| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Handle tied ranks |
| Difficulty | Standard +0.3 This is a straightforward application of Spearman's rank correlation coefficient with standard hypothesis testing. Part (a) requires ranking data and applying the formula (routine calculation), part (b) is standard hypothesis test procedure with critical value lookup, and part (c) asks for recall of how to handle ties. No novel problem-solving or complex reasoning required—slightly easier than average due to small dataset and direct application of learned procedures. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Club | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) |
| Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Average | 37 | 38 | 19 | 27 | 34 | 26 | 22 | 32 |
| Answer | Marks | Guidance |
|---|---|---|
| Attendance ranks 2, 1, 8, 5, 3, 6, 7, 4 | B1 | |
| \(\sum d^2 = 48\) | M1 A1 | Attempt to find \(\sum d^2\) |
| \(r_s = 1 - \frac{6 \times 48}{8 \times 63}\) | M1 | Substitution of their \(\sum d^2\) |
| \(= 0.4286\) | A1 ft | awrt 0.429 |
| (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0 : \rho = 0; H_1 : \rho \neq 0\) | B1 | both |
| With \(n=8\), critical value is 0.7381 | B1 | |
| Since 0.429 is not in the critical region (\(\rho < -0.7381\) or \(\rho > 0.7381\)) then there is no evidence to reject \(H_0\), and it can be concluded that at the 5% level there is no evidence of correlation between league position and attendance | M1 | Correct comparison |
| A1 ft | Conclusion | |
| (4 marks) |
| Answer | Marks |
|---|---|
| Share ranks evenly. | B1 |
| Use product moment correlation coefficient on ranks. | B1 |
| (2 marks) |
## (a)
Attendance ranks 2, 1, 8, 5, 3, 6, 7, 4 | B1 |
$\sum d^2 = 48$ | M1 A1 | Attempt to find $\sum d^2$
$r_s = 1 - \frac{6 \times 48}{8 \times 63}$ | M1 | Substitution of their $\sum d^2$
$= 0.4286$ | A1 ft | awrt 0.429
| (5 marks) |
## (b)
$H_0 : \rho = 0; H_1 : \rho \neq 0$ | B1 | both
With $n=8$, critical value is 0.7381 | B1 |
Since 0.429 is not in the critical region ($\rho < -0.7381$ or $\rho > 0.7381$) then there is no evidence to reject $H_0$, and it can be concluded that at the 5% level there is no evidence of correlation between league position and attendance | M1 | Correct comparison
| A1 ft | Conclusion
| (4 marks) |
## (c)
Share ranks evenly. | B1 |
Use product moment correlation coefficient on ranks. | B1 |
| (2 marks) |At the end of a season a league of eight ice hockey clubs produced the following table showing the position of each club in the league and the average attendances (in hundreds) at home matches.
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
Club & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ \\
\hline
Position & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
Average & 37 & 38 & 19 & 27 & 34 & 26 & 22 & 32 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate the Spearman rank correlation coefficient between position in the league and average home attendance. [5]
\item Stating clearly your hypotheses and using a 5\% two-tailed test, interpret your rank correlation coefficient. [4]
\end{enumerate}
Many sets of data include tied ranks.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Explain briefly how tied ranks can be dealt with. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q4 [11]}}