| Exam Board | Edexcel |
|---|---|
| Module | FS2 AS (Further Statistics 2 AS) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| 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 formula and critical value lookup. Part (a) requires direct substitution into the standard formula, part (b) involves reading from tables, and part (c) requires recognizing that adding a tied rank doesn't change the correlation coefficient. All steps are routine for Further Statistics students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(r_s = 1 - \dfrac{6 \times 84}{10(10^2 - 1)}\) | B1 | awrt 0.491; allow \(\dfrac{27}{55}\) |
| \(= 0.4909\ldots\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((1>) r_s > 0.5636\) | B1 | Correct critical region with 0.5636 or better; condone use of \(\rho\) instead of \(r_s\) |
| 0.491 is not in the critical region; there is insufficient evidence of agreement between their film ranks. | B1ft | Correct ft contextualised conclusion (must include film or ranks) based on their (a) and their CR; allow ft on their CV if a CR is not stated |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{new } r_s = 1 - \dfrac{6 \times 84}{11(11^2 - 1)}\) | M1 | Use of formula with same \(\sum d^2\) and 11 |
| \(= 0.61818\ldots\) awrt 0.618 | A1 | allow \(\dfrac{34}{55}\) |
| new critical value is \(0.5364\) | B1 | 0.5364 or better |
| There is now sufficient evidence of agreement between their film ranks. | A1 | Fully correct solution with awrt 0.618 and contextualised conclusion (must include film or ranks) |
## Question 1:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $r_s = 1 - \dfrac{6 \times 84}{10(10^2 - 1)}$ | B1 | awrt **0.491**; allow $\dfrac{27}{55}$ |
| $= 0.4909\ldots$ | | |
**Total: (1 mark)**
---
### Part (b)(i) and (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(1>) r_s > 0.5636$ | B1 | Correct critical region with 0.5636 or better; condone use of $\rho$ instead of $r_s$ |
| 0.491 is not in the critical region; there is insufficient evidence of agreement between their film ranks. | B1ft | Correct ft contextualised conclusion (must include film or ranks) based on their (a) and their CR; allow ft on their CV if a CR is not stated |
**Total: (2 marks)**
---
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{new } r_s = 1 - \dfrac{6 \times 84}{11(11^2 - 1)}$ | M1 | Use of formula with same $\sum d^2$ and 11 |
| $= 0.61818\ldots$ awrt **0.618** | A1 | allow $\dfrac{34}{55}$ |
| new critical value is $0.5364$ | B1 | 0.5364 or better |
| There is now sufficient evidence of agreement between their film ranks. | A1 | Fully correct solution with awrt 0.618 and contextualised conclusion (must include film or ranks) |
**Total: (4 marks)**\begin{enumerate}
\item Abena and Meghan are both given the same list of 10 films.
\end{enumerate}
Each of them ranks the 10 films from most favourite to least favourite.\\
For the differences, $d$, between their ranks for these 10 films, $\sum d ^ { 2 } = 84$\\
(a) Calculate Spearman's rank correlation coefficient between Abena's ranks and Meghan's ranks.
A test is carried out at the 5\% level of significance to see if there is agreement between their ranks for the films.
The hypotheses for the test are
$$\mathrm { H } _ { 0 } : \rho _ { \mathrm { S } } = 0 \quad \mathrm { H } _ { 1 } : \rho _ { \mathrm { S } } > 0$$
(b) (i) Find the critical region for the test.\\
(ii) State the conclusion of the test.
An 11th film is added to the list. Abena and Meghan both agree that this film is their least favourite.
A new test is carried out at the $5 \%$ level of significance using the same hypotheses.\\
(c) Determine the conclusion of this test. You should state the test statistic and the critical value used.
\hfill \mbox{\textit{Edexcel FS2 AS 2022 Q1 [7]}}