| Exam Board | OCR |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2018 |
| Session | March |
| Marks | 8 |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Critical region or probability |
| Difficulty | Challenging +1.2 This question requires understanding of Spearman's rank correlation coefficient and hypothesis testing, but the actual calculations are straightforward. Part (i) involves counting permutations that yield r_s ≥ 27/28 (which is very close to 1, so only a few permutations qualify) among all 7! possibilities. Part (ii) simply requires recognizing that the significance level equals the probability found in part (i). While this is Further Maths content, it's a relatively standard application of the distribution of r_s under random ranking with minimal computational complexity. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - \frac{62d^2}{7 \times 48} = \frac{27}{28} \Rightarrow \sum d^2 = 2\) | M1 | 3.1b |
| \(\Rightarrow d_i = 0, 0, 0, 0, 0, 1, 1\) in some order | A1 | 1.1 |
| 6 possibilities | A1 | 2.2a |
| \(\frac{1+6}{7!}\) | M1 | 1.1a |
| \(= \frac{1}{720}\) or \(0.0013888\ldots\) | A1 | 1.1a |
| A1 | 2.1 | Www, exact or art 0.00139 |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.1388\ldots\%\) | B1 | 3.1b |
### Part (i)
$1 - \frac{62d^2}{7 \times 48} = \frac{27}{28} \Rightarrow \sum d^2 = 2$ | **M1** | 3.1b | Use $1 - \frac{62d^2}{7 \times 48} = \frac{27}{28}$; correct $\sum d^2$
$\Rightarrow d_i = 0, 0, 0, 0, 0, 1, 1$ in some order | **A1** | 1.1
6 possibilities | **A1** | 2.2a | Stated or implied
$\frac{1+6}{7!}$ | **M1** | 1.1a | Divide by 7!
$= \frac{1}{720}$ or $0.0013888\ldots$ | **A1** | 1.1a | 1 included, www
| **A1** | 2.1 | Www, exact or art 0.00139
### Part (ii)
$0.1388\ldots\%$ | **B1** | 3.1b | Allow 0.00139 etc, or same as (i) | Not 0.00139%8 In a competition, entrants have to give ranks from 1 to 7 to each of seven resorts. The correct ranks for the resorts are decided by an expert.\\
(i) One competitor chooses his ranks randomly. By considering all the possible rankings, find the probability that the value of Spearman's rank correlation coefficient $r _ { s }$ between the competitor's ranks and the expert's ranks is at least $\frac { 27 } { 28 }$.\\
(ii) Another competitor ranks the seven resorts. A significance test is carried out to test whether there is evidence that this competitor is merely guessing the rank order of the seven resorts. The critical region is $r _ { s } \geqslant \frac { 27 } { 28 }$. State the significance level of the test.
\section*{END OF QUESTION PAPER}
\hfill \mbox{\textit{OCR FS1 AS 2018 Q8 [8]}}