Match List - I with List - II. List - I — Total sum of squares (TSS) and…
2024
Match List - I with List - II.
List - I — Total sum of squares (TSS) and Residual sum of squares | List - II — R2 of Regression Model |
|---|---|
(A) RSS = 5, TSS = 20 | (I) 0.62 |
(B) RSS = 5, TSS = 25 | (II) 0.75 |
(C) RSS = 10, TSS = 25 | (III) 0.8 |
(D) RSS = 12, TSS = 32 | (IV) 0.6 |
Choose the correct answer from the options given below :
Answer: B. (A)-(II), (B)-(III), (C)-(IV), (D)-(I) — Concept: In a fitted regression the total variation of the dependent variable splits into an explained part and an unexplained part, so TSS = ESS + RSS, where…
- A.
(A)-(I), (B)-(II), (C)-(III), (D)-(IV)
- B.
(A)-(II), (B)-(III), (C)-(IV), (D)-(I)
- C.
(A)-(III), (B)-(IV), (C)-(I), (D)-(II)
- D.
(A)-(IV), (B)-(I), (C)-(II), (D)-(III)
Show answer & explanation
Correct answer: B
Concept: In a fitted regression the total variation of the dependent variable splits into an explained part and an unexplained part, so TSS = ESS + RSS, where TSS is the total sum of squares, ESS the explained sum of squares and RSS the residual sum of squares. The coefficient of determination measures the share of total variation the model explains, so R2 = ESS/TSS = 1 − RSS/TSS. It is a pure ratio with no units, and it grows as the residual sum of squares becomes small relative to the total sum of squares.
Application: Apply R2 = 1 − RSS/TSS to each model listed in List - I.
(A) RSS = 5, TSS = 20 gives R2 = 1 − 5/20 = 1 − 0.25 = 0.75, the List - II entry (II).
(B) RSS = 5, TSS = 25 gives R2 = 1 − 5/25 = 1 − 0.20 = 0.80, the List - II entry (III).
(C) RSS = 10, TSS = 25 gives R2 = 1 − 10/25 = 1 − 0.40 = 0.60, the List - II entry (IV).
(D) RSS = 12, TSS = 32 gives R2 = 1 − 12/32 = 1 − 0.375 = 0.625, which the paper prints as the List - II entry (I) 0.62, that is the exact value carried to two decimal places without rounding the final 5 upward.
Cross-check: Recompute each value from the other side of the identity, as the explained share ESS/TSS with ESS = TSS − RSS.
Model (List - I) | ESS = TSS − RSS | ESS/TSS | List - II entry |
|---|---|---|---|
(A) RSS = 5, TSS = 20 | 20 − 5 = 15 | 15/20 = 0.75 | (II) 0.75 |
(B) RSS = 5, TSS = 25 | 25 − 5 = 20 | 20/25 = 0.80 | (III) 0.8 |
(C) RSS = 10, TSS = 25 | 25 − 10 = 15 | 15/25 = 0.60 | (IV) 0.6 |
(D) RSS = 12, TSS = 32 | 32 − 12 = 20 | 20/32 = 0.625 | (I) 0.62 |
The values for (A), (B) and (C) reproduce their List - II entries exactly, and the one remaining entry 0.62 is the printed form of the 0.625 obtained for (D). The order is also consistent with the rule that a smaller residual share leaves a larger explained share: RSS/TSS of 0.20 for (B) is the smallest and yields the largest R2, while RSS/TSS of 0.40 for (C) is the largest and yields the smallest.
Result: The correct matching is (A)-(II), (B)-(III), (C)-(IV), (D)-(I).