| Exam Board | Edexcel |
|---|---|
| Module | FS2 (Further Statistics 2) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Handle tied ranks |
| Difficulty | Standard +0.3 This is a straightforward application of tied ranks in Spearman's correlation. Part (a) requires simple averaging of tied ranks (routine), part (b) tests recall of when to use the full formula vs the simplified version, and part (c) is a standard hypothesis test with critical value lookup. All steps are procedural with no novel insight required, making it slightly easier than average. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Student | History mark | Geography mark | History rank | Geography rank |
| A | 76 | 58 | 1 | 3 |
| B | 70 | 60 | 2 | 2 |
| C | 64 | 57 | \(s\) | \(t\) |
| D | 64 | 63 | \(s\) | 1 |
| E | 64 | 57 | \(s\) | \(t\) |
| F | 59 | 50 | 6 | 7 |
| G | 55 | 52 | 7 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(s = 4\) | B1 | cao |
| \(t = 4.5\) | B1 | cao |
| (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Because there are tied ranks | B1 | Correct explanation |
| (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: \rho_s = 0 \quad H_1: \rho_s > 0\) | B1 | Both hypotheses correct with correct notation (must use \(\rho_s\) or \(\rho\)) |
| \(CV = 0.7143\) | B1 | Correct critical value 0.7143 or better |
| \(r_s = 0.7106\) does not lie in the critical region | M1 | Drawing a correct inference using their CV and 0.7106 |
| There is insufficient evidence to suggest that the higher the rank in the History test, the higher the rank in the Geography test (oe) | A1 | Drawing a correct inference (condone "marks" instead of ranks) in context using their CV and 0.7106 |
| (4 marks) |
## Question 1:
### Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $s = 4$ | B1 | cao |
| $t = 4.5$ | B1 | cao |
| **(2 marks)** | | |
### Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| Because there are tied ranks | B1 | Correct explanation |
| **(1 mark)** | | |
### Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \rho_s = 0 \quad H_1: \rho_s > 0$ | B1 | Both hypotheses correct with correct notation (must use $\rho_s$ or $\rho$) |
| $CV = 0.7143$ | B1 | Correct critical value 0.7143 or better |
| $r_s = 0.7106$ does not lie in the critical region | M1 | Drawing a correct inference using their CV and 0.7106 |
| There is insufficient evidence to suggest that the higher the rank in the History test, the higher the rank in the Geography test (oe) | A1 | Drawing a correct inference (condone "marks" instead of ranks) in context using their CV and 0.7106 |
| **(4 marks)** | | |
---\begin{enumerate}
\item Anisa is investigating the relationship between marks on a History test and marks on a Geography test. She collects information from 7 students. She wants to calculate the Spearman's rank correlation coefficient for the 7 students so she ranks their performance on each test.
\end{enumerate}
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
Student & History mark & Geography mark & History rank & Geography rank \\
\hline
A & 76 & 58 & 1 & 3 \\
\hline
B & 70 & 60 & 2 & 2 \\
\hline
C & 64 & 57 & $s$ & $t$ \\
\hline
D & 64 & 63 & $s$ & 1 \\
\hline
E & 64 & 57 & $s$ & $t$ \\
\hline
F & 59 & 50 & 6 & 7 \\
\hline
G & 55 & 52 & 7 & 6 \\
\hline
\end{tabular}
\end{center}
(a) Write down the value of $s$ and the value of $t$
The full product moment correlation coefficient (pmcc) formula is used with the ranks to calculate the Spearman's rank correlation coefficient instead of $r _ { s } = 1 - \frac { 6 \Sigma d ^ { 2 } } { n \left( n ^ { 2 } - 1 \right) }$ and the value obtained is 0.7106 to 4 significant figures.\\
(b) Explain why the full pmcc formula is used to carry out the calculation.\\
(c) Stating your hypotheses clearly, test whether or not there is evidence to suggest that the higher a student ranks in the History test, the higher the student ranks in the Geography test. Use a $5 \%$ level of significance.
\hfill \mbox{\textit{Edexcel FS2 2021 Q1 [7]}}