| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2021 |
| Session | January |
| Marks | 9 |
| 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. Students must rank two datasets, calculate rs using the formula, compare to critical values from tables, and interpret causation vs correlation. While it requires careful arithmetic and multiple steps (ranking, calculation, hypothesis test), it follows a completely standard procedure taught in S3 with no novel problem-solving required. The causation question in part (c) is a routine interpretation point. Slightly easier than average due to its procedural nature. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Student | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | \(K\) |
| No. of objects | 8 | 11 | 9 | 15 | 17 | 6 | 10 | 14 | 12 | 13 | 5 |
| \% in maths test | 30 | 62 | 57 | 80 | 75 | 43 | 65 | 51 | 48 | 55 | 32 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Objects rank: 9, 6, 8, 2, 1, 10, 7, 3, 5, 4, 11; Maths rank: 11, 4, 5, 1, 2, 9, 3, 7, 8, 6, 10 | M1, M1 | 1st M1 for attempt to rank one row with at least 5 correct (could be reversed); 2nd M1 for both rows ranked with at least 5 correct in each row (one or both reversed) |
| \(\sum d^2 = 4+4+9+1+1+1+16+16+9+4+1 = 66\) | M1 | 3rd M1 for attempt to calculate \(\sum d^2\) with at least 5 correct |
| \(r_s = 1 - \frac{6 \times "66"}{11(11^2-1)} = 0.7\) | dM1; A1 | 4th dM1 (dependent on at least one M1) for use of correct formula with \(\sum d^2\); A1 for 0.7 or exact equivalent |
| Subtotal: (5 marks) | ||
| \(H_0: \rho = 0\); \(H_1: \rho > 0\) | B1, B1 | 1st B1 for both hypotheses in terms of \(\rho\) or \(\rho_s\) [If \(r_s < 0\) in (a) allow \(H_1: \rho < 0\)]; 2nd B1 for critical value 0.5364 (sign compatible with \(r_s\)). [If \(r_s < 0\) in (a) need \(-0.5364\)]. Allow 0.6182 if 1st B0 for \(H_1: \rho \neq 0\) |
| Significant result so there is evidence to support the teacher's belief OR there is evidence of a positive correlation between short term memory and mathematical ability OR evidence that students with strong maths ability also have good short term memory | B1 | 3rd B1 for correct conclusion in context. Penalise contradictory comments e.g. "not significant so supports teacher's belief" [No ft] |
| Subtotal: (3 marks) | ||
| Data shows positive correlation but does not necessarily imply that enhanced short term memory causes increase in mathematical ability | B1 | B1 for comment that states correlation does not imply causation. Need to see "cause" or "causation" clearly mentioned |
| Subtotal: (1 mark) |
| Answer/Working | Marks | Guidance |
|---|---|---|
| Objects rank: 9, 6, 8, 2, 1, 10, 7, 3, 5, 4, 11; Maths rank: 11, 4, 5, 1, 2, 9, 3, 7, 8, 6, 10 | M1, M1 | 1st M1 for attempt to rank one row with at least 5 correct (could be reversed); 2nd M1 for both rows ranked with at least 5 correct in each row (one or both reversed) |
| $\sum d^2 = 4+4+9+1+1+1+16+16+9+4+1 = 66$ | M1 | 3rd M1 for attempt to calculate $\sum d^2$ with at least 5 correct |
| $r_s = 1 - \frac{6 \times "66"}{11(11^2-1)} = 0.7$ | dM1; A1 | 4th dM1 (dependent on at least one M1) for use of correct formula with $\sum d^2$; A1 for 0.7 or exact equivalent |
| **Subtotal: (5 marks)** | | |
| $H_0: \rho = 0$; $H_1: \rho > 0$ | B1, B1 | 1st B1 for both hypotheses in terms of $\rho$ or $\rho_s$ [If $r_s < 0$ in (a) allow $H_1: \rho < 0$]; 2nd B1 for critical value 0.5364 (sign compatible with $r_s$). [If $r_s < 0$ in (a) need $-0.5364$]. Allow 0.6182 if 1st B0 for $H_1: \rho \neq 0$ |
| Significant result so there is evidence to support the teacher's belief OR there is evidence of a positive correlation between short term memory and mathematical ability OR evidence that students with strong maths ability also have good short term memory | B1 | 3rd B1 for correct conclusion in context. Penalise contradictory comments e.g. "not significant so supports teacher's belief" [No ft] |
| **Subtotal: (3 marks)** | | |
| Data shows positive correlation but does not necessarily imply that enhanced short term memory causes increase in mathematical ability | B1 | B1 for comment that states correlation does not imply causation. Need to see "cause" or "causation" clearly mentioned |
| **Subtotal: (1 mark)** | | |
**Total: [9 marks]**
---2. A teacher believes that those of her students with strong mathematical ability may also have enhanced short-term memory. She shows a random sample of 11 students a tray of different objects for eight seconds and then asks them to write down as many of the objects as they can remember. The results, along with their percentage score in a recent mathematics test, are given in the table below.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Student & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ & $J$ & $K$ \\
\hline
No. of objects & 8 & 11 & 9 & 15 & 17 & 6 & 10 & 14 & 12 & 13 & 5 \\
\hline
\% in maths test & 30 & 62 & 57 & 80 & 75 & 43 & 65 & 51 & 48 & 55 & 32 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate Spearman's rank correlation coefficient for these data. Show your working clearly.
\item Stating your hypotheses clearly, carry out a suitable test to assess the teacher's belief. Use a $5 \%$ level of significance and state your critical value.
The teacher shows these results to her class and argues that spending more time trying to improve their short-term memory would improve their mathematical ability.
\item Explain whether or not you agree with the teacher's argument.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2021 Q2 [9]}}