| Exam Board | Edexcel |
|---|---|
| Module | FS2 (Further Statistics 2) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Reconstruct missing ranks |
| Difficulty | Challenging +1.8 This question requires working backwards from a given correlation coefficient to reconstruct missing data—a non-standard problem requiring algebraic manipulation of the Spearman's formula and systematic reasoning about constraints. Part (a) is routine hypothesis testing, but part (b) demands problem-solving beyond typical textbook exercises, involving solving for three unknowns given one equation and logical constraints about permutations of ranks. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Athlete | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) |
| Position in 100 m sprint | 4 | 6 | 7 | 9 | 2 | 8 | 3 | 1 | 5 |
| Position in long jump | 5 | 4 | 9 | 3 | 1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \rho_s = 0 \quad H_1: \rho_s > 0\) | B1 | Both hypotheses correct written using the notation \(\rho\) |
| \(CV = 0.6\) | B1 | awrt 0.6 |
| \(r_s = 0.85\) does lie in the critical region | M1 | Drawing a correct inference using their CV and the value of \(r_s\) |
| There is evidence to suggest that there is a relationship between the position in the 100m sprint and the position in the long jump | A1 | Drawing a correct inference in context using their CV and the value of \(r_s\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(1 - \dfrac{6\sum d^2}{9(80)} = 0.85\) | M1 | For realising they need to equate this expression to 0.85 to find \(\sum d^2\) |
| \(\sum d^2 = 18\) | A1 | 18 |
| \(\sum d^2\) needed is \(18 - 15 = 3\) | M1 | For \(\sum d^2 = 3\) |
| Since \(\sum d^2 = 3\) for the 3 missing places each place must contribute 1, therefore \(B\) must be in position 5 or 7. However, 5 has already been used so they must be position 7 | A1 | For using the information with \(\sum d^2\) to deduce each must contribute 1 and explain why \(B\) must be position 7 |
| \(C\) is \(6^{\text{th}}\) and \(D\) is \(8^{\text{th}}\) | A1 | \(C\ 6^{\text{th}}\), \(D\ 8^{\text{th}}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The \(\sum d^2\) will not change but the value of \(n\) will decrease therefore | M1 | Complete explanation why it decreases |
| Spearman's rank correlation will decrease | A1 | Using the information given to deduce that it decreases |
## Question 8:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \rho_s = 0 \quad H_1: \rho_s > 0$ | B1 | Both hypotheses correct written using the notation $\rho$ |
| $CV = 0.6$ | B1 | awrt 0.6 |
| $r_s = 0.85$ does lie in the critical region | M1 | Drawing a correct inference using their CV and the value of $r_s$ |
| There is evidence to suggest that there is a relationship between the position in the 100m sprint and the position in the long jump | A1 | Drawing a correct inference in context using their CV and the value of $r_s$ |
**Total: (4)**
---
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $1 - \dfrac{6\sum d^2}{9(80)} = 0.85$ | M1 | For realising they need to equate this expression to 0.85 to find $\sum d^2$ |
| $\sum d^2 = 18$ | A1 | 18 |
| $\sum d^2$ needed is $18 - 15 = 3$ | M1 | For $\sum d^2 = 3$ |
| Since $\sum d^2 = 3$ for the 3 missing places each place must contribute 1, therefore $B$ must be in position 5 or 7. However, 5 has already been used so they must be position 7 | A1 | For using the information with $\sum d^2$ to deduce each must contribute 1 and explain why $B$ must be position 7 |
| $C$ is $6^{\text{th}}$ and $D$ is $8^{\text{th}}$ | A1 | $C\ 6^{\text{th}}$, $D\ 8^{\text{th}}$ |
**SC:** B7, C6, D8 with no reasons B1 marks as final A1 on open
**Total: (5)**
---
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The $\sum d^2$ will not change but the value of $n$ will decrease therefore | M1 | Complete explanation why it decreases |
| Spearman's rank correlation will decrease | A1 | Using the information given to deduce that it decreases |
**Total: (2)**
**Overall Total: (11 marks)**8 Nine athletes, $A , B , C , D , E , F , G , H$ and $I$, competed in both the 100 m 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
\begin{enumerate}[label=(\alph*)]
\item Stating your hypotheses clearly, test whether or not there is evidence to suggest that the higher an athlete's position is in the 100 m sprint, the higher their position is in the long jump. Use a $5 \%$ level of significance.
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 100 m 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,
\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.
\item Without recalculating the coefficient, explain how Spearman's rank correlation coefficient would change if athlete $H$ was disqualified from both the 100 m sprint and the long jump.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FS2 2019 Q8 [11]}}