| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Hypothesis test for association |
| Difficulty | Standard +0.3 This is a straightforward application of Spearman's rank correlation coefficient with clear data, followed by a standard hypothesis test lookup and interpretation. Part (c) requires basic understanding of when to use rank vs product moment correlation, but no deep insight. Slightly easier than average due to small dataset (n=8) and routine procedure. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Temperature \(\left( { } ^ { \circ } \mathrm { C } \right)\) | 16 | 9 | 11 | 5 | 7 | 21 | 12 | 15 |
| Finishing position | 2 | 15 | 5 | 19 | 10 | 4 | 6 | 11 |
| Answer | Marks | Guidance |
|---|---|---|
| temp. | 16 | 9 |
| position | 2 | 15 |
| temp. rank | 2 | 6 |
| pos'n rank | 1 | 7 |
| \(d^2\) | 1 | 1 |
| \(\sum d^2 = 20\) | M2 A2 | |
| \(r_s = 1 - \frac{6 \times 20}{8 \times 63} = 0.7619\) | M1 A1 | |
| (b) \(H_0: \rho = 0\); \(H_1: \rho > 0\). \(n = 8\), 5% level \(\therefore\) C.R. is \(r_s > 0.6429\). \(0.7619 > 0.6429 \therefore\) significant. There is evidence that she will do better at higher temperatures | B1, M1 A1, A1 | |
| (c) e.g. this would not answer her query which relates to how well she does compared to others, all runners may be slower in higher temps | B2 | Total: 12 marks |
(a)
| temp. | 16 | 9 | 11 | 5 | 7 | 21 | 12 | 15 |
|-------|----|----|----|----|----|----|----|----|
| position | 2 | 15 | 5 | 19 | 10 | 4 | 6 | 11 |
| temp. rank | 2 | 6 | 5 | 8 | 7 | 1 | 4 | 3 |
| pos'n rank | 1 | 7 | 3 | 8 | 5 | 2 | 4 | 6 |
| $d^2$ | 1 | 1 | 4 | 0 | 4 | 1 | 0 | 9 |
$\sum d^2 = 20$ | M2 A2 |
$r_s = 1 - \frac{6 \times 20}{8 \times 63} = 0.7619$ | M1 A1 |
(b) $H_0: \rho = 0$; $H_1: \rho > 0$. $n = 8$, 5% level $\therefore$ C.R. is $r_s > 0.6429$. $0.7619 > 0.6429 \therefore$ significant. There is evidence that she will do better at higher temperatures | B1, M1 A1, A1 |
(c) e.g. this would not answer her query which relates to how well she does compared to others, all runners may be slower in higher temps | B2 | Total: 12 marks5. A marathon runner believes that she is more likely to win a medal at her national championships the higher the temperature is on the day of the race.
She records the temperature at the start of each of eight races against fields of a similar standard and her finishing position in each race. Her results are shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
Temperature $\left( { } ^ { \circ } \mathrm { C } \right)$ & 16 & 9 & 11 & 5 & 7 & 21 & 12 & 15 \\
\hline
Finishing position & 2 & 15 & 5 & 19 & 10 & 4 & 6 & 11 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate Spearman's rank correlation coefficient for these data.
\item Using a 5\% level of significance and stating your hypotheses clearly, interpret your result.
Another runner suggests that she should use her time in each race instead of her finishing position and calculate the product moment correlation coefficient for the data.
\item Comment on this suggestion.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q5 [12]}}