| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 12 |
| 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 careful but routine calculation of rank differences and the correlation formula (6 marks reflects computational work, not conceptual difficulty). Part (b) is a standard one-tailed test comparing to critical values. Part (c) tests understanding of tied ranks, requiring only brief explanation. While requiring attention to detail in matching wines between rows, this is a textbook S3 exercise with no novel problem-solving or insight required—slightly easier than average due to its purely procedural nature. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Enthusiast | \(D\) | \(C\) | \(J\) | \(A\) | \(G\) | \(F\) | \(B\) | \(E\) | \(I\) | \(H\) |
| Price | \(A\) | \(C\) | \(D\) | \(H\) | \(J\) | \(B\) | \(F\) | \(I\) | \(G\) | \(E\) |
| Answer | Marks | Guidance |
|---|---|---|
| bottle | A | B |
| enth. rank | 4 | 7 |
| price rank | 1 | 6 |
| \(d^2\) | 9 | 1 |
| \(\sum d^2 = 76\) | M2 A2 | |
| \(r_s = 1 - \frac{6 \times 76}{10 \times 99} = 0.5394\) | M1 A1 | |
| Part (b): \(H_0: \rho = 0\); \(H_1: \rho > 0\) | B1 | |
| \(n = 10\), 5% level \(\therefore\) C.R. is \(r_s > 0.5636\) | M1 A1 | |
| \(0.5394 < 0.5636\) \(\therefore\) not significant; there is no evidence of positive correlation | A1 | |
| Part (c): share ranks, both 6.5, use pmcc | B2 | (12 marks total) |
**Part (a):**
| bottle | A | B | C | D | E | F | G | H | I | J |
|--------|---|---|---|---|---|---|---|---|---|---|
| enth. rank | 4 | 7 | 2 | 1 | 8 | 6 | 5 | 10 | 9 | 3 |
| price rank | 1 | 6 | 2 | 3 | 10 | 7 | 9 | 4 | 8 | 5 |
| $d^2$ | 9 | 1 | 0 | 4 | 4 | 1 | 16 | 36 | 1 | 4 |
$\sum d^2 = 76$ | M2 A2 |
$r_s = 1 - \frac{6 \times 76}{10 \times 99} = 0.5394$ | M1 A1 |
**Part (b):** $H_0: \rho = 0$; $H_1: \rho > 0$ | B1 |
$n = 10$, 5% level $\therefore$ C.R. is $r_s > 0.5636$ | M1 A1 |
$0.5394 < 0.5636$ $\therefore$ not significant; there is no evidence of positive correlation | A1 |
**Part (c):** share ranks, both 6.5, use pmcc | B2 | (12 marks total)
---In a competition, a wine-enthusiast has to rank ten bottles of wine, $A$ to $J$, in order starting with the one he thinks is the most expensive. The table below shows his rankings and the actual order according to price.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Rank & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Enthusiast & $D$ & $C$ & $J$ & $A$ & $G$ & $F$ & $B$ & $E$ & $I$ & $H$ \\
\hline
Price & $A$ & $C$ & $D$ & $H$ & $J$ & $B$ & $F$ & $I$ & $G$ & $E$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate Spearman's rank correlation coefficient for these data. [6]
\item Stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence of positive correlation. [4]
\item Explain briefly how you would have been able to carry out the test if bottles $B$ and $F$ had the same price. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q5 [12]}}