| Exam Board | Edexcel |
|---|---|
| Module | FS2 AS (Further Statistics 2 AS) |
| Year | 2018 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Hypothesis test for positive correlation |
| Difficulty | Standard +0.8 This is a Further Maths Statistics question requiring calculation of Spearman's rank correlation coefficient (with tied ranks), conducting two different hypothesis tests (Spearman's and PMCC), and comparing conclusions. While methodical, it requires knowledge of critical values, handling ties, and interpreting multiple tests—more demanding than standard A-level but routine for Further Maths FS2. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Competitor | A | B | C | D | E | F |
| Height (m) | 2.05 | 1.93 | 2.02 | 1.96 | 1.81 | 2.02 |
| V349 SIHI NI IMIMM ION OC | VJYV SIHIL NI LIIIM ION OO | VJYV SIHIL NI JIIYM ION OC |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Rank first row using tied ranks (at least 4 correct): A=1, B=5, C=2.5, D=4, E=6, F=2.5 | M1 | 1.1b |
| Rank second row (at least 4 correct): A=2, B=5, C=1, D=4, E=6, F=3 | M1 | 1.1b |
| \([\sum h^2 = 90.5,\ \sum l^2 = 91,\ \sum hl = 89]\) | ||
| Use of pmcc: \(r_s = \dfrac{89 - \frac{21 \times 21}{6}}{\sqrt{(90.5 - \frac{21^2}{6})(91 - \frac{21^2}{6})}}\) | M1 | 1.1b |
| \(r = \text{awrt}\ \mathbf{0.899}\) | A1 | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \rho_s = 0 \quad H_1: \rho_s > 0\) | B1 | 2.5 |
| Critical value \(\rho_s = 0.8286\) | B1 | 1.1b |
| \(r_s = 0.899\) lies in the critical region / reject \(H_0\) | M1 | 2.1 |
| There is positive rank correlation between high jump and long jump results | A1ft | 2.2b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([H_0: \rho = 0 \quad H_1: \rho > 0]\); \(0.678 <\) Critical value \(\rho = 0.7293\) | M1 | 2.1 |
| There is no evidence of (positive) correlation between high jump and long jump | A1 | 2.2b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The test in part (c) requires the data to come from a bivariate normal distribution | B1 | 2.3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Although there is evidence of a positive correlation between the ranks, the data does not appear to fit a linear pattern | B1 | 2.4 |
# Question 3:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Rank first row using tied ranks (at least 4 correct): A=1, B=5, C=2.5, D=4, E=6, F=2.5 | M1 | 1.1b |
| Rank second row (at least 4 correct): A=2, B=5, C=1, D=4, E=6, F=3 | M1 | 1.1b |
| $[\sum h^2 = 90.5,\ \sum l^2 = 91,\ \sum hl = 89]$ | | |
| Use of pmcc: $r_s = \dfrac{89 - \frac{21 \times 21}{6}}{\sqrt{(90.5 - \frac{21^2}{6})(91 - \frac{21^2}{6})}}$ | M1 | 1.1b |
| $r = \text{awrt}\ \mathbf{0.899}$ | A1 | 1.1b |
**(4 marks)**
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \rho_s = 0 \quad H_1: \rho_s > 0$ | B1 | 2.5 |
| Critical value $\rho_s = 0.8286$ | B1 | 1.1b |
| $r_s = 0.899$ lies in the critical region / reject $H_0$ | M1 | 2.1 |
| There is **positive** rank correlation between **high jump** and **long jump** results | A1ft | 2.2b |
**(4 marks)**
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[H_0: \rho = 0 \quad H_1: \rho > 0]$; $0.678 <$ Critical value $\rho = 0.7293$ | M1 | 2.1 |
| There is no evidence of (positive) correlation between high jump and long jump | A1 | 2.2b |
**(2 marks)**
## Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The test in part (c) requires the data to come from a bivariate normal distribution | B1 | 2.3 |
**(1 mark)**
## Part (e):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Although there is evidence of a positive correlation between the ranks, the data does not appear to fit a linear pattern | B1 | 2.4 |
**(1 mark)**
---\begin{enumerate}
\item The table below shows the heights cleared, in metres, for each of 6 competitors in a high jump competition.
\end{enumerate}
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Competitor & A & B & C & D & E & F \\
\hline
Height (m) & 2.05 & 1.93 & 2.02 & 1.96 & 1.81 & 2.02 \\
\hline
\end{tabular}
\end{center}
These 6 competitors also took part in a long jump competition and finished in the following order, with C jumping the furthest.\\
C\\
A\\
F\\
D\\
B\\
E\\
(a) Calculate Spearman's rank correlation coefficient for these data.\\
(b) Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not there is a positive correlation between results in the high jump and results in the long jump.
The product moment correlation coefficient between the height of the high jump and the length of the long jump for each competitor is found to be 0.678\\
(c) Use this value to test, at the $5 \%$ level of significance, for evidence of positive correlation between results in the high jump and results in the long jump.\\
(d) State the condition required for the test in part (c) to be valid.\\
(e) Explain what your conclusions in part (b) and part (c) suggest about the relationship between results in the high jump and results in the long jump.
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
V349 SIHI NI IMIMM ION OC & VJYV SIHIL NI LIIIM ION OO & VJYV SIHIL NI JIIYM ION OC \\
\hline
\hline
\end{tabular}
\end{center}
\hfill \mbox{\textit{Edexcel FS2 AS 2018 Q3 [12]}}