| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Effect of data changes |
| Difficulty | Standard +0.8 Part (a) is a routine calculation of Spearman's rank correlation coefficient requiring ranking and formula application. Part (b) requires conceptual understanding of how the formula behaves when data changes—students must reason about the effect on Σd² and n when a concordant pair is added, which goes beyond mechanical calculation to require insight into the formula's structure. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Person | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) |
| Height (m) | 1.72 | 1.63 | 1.77 | 1.68 | 1.74 |
| Mass (kg) | 75 | 62 | 64 | 60 | 70 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Ranks: \(3\ 5\ 1\ 4\ 2\) and \(3\ 1\ 5\ 2\ 4\); \(1\ 4\ 3\ 5\ 2\) and \(5\ 2\ 3\ 1\ 4\) | M1, A1 | Attempt ranks for both variables; Correct ranks; May be implied by \(\Sigma d^2 = 10\) |
| \(\Sigma d^2\) attempted \((= 10)\) | M1 | \(S_{xx}\) or \(S_{yy} = 55 - \frac{15^2}{5}\) \((=10)\) or \(S_{xy} = 50 - \frac{15^2}{5}\) \((=5)\) |
| \(r_s = 1 - \frac{6\Sigma d^2}{5(5^2-1)}\), dep \(\geq\) M1 gained | M1 | \(\frac{5}{\sqrt{10 \times 10}}\) |
| \(= 0.5\) | A1 [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(n(n^2-1)\) greater or increases or becomes \((n+1)((n+1)^2-1)\) | B1ind | or "denom increases" or "\(\div\) by larger number" or "fraction decreases" or "value taken from 1 decreases" oe |
| \(\Sigma d^2\) unchanged (or not increase); Allow \(d^2\) unchanged | B1ind | or \(d=0\) or \(d^2=0\) or the difference is 0 |
| \(r_s\) greater | B1 [3] | dep \(\geq\) B1 or no explanation |
## Question 4:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Ranks: $3\ 5\ 1\ 4\ 2$ and $3\ 1\ 5\ 2\ 4$; $1\ 4\ 3\ 5\ 2$ and $5\ 2\ 3\ 1\ 4$ | M1, A1 | Attempt ranks for both variables; Correct ranks; May be implied by $\Sigma d^2 = 10$ | If alphabetical order used for one or both sets: M0A0; if $\Sigma d^2 = 14$ or $16$, check carefully |
| $\Sigma d^2$ attempted $(= 10)$ | M1 | $S_{xx}$ or $S_{yy} = 55 - \frac{15^2}{5}$ $(=10)$ or $S_{xy} = 50 - \frac{15^2}{5}$ $(=5)$ | |
| $r_s = 1 - \frac{6\Sigma d^2}{5(5^2-1)}$, dep $\geq$ M1 gained | M1 | $\frac{5}{\sqrt{10 \times 10}}$ | |
| $= 0.5$ | A1 [5] | | |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $n(n^2-1)$ greater or increases or becomes $(n+1)((n+1)^2-1)$ | B1ind | or "denom increases" or "$\div$ by larger number" or "fraction decreases" or "value taken from 1 decreases" oe | Allow increases to $6\times35$; NOT just "$n$ increases" |
| $\Sigma d^2$ unchanged (or not increase); Allow $d^2$ unchanged | B1ind | or $d=0$ or $d^2=0$ or the difference is 0 | NOT $n(n^2-1)$ changes; NOT "difference is unchanged" |
| $r_s$ greater | B1 [3] | dep $\geq$ B1 or no explanation | Use of incorrect formula: max B1B1B0 (B0 for $r_s$ greater) |
---4
\begin{enumerate}[label=(\alph*)]
\item The table gives the heights and masses of 5 people.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Person & $A$ & $B$ & $C$ & $D$ & $E$ \\
\hline
Height (m) & 1.72 & 1.63 & 1.77 & 1.68 & 1.74 \\
\hline
Mass (kg) & 75 & 62 & 64 & 60 & 70 \\
\hline
\end{tabular}
\end{center}
Calculate Spearman's rank correlation coefficient.
\item In an art competition the value of Spearman's rank correlation coefficient, $r _ { s }$, calculated from two judges' rankings was 0.75 . A late entry for the competition was received and both judges ranked this entry lower than all the others. By considering the formula for $r _ { s }$, explain whether the new value of $r _ { s }$ will be less than 0.75 , equal to 0.75 , or greater than 0.75 .
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2012 Q4 [8]}}