Answer: 0.7143

The rank correlation coefficient (Spearman's rank correlation) is calculated using the formula: $$\hat{\rho}_s = 1 - \frac{6 \sum_{i=1}^{n} (R_{X_i} - R_{Y_i})^2}{n(n^2 - 1)}$$ First, we need to rank the X and Y values separately: **Ranking X values (ascending order):** - -0.6 (smallest) → Rank 6 - -0.5 → Rank 5 - 0.5 → Rank 4 - 1.3 → Rank 3 - 3.1 → Rank 2 - 4.3 (largest) → Rank 1 **Ranking Y values (ascending order):** - -5.4 (smallest) → Rank 6 - -2.3 → Rank 5 - 2.1 → Rank 4 - 2.5 → Rank 3 - 3.0 → Rank 2 - 6.5 (largest) → Rank 1 Now we calculate the squared differences between ranks: | i | X | Y | Rₓ | Rᵧ | (Rₓ − Rᵧ)² | |---|-----|------|----|----|------------| | 1 | 0.5 | 2.5 | 4 | 3 | 1 | | 2 | 1.3 | 6.5 | 3 | 1 | 4 | | 3 | -0.5| -2.3 | 5 | 5 | 0 | | 4 | -0.6| -5.4 | 6 | 6 | 0 | | 5 | 4.3 | 3.0 | 1 | 2 | 1 | | 6 | 3.1 | 2.1 | 2 | 4 | 4 | | | | | | Sum| 10 | Now plug into the formula: $$\hat{\rho}_s = 1 - \frac{6 \times 10}{6(6^2 - 1)} = 1 - \frac{60}{6 \times 35} = 1 - \frac{60}{210} = 1 - 0.2857 = 0.7143$$ Therefore, the rank correlation coefficient is 0.7143, which corresponds to option D.

Author: Nikitesh Somanthe