| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Determine ranks from coefficient |
| Difficulty | Moderate -0.8 Part (i) is a straightforward application of Spearman's rank correlation coefficient formula requiring ranking of grades and times, calculating differences, and substituting into the formula - pure procedural recall with no conceptual challenge. Part (ii) requires minimal insight that swapping two ranks with equal d² contributions preserves r_s, but this is a simple pattern recognition task. Both parts are routine S1 exercises with no problem-solving or novel thinking required. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Student | Ann | Bill | Caz | Den | Ed |
| Time revising | 0 | 60 | 35 | 100 | 45 |
| Grade | C | D | E | B | A |
| Competitor | P | Q | R | S |
| Judge 1 rank | 1 | 2 | 3 | 4 |
| Judge 2 rank | 3 | 2 | 1 | 4 |
### (i)
**Answer:**
- Ranks: $5\ 2\ 4\ 1\ 3$ or $A\ B\ C\ D\ E$ (grades) over $3\ 4\ 5\ 2\ 1$ or $3\ 1\ 5\ 2\ 4$
- $d^2 4\ 4\ 1\ 1\ 4$
- $\Sigma d^2 = 14$ (= 14)
- $1 - \frac{6\times"14"}{5\times(5^2-1)} = 0.3$ oe
**Marks:** M1, A1, M1, M1, A1[5]
**Guidance:**
- Attempt ranks
- Correct ranks; allow both sets reversed. Can be implied by eg $\Sigma f = 14$
- Attempt $\Sigma d^2$ dep 1st M1
- ft $\Sigma d^2$ dep 1st M1
- If one set reversed, $r_s = -0.3$ M1A0M1A0
- Use PMCC on ranks: 1st M1A1 as main scheme then: $\Sigma x = \Sigma y = 15$, $\Sigma x^2 = \Sigma y^2 = 55$, $\Sigma xy = 48$
- $S_{xx} = S_{yy} = 10$, $S_{xy} = 3$, allow one arith error M1
- $r = 3/\sqrt{(10\times10)}$ allow one arith error M1
- $= 0.3$ A1
### (ii)
**Answer:**
- $\Sigma d^2 = 8$ or '2 the same and 2 differ by 2' or $1\ 4\ 3\ 2$
**Marks:** M1, A1[2]
**Guidance:**
- May be implied
- Allow $d^2 = 8$ or similar
---\begin{enumerate}[label=(\roman*)]
\item The table shows the times, in minutes, spent by five students revising for a test, and the grades that they achieved in the test.
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Student & Ann & Bill & Caz & Den & Ed \\
\hline
Time revising & 0 & 60 & 35 & 100 & 45 \\
\hline
Grade & C & D & E & B & A \\
\hline
\end{tabular}
Calculate Spearman's rank correlation coefficient. [5]
\item The table below shows the ranks given by two judges to four competitors.
\begin{tabular}{|c|c|c|c|c|}
\hline
Competitor & P & Q & R & S \\
\hline
Judge 1 rank & 1 & 2 & 3 & 4 \\
\hline
Judge 2 rank & 3 & 2 & 1 & 4 \\
\hline
\end{tabular}
Spearman's rank correlation coefficient for these ranks is denoted by $r_s$. With the same set of ranks for Judge 1, write down a different set of ranks for Judge 2 which gives the same value of $r_s$. There is no need to find the value of $r_s$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2013 Q2 [7]}}