| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Hypothesis test for association |
| Difficulty | Standard +0.3 This is a straightforward application of Spearman's rank correlation coefficient hypothesis test with data already in rank form. Students need to calculate differences, apply the standard formula, and compare to critical values. While it's a Further Maths topic (making it slightly above average), it requires only routine procedural steps with no conceptual challenges or novel problem-solving. |
| Spec | 5.08f Hypothesis test: Spearman rank |
| Runner | A | B | C | D | E | F | G | H |
| First race | 3 | 1 | 5 | 6 | 2 | 8 | 7 | 4 |
| Second race | 4 | 3 | 8 | 7 | 2 | 5 | 6 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \rho_s = 0,\ H_1: \rho_s > 0\), where \(\rho_s\) is the population value of Spearman's rank correlation coefficient | B2 | One error e.g. \(\rho_s\) not defined in terms of population: B1. Allow \(\rho\) or \(r_s\) rather than \(\rho_s\). SR: \(H_0\): No association between orders in the two races, \(H_1\): positive association: B1 max |
| Answer | Marks | Guidance |
|---|---|---|
| \(r_s = 1 - \frac{6\Sigma d^2}{8\times63}\) or \(\frac{3.125}{\sqrt{5.25^2}}\) | M1 | Use \(\Sigma d^2\) and correct formula. Allow use of pmcc formula here |
| \(= 0.595(238)\) or \(\frac{25}{42}\) | A1 | Exact or awrt \(0.595\) |
| \(< 0.6429\) | A1 | Not \(< 0.6215\) (pmcc) |
| Do not reject \(H_0\). Insufficient evidence that those who do well in one race tend to do well in the other. | M1ft, A1ft [7] | Consistent, ft on their TS. Contextualised, not over-assertive. Not "sig evidence that \(H_0\) true". Needs recognisable attempt at \(r_s\) and \(-1 \leq r_s \leq 1\). Allow from \(r_s < 0.7381\) (2-tailed test) or from \(< 0.6215\) (pmcc) |
# Question 2:
$H_0: \rho_s = 0,\ H_1: \rho_s > 0$, where $\rho_s$ is the population value of Spearman's rank correlation coefficient | **B2** | One error e.g. $\rho_s$ not defined in terms of population: B1. Allow $\rho$ or $r_s$ rather than $\rho_s$. SR: $H_0$: No association between orders in the two races, $H_1$: positive association: B1 max
$\Sigma d^2 = 1+4+9+1+0+9+1+9 = 34$
$r_s = 1 - \frac{6\Sigma d^2}{8\times63}$ or $\frac{3.125}{\sqrt{5.25^2}}$ | **M1** | Use $\Sigma d^2$ and correct formula. Allow use of pmcc formula here
$= 0.595(238)$ or $\frac{25}{42}$ | **A1** | Exact or awrt $0.595$
$< 0.6429$ | **A1** | Not $< 0.6215$ (pmcc)
Do not reject $H_0$. Insufficient evidence that those who do well in one race tend to do well in the other. | **M1ft, A1ft [7]** | Consistent, ft on their TS. Contextualised, not over-assertive. Not "sig evidence that $H_0$ true". Needs recognisable attempt at $r_s$ and $-1 \leq r_s \leq 1$. Allow from $r_s < 0.7381$ (2-tailed test) or from $< 0.6215$ (pmcc)
---2 Eight runners took part in two races. The positions in which the runners finished in the two races are shown in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
Runner & A & B & C & D & E & F & G & H \\
\hline
First race & 3 & 1 & 5 & 6 & 2 & 8 & 7 & 4 \\
\hline
Second race & 4 & 3 & 8 & 7 & 2 & 5 & 6 & 1 \\
\hline
\end{tabular}
\end{center}
Test at the 5\% significance level whether those runners who do better in one race tend to do better in the other.
\hfill \mbox{\textit{OCR Further Statistics AS 2022 Q2 [7]}}