| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2021 |
| Session | January |
| Marks | 7 |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Reconstruct missing ranks |
| Difficulty | Challenging +1.2 This question requires systematic application of Spearman's formula with algebraic manipulation to find missing ranks, which is more challenging than routine calculation. However, it's a standard Further Maths Statistics exercise with clear structure: students know to use r_s = 1 - 6Σd²/(n³-n), calculate known d² values, solve for the remaining positions, and verify. Part (b) requires conceptual understanding but is straightforward. The multi-step nature and need to check permutations elevates it above average difficulty, but it remains a textbook-style problem without requiring novel insight. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Athlete | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) |
| Position in 100m sprint | 4 | 6 | 7 | 9 | 2 | 8 | 3 | 1 | 5 |
| Position in long jump | 5 | 4 | 9 | 3 | 1 | 2 |
Nine athletes, $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ and $I$, competed in both the 100m sprint and the long jump. After the two events the positions of each athlete were recorded and Spearman's rank correlation coefficient was calculated and found to be 0.85
The piece of paper the positions were recorded on was mislaid. Although some of the athletes agreed their positions, there was some disagreement between athletes $B$, $C$ and $D$ over their long jump results.
The table shows the results that are agreed to be correct.
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}
\hline
Athlete & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ \\
\hline
Position in 100m sprint & 4 & 6 & 7 & 9 & 2 & 8 & 3 & 1 & 5 \\
\hline
Position in long jump & 5 & & & & 4 & 9 & 3 & 1 & 2 \\
\hline
\end{tabular}
\end{center}
Given that there were no tied ranks,
\begin{enumerate}[label=(\alph*)]
\item find the correct positions of athletes $B$, $C$ and $D$ in the long jump. You must show your working clearly and give reasons for your answers. [5]
\item Without recalculating the coefficient, explain how Spearman's rank correlation coefficient would change if athlete $H$ was disqualified from both the 100m sprint and the long jump. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Statistics 2021 Q7 [7]}}