| Exam Board | Edexcel |
|---|---|
| Module | FS2 AS (Further Statistics 2 AS) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| 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 with standard hypothesis testing. Part (a) requires ranking data and applying the formula (routine calculation), part (b) is a standard one-tailed test using critical value tables, and part (c) requires simple interpretation. The question is slightly above average difficulty only because it involves ranking ties and requires careful arithmetic, but follows a completely standard template with no novel problem-solving required. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Applicant | \(A\) | \(B\) | C | \(D\) | E | \(F\) | G | H | I |
| Task \(\boldsymbol { P }\) | 19 | 16 | 16 | 12 | 8 | 17 | 12 | 12 | 5 |
| Task \(Q\) | 17 | 11 | 14 | 7 | 6 | 18 | 15 | 11 | 10 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Ranking at least one row (at least 4 correct) | M1 | May have no tied ranks |
| Ranking both rows using tied ranks (at least 4 correct) | M1 | |
| \(r_s = \dfrac{"271.25" - \dfrac{45 \times 45}{9}}{\sqrt{\left("282.5" - \dfrac{45^2}{9}\right)\left("284.5" - \dfrac{45^2}{9}\right)}}\) | M1 | Must be using ranks; may be implied by correct answer |
| \(r_s = 0.7907...\) awrt \(0.791\) | A1 | NB 0.796 is M1M1 M1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: \rho = 0\), \(H_1: \rho > 0\) | B1 | Both hypotheses stated in terms of \(\rho\) or \(\rho_s\) |
| Critical value \(\rho = 0.7833\) | B1 | Correct critical value of 0.7833 (use of pmcc 0.7498 is B0) |
| \(r_s = 0.7907\) lies in critical region / reject \(H_0\) / significant | M1 | Comparing their 0.791 with their 0.7833; correct statement |
| Positive correlation between ranks of scores for tasks \(P\) and \(Q\) | A1ft | Correct ft contextual conclusion, no contradictions; must mention "scores" and either "task" or "\(P\) and \(Q\)" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P\) and \(R\) not in agreement so give different information about applicants, or \(P\) and \(Q\), \(Q\) and \(R\) in some agreement so give similar information (oe) | M1 | Correct explanation supporting answer; must have idea of "different" or "similar" tasks |
| \(P\) and \(R\) | A1 | Correct deduction from information, i.e. choosing \(P\) and \(R\) only |
# Question 1:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Ranking at least one row (at least 4 correct) | M1 | May have no tied ranks |
| Ranking both rows using tied ranks (at least 4 correct) | M1 | |
| $r_s = \dfrac{"271.25" - \dfrac{45 \times 45}{9}}{\sqrt{\left("282.5" - \dfrac{45^2}{9}\right)\left("284.5" - \dfrac{45^2}{9}\right)}}$ | M1 | Must be using ranks; may be implied by correct answer |
| $r_s = 0.7907...$ awrt $0.791$ | A1 | NB 0.796 is M1M1 M1A0 |
**ALT:** For use of $\sum d^2 = 1^2+2^2+0.5^2+2^2+1^2+1^2+3^2+0.5^2+2^2 [=24.5]$, then $r_s = 1 - \dfrac{6 \times "24.5"}{9 \times 80}\ (= 0.7958...)$ — M1M1M1A0 (dep on some attempt to rank)
**(4 marks)**
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: \rho = 0$, $H_1: \rho > 0$ | B1 | Both hypotheses stated in terms of $\rho$ or $\rho_s$ |
| Critical value $\rho = 0.7833$ | B1 | Correct critical value of 0.7833 (use of pmcc 0.7498 is B0) |
| $r_s = 0.7907$ lies in critical region / reject $H_0$ / significant | M1 | Comparing their 0.791 with their 0.7833; correct statement |
| Positive correlation between ranks of scores for tasks $P$ and $Q$ | A1ft | Correct ft contextual conclusion, no contradictions; must mention "scores" and either "task" or "$P$ and $Q$" |
**(4 marks)**
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P$ and $R$ not in agreement so give **different** information about applicants, **or** $P$ and $Q$, $Q$ and $R$ in some agreement so give **similar** information (oe) | M1 | Correct explanation supporting answer; must have idea of "different" or "similar" tasks |
| $P$ and $R$ | A1 | Correct deduction from information, i.e. choosing $P$ and $R$ only |
**(2 marks)**
**Total: 10 marks**\begin{enumerate}
\item Every applicant for a job at Donala is given three different tasks, $P , Q$ and $R$.
\end{enumerate}
For each task the applicant is awarded a score.\\
The scores awarded to 9 of the applicants, for the tasks $P$ and $Q$, are given below.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
Applicant & $A$ & $B$ & C & $D$ & E & $F$ & G & H & I \\
\hline
Task $\boldsymbol { P }$ & 19 & 16 & 16 & 12 & 8 & 17 & 12 & 12 & 5 \\
\hline
Task $Q$ & 17 & 11 & 14 & 7 & 6 & 18 & 15 & 11 & 10 \\
\hline
\end{tabular}
\end{center}
(a) Calculate Spearman's rank correlation coefficient for the scores awarded for the tasks $P$ and $Q$.\\
(b) Test, at the $1 \%$ level of significance, whether or not there is evidence for a positive correlation between the ranks of scores for tasks $P$ and $Q$. You should state your hypotheses and critical value clearly.
The Spearman's rank correlation coefficient for $P$ and $R$ is 0.290 and for $Q$ and $R$ is 0.795
The manager of Donala wishes to reduce the number of tasks given to job applicants from three to two.\\
(c) Giving a reason for your answer, state which 2 tasks you would recommend the manager uses.
\hfill \mbox{\textit{Edexcel FS2 AS 2023 Q1 [10]}}