1. **State the problem:** We are given two sets of marks for Statistics I and Statistics II and need to calculate the Spearman's rank correlation coefficient to measure the strength and direction of the association between the two variables. 2. **Formula:** The Spearman's rank correlation coefficient $r_s$ is given by: $$r_s = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}$$ where $d_i$ is the difference between the ranks of each pair and $n$ is the number of pairs. 3. **Assign ranks:** - Statistics I marks: 80, 60, 65, 50, 35, 30, 90 - Statistics II marks: 80, 50, 60, 55, 45, 30, 95 Ranks for Statistics I (highest mark gets rank 1): - 90 (rank 1), 80 (rank 2), 65 (rank 3), 60 (rank 4), 50 (rank 5), 35 (rank 6), 30 (rank 7) Ranks for Statistics II: - 95 (rank 1), 80 (rank 2), 60 (rank 3), 55 (rank 4), 50 (rank 5), 45 (rank 6), 30 (rank 7) 4. **Calculate differences $d_i$ and $d_i^2$:** | Student | Stat I Rank | Stat II Rank | $d_i$ | $d_i^2$ | |---------|-------------|--------------|-------|---------| | 1 | 2 | 2 | 0 | 0 | | 2 | 4 | 5 | -1 | 1 | | 3 | 3 | 3 | 0 | 0 | | 4 | 5 | 4 | 1 | 1 | | 5 | 6 | 6 | 0 | 0 | | 6 | 7 | 7 | 0 | 0 | | 7 | 1 | 1 | 0 | 0 | Sum of $d_i^2 = 1 + 1 = 2$ 5. **Calculate $r_s$:** $$r_s = 1 - \frac{6 \times 2}{7(7^2 - 1)} = 1 - \frac{12}{7(49 - 1)} = 1 - \frac{12}{7 \times 48} = 1 - \frac{12}{336} = 1 - 0.0357 = 0.9643$$ 6. **Interpretation:** The Spearman's rank correlation coefficient is approximately $0.964$, which indicates a very strong positive correlation between the marks in Statistics I and Statistics II. This means students who scored high in Statistics I also tended to score high in Statistics II.