| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | Calculate PMCC from summary statistics |
| Difficulty | Standard +0.3 This is a standard S3 correlation question requiring formula application for PMCC, a one-tailed hypothesis test using critical values, and stating the bivariate normality assumption. All steps are routine and follow textbook procedures with no conceptual challenges beyond correct formula substitution and table lookup. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Answer | Marks |
|---|---|
| (a) \(S_{pp} = 70932 − \frac{1126^2}{65} = 1783.2\) | M1 |
| \(S_{ff} = 19213 − \frac{511^2}{20} = 6156.95\) | M1 |
| \(S_{pf} = 27188 − \frac{1126×511}{20} = \text{−}2858.8\) | M1 |
| \(r = \frac{\text{−}2858.8}{\sqrt{1783.2×6156.95}} = \text{−}0.8628\) | M1 A1 |
| (b) \(H_0 : \rho = 0\) \(\quad\) \(H_1 : \rho < 0\) | B1 |
| \(n = 20\), 1% level \(\therefore\) C.R. is \(r < \text{−}0.5155\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| there is evidence that people with lower rest pulse are fitter | A1 | |
| (c) variables need to be jointly normally distributed | B1 | |
| e.g. it seems reasonable that the fitness of those with a given rest pulse should follow a normal dist. and vice versa | B1 | (11) |
**(a)** $S_{pp} = 70932 − \frac{1126^2}{65} = 1783.2$ | M1 |
$S_{ff} = 19213 − \frac{511^2}{20} = 6156.95$ | M1 |
$S_{pf} = 27188 − \frac{1126×511}{20} = \text{−}2858.8$ | M1 |
$r = \frac{\text{−}2858.8}{\sqrt{1783.2×6156.95}} = \text{−}0.8628$ | M1 A1 |
**(b)** $H_0 : \rho = 0$ $\quad$ $H_1 : \rho < 0$ | B1 |
$n = 20$, 1% level $\therefore$ C.R. is $r < \text{−}0.5155$ | M1 A1 |
−0.8628 < −0.5155 $\therefore$ significant
there is evidence that people with lower rest pulse are fitter | A1 |
**(c)** variables need to be jointly normally distributed | B1 |
e.g. it seems reasonable that the fitness of those with a given rest pulse should follow a normal dist. and vice versa | B1 | (11)A sports scientist wishes to examine the link between resting pulse and fitness. He records the resting pulse, $p$, of 20 volunteers and the length of time, $t$ minutes, that each one can run comfortably at 4 metres per second on a treadmill. The results are summarised by
$$\Sigma p = 1176, \quad \Sigma t = 511, \quad \Sigma p^2 = 70932, \quad \Sigma t^2 = 19213, \quad \Sigma pt = 27188.$$
\begin{enumerate}[label=(\alph*)]
\item Calculate the product moment correlation coefficient for these data. [5 marks]
\item Stating your hypotheses clearly, test at the 1\% level of significance whether there is evidence of people with a lower resting pulse having a higher level of fitness as measured by the test. [4 marks]
\item State an assumption necessary to carry out the test in part (b) and comment on its validity in this case. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q7 [11]}}