum X^2 = 4 + 1 + 0 + 1 + 4 = 10\)
\( \displaystyle a = \frac{120}{5} = 24 \)
\( \displaystyle b = \frac{39}{10} = 3.9 \)
For 2024, X = 2:
\(y = 24 + 3.9 \times 2 = 31.8\)
The 3-year moving average for 2022 is the mean of 2021, 2022 and 2023:
\( \displaystyle \frac{21 + 23 + 28}{3} = \frac{72}{3} = 24 \)
So a = 24, the trend rises by b = 3.9 every year, the trend value for 2024 is 31.8, and the 3-year moving average for 2022 is 24.
Checking the statements
(A) \(a = 24\) — Correct. a = 120 ÷ 5 = 24.
(B) The trend value of the profit rises by 7.8 lakh Rs. every year. — Incorrect. It rises by b = 3.9 each year; 7.8 is the rise over two years.
(C) The trend value of the profit for the year 2024 is 31.8 lakh Rs. — Correct. 24 + 3.9 × 2 = 31.8.
(D) The 3-year moving average of the profit for the year 2022 is 23 lakh Rs. — Incorrect. It is 72 ÷ 3 = 24; 23 is the actual profit of 2022.