解题
To calculate Spearman's rank correlation coefficient, both variables must first be ranked independently, in the same direction (here, rank 1 given to the smallest value in each column, rising to rank 8 for the largest). Ranking distance from the CBD, since the values 50, 150, 300, 450, 600, 800, 1000, 1300 are already listed in increasing order, they are simply ranked 1 to 8 in order: 1, 2, 3, 4, 5, 6, 7, 8. Ranking pedestrian count (118, 102, 85, 90, 55, 40, 35, 15), the smallest value, 15, is ranked 1, the next smallest, 35, is ranked 2, then 40 is ranked 3, then 55 is ranked 4; the remaining four values need care because 85 and 90 are close together: ordering all eight values from smallest to largest gives 15, 35, 40, 55, 85, 90, 102, 118, so 85 (the 5th smallest) is ranked 5, 90 (the 6th smallest) is ranked 6, 102 is ranked 7, and 118 (the largest) is ranked 8; reading the count ranks back in survey-point order (points 1 to 8, counts 118, 102, 85, 90, 55, 40, 35, 15) therefore gives ranks 8, 7, 5, 6, 4, 3, 2, 1. Next, the rank difference d is calculated for each survey point by subtracting the count rank from the distance rank: point 1, 1 - 8 = -7; point 2, 2 - 7 = -5; point 3, 3 - 5 = -2; point 4, 4 - 6 = -2; point 5, 5 - 4 = 1; point 6, 6 - 3 = 3; point 7, 7 - 2 = 5; point 8, 8 - 1 = 7. Each d value is then squared (which makes all values positive and removes the effect of direction, since the formula only needs the total squared difference): (-7)^2 = 49, (-5)^2 = 25, (-2)^2 = 4, (-2)^2 = 4, (1)^2 = 1, (3)^2 = 9, (5)^2 = 25, (7)^2 = 49; summing these eight values, 49 + 25 + 4 + 4 + 1 + 9 + 25 + 49 = 166, so the sum of d^2 is 166. With n = 8 survey points, n^2 - 1 = 64 - 1 = 63, so n(n^2-1) = 8 x 63 = 504, and 6 x sum of d^2 = 6 x 166 = 996; substituting into the formula, rs = 1 - (996/504) = 1 - 1.9762 (to 4dp) = -0.9762, which rounds to -0.976 to 3 significant figures. Self-verifying this result by a second route: since every rank difference is small (the largest |d| is 7, out of a maximum possible span of 7 for n=8) and the pattern of ranks is almost perfectly reversed between the two variables (rank 1 pairs with rank 8, rank 2 with rank 7, and so on, with only points 3 and 4 swapped from a perfect reversal), the result should be very close to rs = -1 (a perfect negative correlation), which is exactly what the calculated value of -0.976 shows, confirming the arithmetic is consistent with the structure of the ranked data. Answer: rs = -0.976 (to 3 significant figures), indicating an almost perfect negative correlation between distance from the CBD and pedestrian count.
评分标准
Point/method marking, maximum [7]. [1] correct ranking of distance (1-8, in order given); [1] correct ranking of pedestrian count (8,7,5,6,4,3,2,1); [1] correct calculation of all 8 d values; [1] correct calculation of all 8 d^2 values; [1] correct sum of d^2 (166); [1] correct substitution into the formula (1 - (6x166)/(8x63)); [1] correct final answer, rs = -0.976 (accept -0.98 or -0.976 to -0.9762, allow follow-through marks if an earlier ranking error is carried through consistently and correctly to a final answer using the correct method). Total [7].