| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Calculate Spearman's coefficient only |
| Difficulty | Easy -1.2 This question tests basic recall and application of Spearman's rank correlation coefficient formula. Parts (i)(a) and (i)(b) require only recognition of perfect positive and perfect negative correlation (trivial pattern matching), while part (ii) involves straightforward calculation using the standard formula with n=4. The interpretation in part (iii) is elementary vocabulary. This is below average difficulty for A-level as it's purely procedural with small datasets and no problem-solving required. |
| Spec | 5.08e Spearman rank correlation |
| Judge \(A\) ranks | 1 | 2 | 3 | 4 |
| Judge \(C\) ranks | 4 | 3 | 2 | 1 |
| Judge \(A\) ranks | 1 | 2 | 3 | 4 |
| Judge \(B\) ranks | 1 | 2 | 3 | 4 |
| Judge \(A\) ranks | 1 | 2 | 3 | 4 |
| Judge \(D\) ranks | 2 | 4 | 1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 1 | B1 | NOT close to 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(-1\) | B1 | NOT close to \(-1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\Sigma d^2\) attempted \((= 10)\) | M1 | if \(\Sigma d^2 = 10\), may be implied by next line; if \(\Sigma d^2 \neq 10\), must see working. \(S_{xx}\) or \(S_{yy} = 30 - \frac{100}{4}\) \((= 5)\) or \(S_{xy} = 25 - \frac{100}{4}\) \((= 0)\): M1 |
| \(1 - \frac{6 \times \Sigma d^2}{4(4^2-1)}\) | M1 dep M1 | \(\frac{0}{\sqrt{5\times5}}\): M1 |
| \(= 0\) | A1 | Use of \((\Sigma d)^2\): M0M0A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Part ia: Total (or perfect or max or complete) agreement; They have the same opinions/ranks/numbers etc; They were identical | B1 | NOT: "They agree" or "Strongly agree"; "They agree most ranks"; "Similar rankings"; "As A's ranks increase so do B's"; "Perfect relnship" |
| Part ib: Opposite/reverse opinions/views/marks/ranks/decisions/results | B1 | Total (or max or complete or perfect) disagreement; A's highest is B's lowest; "Opposite" seen is sufficient. NOT: "Don't agree any ranks"; "Disagree or Strongly disagree"; "Disagree on all ranks"; "Perfect neg relnship" |
| Answer | Marks | Guidance |
|---|---|---|
| Either NO relationship or similar | No relationship/pattern/link/similarity between opinions/views/marks/ranks/decisions/results; opinions/etc not related; scoring appears random. NOT: "Different views"; "Don't agree but some rel'nshp"; "Ranks all different"; "No corr'n betw judges' views"; "Don't agree"; "nothing in common at all" | |
| Or indicate BOTH agreement & disagreement or NEITHER agree nor disagree | Neither agree nor disagree; Both agree & disagree; Agree for some, disagree for others; mixed/varied opinions on the ranks. NOT: "not much in common"; "completely different orders"; "opinions completely different"; "half way between (a) and (b)" | |
| Or DIFFERENT but NOT OPPOSITE | B1 | All three parts: Must refer to (or imply) opinions/views/marks/ranks/scores or (dis)agreement, or relationship or pattern; NOT just corr'n. Ignore all other |
## Question 5:
### Part (i)(a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 1 | B1 | NOT close to 1 |
### Part (i)(b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-1$ | B1 | NOT close to $-1$ |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\Sigma d^2$ attempted $(= 10)$ | M1 | if $\Sigma d^2 = 10$, may be implied by next line; if $\Sigma d^2 \neq 10$, must see working. $S_{xx}$ or $S_{yy} = 30 - \frac{100}{4}$ $(= 5)$ or $S_{xy} = 25 - \frac{100}{4}$ $(= 0)$: M1 |
| $1 - \frac{6 \times \Sigma d^2}{4(4^2-1)}$ | M1 dep M1 | $\frac{0}{\sqrt{5\times5}}$: M1 |
| $= 0$ | A1 | Use of $(\Sigma d)^2$: M0M0A0 |
# Question 5(iii):
**Part ia:** Total (or perfect or max or complete) agreement; They have the same opinions/ranks/numbers etc; They were identical | B1 | NOT: "They agree" or "Strongly agree"; "They agree most ranks"; "Similar rankings"; "As A's ranks increase so do B's"; "Perfect relnship"
**Part ib:** Opposite/reverse opinions/views/marks/ranks/decisions/results | B1 | Total (or max or complete or perfect) disagreement; A's highest is B's lowest; "Opposite" seen is sufficient. NOT: "Don't agree any ranks"; "Disagree or Strongly disagree"; "Disagree on all ranks"; "Perfect neg relnship"
**Part ii:** For $r = 0$ must state or imply:
Either NO relationship or similar | | No relationship/pattern/link/similarity between opinions/views/marks/ranks/decisions/results; opinions/etc not related; scoring appears random. NOT: "Different views"; "Don't agree but some rel'nshp"; "Ranks all different"; "No corr'n betw judges' views"; "Don't agree"; "nothing in common at all"
Or indicate BOTH agreement & disagreement or NEITHER agree nor disagree | | Neither agree nor disagree; Both agree & disagree; Agree for some, disagree for others; mixed/varied opinions on the ranks. NOT: "not much in common"; "completely different orders"; "opinions completely different"; "half way between (a) and (b)"
Or DIFFERENT but NOT OPPOSITE | B1 | All three parts: Must refer to (or imply) opinions/views/marks/ranks/scores or (dis)agreement, or relationship or pattern; NOT just corr'n. Ignore all other
**Total: [3]**
---5 (i) Write down the value of Spearman's rank correlation coefficent, $r _ { s }$, for the following sets of ranks.
All the discs are replaced in the bag. Tony now takes three discs from the bag at random without replacement.\\
(iii) Given that the first disc Tony takes is red, find the probability that the third disc Tony takes is also red.\\[0pt]
[2\\
(i) Write down the value of Spearman's rank correlation coefficent, $r _ { s }$, for the following sets of ranks.\\
(b)
\begin{center}
\begin{tabular}{ | l | l | l | l | l | }
\hline
Judge $A$ ranks & 1 & 2 & 3 & 4 \\
\hline
Judge $C$ ranks & 4 & 3 & 2 & 1 \\
\hline
\end{tabular}
\end{center}
(a)\\
(a)
\begin{center}
\begin{tabular}{ | l | l | l | l | l | }
\hline
Judge $A$ ranks & 1 & 2 & 3 & 4 \\
\hline
Judge $B$ ranks & 1 & 2 & 3 & 4 \\
\hline
\end{tabular}
\end{center}
(ii) Calculate the value of $r _ { s }$ for the following ranks.
\begin{center}
\begin{tabular}{ | l | l | l | l | l | }
\hline
Judge $A$ ranks & 1 & 2 & 3 & 4 \\
\hline
Judge $D$ ranks & 2 & 4 & 1 & 3 \\
\hline
\end{tabular}
\end{center}
(iii) For each of parts (i)(a), (i)(b) and (ii), describe in everyday terms the relationship between the two judges' opinions.
\hfill \mbox{\textit{OCR S1 2012 Q5 [8]}}