A straight-line trend \(y = a + bX\), where X = year − 2022, is fitted by the method of least squares to the yearly profit (in lakh Rs.) of a company for the five years 2020 to 2024. For the data, \(\sum y = 110\), \(\sum Xy = 30\) and \(\sum X^2 = 10\). Which of the following statements are correct?
(A) \(a = 22\)
(B) The sum of the five trend values is 110.
(C) The trend value of the profit for the year 2025 is 30 lakh Rs.
(D) The trend value of the profit rises by 3 lakh Rs. every year.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
The origin 2022 is the middle year, so \(\sum X = 0\) and the normal equations give
\( \displaystyle a = \frac{\sum y}{n} = \frac{110}{5} = 22 \)
\( \displaystyle b = \frac{\sum Xy}{\sum X^2} = \frac{30}{10} = 3 \)
The trend line is \(y = 22 + 3X\). For 2025, X = 3:
\(y = 22 + 3 \times 3 = 31\)
For the least squares line \(\sum (y - y_t) = 0\), so the trend values add up to \(\sum y = 110\).
So a = 22, b = 3 (a rise of 3 lakh Rs. per year), the trend values total 110, and the trend value for 2025 is 31 lakh Rs.
Checking the statements
(A) \(a = 22\) — Correct. a = 110 ÷ 5 = 22.
(B) The sum of the five trend values is 110. — Correct. The trend values total the same as the actual values.
(C) The trend value of the profit for the year 2025 is 30 lakh Rs. — Incorrect. It is 22 + 3 × 3 = 31 lakh Rs.
(D) The trend value of the profit rises by 3 lakh Rs. every year. — Correct. b = 30 ÷ 10 = 3 per year.