| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2020 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Justify use of Spearman's |
| Difficulty | Standard +0.3 This is a straightforward application of Spearman's rank correlation with only 6 data pairs. Part (a) requires recognizing that the data are totals/ordinal rather than bivariate normal, part (b) involves routine ranking and table lookup with n=6, and part (c) tests definition recall. The calculations are simple and the conceptual demands are minimal for Further Statistics students. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Aspect | Couples | Single travellers |
| Organisation | 884 | 867 |
| Travel | 710 | 633 |
| Food | 692 | 675 |
| Leader | 898 | 898 |
| Included visits | 561 | 736 |
| Optional visits | 683 | 712 |
| Answer | Marks | Guidance |
|---|---|---|
| Test is for rankings/rankings arbitrary/not bivariate normal etc | B1 [1] | OE |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \rho_s = 0,\ H_1: \rho_s > 0\), where \(\rho_s\) is the population rank correlation coefficient | B1 | Allow \(\rho_s\) not defined; allow \(\rho\). Allow: \(H_0\): no association between rankings; \(H_1\): positive association (but *not* \(H_1\): association) |
| Ranks \(5\ 4\ 3\ 6\ 1\ 2\) and \(5\ 1\ 2\ 6\ 4\ 3\) | ||
| \(\Sigma d^2 = 20\) | B1 | |
| \(r_s = 1 - \frac{6 \times 20}{6 \times 35}\) | M1 | |
| \(= 3/7\) or \(0.42857\ldots\) | A1 | Exact or awrt \(0.429\) |
| \(< 0.9429\) | B1 | |
| Do not reject \(H_0\) | M1ft | FT on their \(\Sigma d^2\) only |
| Insufficient evidence of association between ranking given by the two categories | A1ft [7] | Contextualised, not too assertive |
| Answer | Marks | Guidance |
|---|---|---|
| Not dependent on any distributional assumptions | B1 [1] | Oe (*cf.* Specification, 5.08f) |
# Question 4:
## Part (a)
Test is for rankings/rankings arbitrary/not bivariate normal etc | **B1** [1] | OE
## Part (b)
$H_0: \rho_s = 0,\ H_1: \rho_s > 0$, where $\rho_s$ is the population rank correlation coefficient | **B1** | Allow $\rho_s$ not defined; allow $\rho$. Allow: $H_0$: no association between rankings; $H_1$: positive association (but *not* $H_1$: association)
Ranks $5\ 4\ 3\ 6\ 1\ 2$ and $5\ 1\ 2\ 6\ 4\ 3$ | |
$\Sigma d^2 = 20$ | **B1** |
$r_s = 1 - \frac{6 \times 20}{6 \times 35}$ | **M1** |
$= 3/7$ or $0.42857\ldots$ | **A1** | Exact or awrt $0.429$
$< 0.9429$ | **B1** |
Do not reject $H_0$ | **M1ft** | FT on their $\Sigma d^2$ only
Insufficient evidence of association between ranking given by the two categories | **A1ft** [7] | Contextualised, not too assertive
## Part (c)
Not dependent on any distributional assumptions | **B1** [1] | Oe (*cf.* Specification, 5.08f)
---4 After a holiday organised for a group, the company organising the holiday obtained scores out of 10 for six different aspects of the holiday. The company obtained responses from 100 couples and 100 single travellers. The total scores for each of the aspects are given in the following table.
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Aspect & Couples & Single travellers \\
\hline
Organisation & 884 & 867 \\
\hline
Travel & 710 & 633 \\
\hline
Food & 692 & 675 \\
\hline
Leader & 898 & 898 \\
\hline
Included visits & 561 & 736 \\
\hline
Optional visits & 683 & 712 \\
\hline
\end{tabular}
\end{center}
Fred wishes to test whether there is significant positive correlation between the scores given by the two categories.
\begin{enumerate}[label=(\alph*)]
\item Explain why it is probably not appropriate to use Pearson's product-moment correlation coefficient.
\item Carry out an appropriate test at the $1 \%$ level.
\item Explain what is meant by the statement that the test carried out in part (b) is a non-parametric test.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics AS 2020 Q4 [9]}}