For different sets of datasets X and Y, the correlation coefficient between X…
2025
For different sets of datasets X and Y, the correlation coefficient between X and Y (denoted as R) and the standard deviation of X and Y (denoted as S(X) and S(Y)) are given in A–D below. Estimate the covariance between X and Y for each dataset and arrange them in ascending order.
A. R = 0.5, S(X) = 8, S(Y) = 10
B. R = 4/7, S(X) = 7, S(Y) = 9
C. R = 0.7, S(X) = 12, S(Y) = 5
D. R = 19/42, S(X) = 14, S(Y) = 6
Choose the correct answer from the options given below:
- A.
B, A, D, C
- B.
C, B, A, D
- C.
B, D, A, C
- D.
A, C, D, B
Show answer & explanation
Correct answer: C
Concept: Covariance between two variables is directly related to their correlation coefficient: since R = Cov(X, Y) ÷ (S(X) × S(Y)), rearranging gives Cov(X, Y) = R × S(X) × S(Y) — the correlation coefficient scaled by the product of the two standard deviations.
Application: Apply this formula to each dataset.
Dataset A: Cov(X, Y) = R × S(X) × S(Y) = 0.5 × 8 × 10 = 40.
Dataset B: Cov(X, Y) = (4/7) × 7 × 9 = 4 × 9 = 36.
Dataset C: Cov(X, Y) = 0.7 × 12 × 5 = 42.
Dataset D: Cov(X, Y) = (19/42) × 14 × 6 = 19 × 2 = 38.
Cross-check: Recomputing dataset D independently confirms the value: 19/42 simplifies against 14 to 19/3, and 19/3 × 6 = 19 × 2 = 38, matching the direct computation above.
Arranging the four covariance values in ascending order places the datasets in the sequence B, D, A, C.