A straight-line trend is fitted by the method of least squares to yearly data, taking the middle of the period as the origin. Match List-I with List-II
| List-I (Number of years and choice of X) | List-II (Value of \(\sum X^2\)) | ||
| (A) | 3 years, X = year − middle year | (I) | 28 |
| (B) | 5 years, X = year − middle year | (II) | 2 |
| (C) | 7 years, X = year − middle year | (III) | 70 |
| (D) | 6 years, \( \displaystyle X = \frac{\text{year} - \text{midpoint of the two middle years}}{0.5}\) | (IV) | 10 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(II), (B)-(IV), (C)-(I), (D)-(III)
With the middle of the period as origin, the X values lie symmetrically about 0, and \(\sum X^2\) is the sum of their squares.
(A) 3 years, X = year − middle year
This matches (II) 2. Here X = −1, 0, 1.
\(\sum X^2 = 1 + 0 + 1 = 2\)
(B) 5 years, X = year − middle year
This matches (IV) 10. Here X = −2, −1, 0, 1, 2.
\(\sum X^2 = 4 + 1 + 0 + 1 + 4 = 10\)
(C) 7 years, X = year − middle year
This matches (I) 28. Here X = −3 to 3.
\(\sum X^2 = 9 + 4 + 1 + 0 + 1 + 4 + 9 = 28\)
(D) 6 years, X = (year − midpoint of the two middle years) ÷ 0.5
This matches (III) 70. Here X = −5, −3, −1, 1, 3, 5.
\(\sum X^2 = 25 + 9 + 1 + 1 + 9 + 25 = 70\)