IB Mathematics SL 4.10 Spearman’s rank correlation coefficient - AI HL Paper 1- Exam Style Questions- New Syllabus

(a)
By inspecting the table, Aris ranks sheepdog B at position 4.

(b)
Organize the ranks for each dog (A to H) :
Aris: \(1, 4, 2, 3, 5, 6, 7, 8\)
Beatrix: \(1, 2, 4, 3, 5, 7, 6, 8\)

Calculate the differences \(d\) and their squares \(d^2\) :
A: \(d = 1-1=0 \rightarrow d^2 = 0\)
B: \(d = 4-2=2 \rightarrow d^2 = 4\)
C: \(d = 2-4=-2 \rightarrow d^2 = 4\)
D: \(d = 3-3=0 \rightarrow d^2 = 0\)
E: \(d = 5-5=0 \rightarrow d^2 = 0\)
F: \(d = 6-7=-1 \rightarrow d^2 = 1\)
G: \(d = 7-6=1 \rightarrow d^2 = 1\)
H: \(d = 8-8=0 \rightarrow d^2 = 0\)
\(\sum d^2 = 10\)

Using the Spearman’s formula \(r_s = 1 – \frac{6 \sum d^2}{n(n^2 – 1)}\) with \(n=8\) :
\(r_s = 1 – \frac{6(10)}{8(63)} = 1 – \frac{60}{504}\)
\(r_s \approx 0.881\) 

(c)
With an \(r_s\) value of approximately \(0.881\), there is a strong positive correlation. This indicates that Aris and Beatrix are in significant agreement regarding the rankings of the sheepdogs