| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Hypothesis test for positive correlation |
| Difficulty | Standard +0.3 This is a straightforward application of Spearman's rank correlation coefficient with standard hypothesis testing (part a), followed by a routine calculation to find the range of x values (part b), and a conceptual question about when to use different correlation coefficients (part c). All parts follow standard procedures taught in Further Statistics with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| 1-tail test | 5\% | 2.5\% | 1\% | 0.5\% | |
| 2-tail test | 10\% | 5\% | 2\% | 1\% | |
| \multirow{3}{*}{\(n\)} | 6 | 0.8286 | 0.8857 | 0.9429 | 1.0000 |
| 7 | 0.7143 | 0.7857 | 0.8929 | 0.9286 | |
| 8 | 0.6429 | 0.7381 | 0.8333 | 0.8810 |
| Contestant | A | B | C | D | \(E\) | \(F\) | G |
| Judge 1 | 64 | 65 | 67 | 78 | 79 | 80 | 86 |
| Judge 2 | 61 | 63 | 78 | 80 | 81 | 90 | \(x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: \rho_s = 0\), \(H_1: \rho_s > 0\), where \(\rho_s\) is the population srcc | B2 | Allow \(r\) or \(\rho\). One error e.g. 2-tailed or symbol not defined in terms of population: B1 |
| \(0.964 > 0.8929\) | B1 | Explicit comparison with 0.893 or better, allow \(\frac{27}{28} > 0.893\) |
| Reject \(H_0\). There is significant evidence that judges agree with each other. | B1 [4] | Full conclusion, contextualised, not over-assertive, needs 0.893 or 0.929. 2-tailed (\(0.964 > 0.9286\)): can give B1B0 B0 B1, max 2/4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - \frac{6\Sigma d^2}{7 \times 48} = \frac{27}{28}\) | M1 | Equation using correct Spearman formula seen, *not* using \((86-x)^2\) |
| \(\Rightarrow \Sigma d^2 = 2\) | A1 | Correctly obtain \(\Sigma d^2 = 2\) |
| Only one pair in wrong order | B1 | Stated or implied, e.g. any number in range [81, 90] mentioned |
| \(\Rightarrow 82 \leq X \leq 89\) \((81 < X < 90)\) | A1 [4] | Or equivalent (allow non-integer answers). SC 27/28 not used: B2 for final answer only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| e.g. PMCC takes into account the actual scores, *or* the data may have a bivariate normal distribution, *or* PMCC measures correlation not association/ranking | B1 [1] | Allow "uses scores, not ranks", but *not* "because the data are scores, not ranks". *Not* "data is bivariate normal", *not* "PMCC takes into account whether the two judges' scores are very different". More than one reason, one of which is wrong: B0 |
# Question 5:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \rho_s = 0$, $H_1: \rho_s > 0$, where $\rho_s$ is the population srcc | **B2** | Allow $r$ or $\rho$. One error e.g. 2-tailed or symbol not defined in terms of population: B1 |
| $0.964 > 0.8929$ | **B1** | Explicit comparison with 0.893 or better, allow $\frac{27}{28} > 0.893$ |
| Reject $H_0$. There is significant evidence that judges agree with each other. | **B1 [4]** | Full conclusion, contextualised, not over-assertive, needs 0.893 or 0.929. 2-tailed ($0.964 > 0.9286$): can give B1B0 B0 B1, max 2/4 |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - \frac{6\Sigma d^2}{7 \times 48} = \frac{27}{28}$ | **M1** | Equation using correct Spearman formula seen, *not* using $(86-x)^2$ |
| $\Rightarrow \Sigma d^2 = 2$ | **A1** | Correctly obtain $\Sigma d^2 = 2$ |
| Only one pair in wrong order | **B1** | Stated or implied, e.g. any number in range [81, 90] mentioned |
| $\Rightarrow 82 \leq X \leq 89$ $(81 < X < 90)$ | **A1 [4]** | Or equivalent (allow non-integer answers). SC 27/28 not used: B2 for final answer only |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. PMCC takes into account the actual scores, *or* the data may have a bivariate normal distribution, *or* PMCC measures correlation not association/ranking | **B1 [1]** | Allow "uses scores, not ranks", but *not* "because the data are scores, not ranks". *Not* "data is bivariate normal", *not* "PMCC takes into account whether the two judges' scores are very different". More than one reason, one of which is wrong: B0 |
---5 In a fashion competition, two judges gave marks to a large number of contestants.
The value of Spearman's rank correlation coefficient, $\mathrm { r } _ { \mathrm { s } }$, between the marks given to 7 randomly chosen contestants is $\frac { 27 } { 28 }$.
\begin{enumerate}[label=(\alph*)]
\item An excerpt from the table of critical values of $\mathrm { r } _ { \mathrm { s } }$ is shown below.
\section*{Critical values of Spearman's rank correlation coefficient}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
& 1-tail test & 5\% & 2.5\% & 1\% & 0.5\% \\
\hline
& 2-tail test & 10\% & 5\% & 2\% & 1\% \\
\hline
\multirow{3}{*}{$n$} & 6 & 0.8286 & 0.8857 & 0.9429 & 1.0000 \\
\hline
& 7 & 0.7143 & 0.7857 & 0.8929 & 0.9286 \\
\hline
& 8 & 0.6429 & 0.7381 & 0.8333 & 0.8810 \\
\hline
\end{tabular}
\end{center}
Test whether there is evidence, at the 1\% significance level, that the judges agree with each another.
The marks given by the two judges to the 7 randomly chosen contestants were as follows, where $x$ is an integer.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
Contestant & A & B & C & D & $E$ & $F$ & G \\
\hline
Judge 1 & 64 & 65 & 67 & 78 & 79 & 80 & 86 \\
\hline
Judge 2 & 61 & 63 & 78 & 80 & 81 & 90 & $x$ \\
\hline
\end{tabular}
\end{center}
\item Use the value $\mathrm { r } _ { \mathrm { s } } = \frac { 27 } { 28 }$ to determine the range of possible values of $x$.
\item Give a reason why it might be preferable to use the product moment correlation coefficient rather than Spearman's rank correlation coefficient in this context.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics AS 2024 Q5 [9]}}