| Exam Board | AQA |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | One-tailed test for positive correlation |
| Difficulty | Moderate -0.3 This is a straightforward application of standard formulas for correlation coefficient and hypothesis testing. Part (a) requires direct substitution into r = S_xy/√(S_xx·S_yy), part (b) is a routine one-tailed test comparing r to critical values from tables, and part (c) asks for basic interpretation. While it requires knowledge of hypothesis testing procedures, there are no conceptual challenges or problem-solving elements beyond following the standard algorithm. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Answer | Marks | Guidance |
|---|---|---|
| Part (a) | ||
| \(r = \frac{1.3781}{\sqrt{7.0036 \times 0.8464}} = 0.56 \text{ to } 0.57\) | M1; A1 | AWFW (0.56602) |
| Part (b) | ||
| \(H_0: \rho = 0\); \(H_1: \rho > 0\) | B1 | Both |
| \(\text{SL } \alpha = 0.01 \text{ (1%)}\); \(\text{CV } r = 0.515 \text{ to } 0.516\) | B1 | AWFW (0.5155) |
| Calculated \(r >\) Tabulated \(r\) | M1 | Comparison |
| Evidence, at 1% level, of a positive correlation between \(x\) and \(y\) | A1↑ | ft on \(r\) and CV; 4 marks |
| Part (b) Special Case | ||
| \(\text{CV } t_{n-2}(0.99) = 2.552\) | (B1) | |
| \(r\sqrt{\frac{n-2}{1-r^2}} = 2.913\) | (B1) | |
| Part (c) | ||
| (Strong) evidence of a positive correlation between best performances of junior athletes in the long jump and in the high jump | B1↑ | ft on (b); or equivalent; 1 mark |
| **Part (a)** |
|---|
| $r = \frac{1.3781}{\sqrt{7.0036 \times 0.8464}} = 0.56 \text{ to } 0.57$ | M1; A1 | AWFW (0.56602) |
| **Part (b)** |
|---|
| $H_0: \rho = 0$; $H_1: \rho > 0$ | B1 | Both |
| $\text{SL } \alpha = 0.01 \text{ (1%)}$; $\text{CV } r = 0.515 \text{ to } 0.516$ | B1 | AWFW (0.5155) |
| Calculated $r >$ Tabulated $r$ | M1 | Comparison |
| Evidence, at 1% level, of a positive correlation between $x$ and $y$ | A1↑ | ft on $r$ and CV; 4 marks |
| **Part (b) Special Case** |
|---|
| $\text{CV } t_{n-2}(0.99) = 2.552$ | (B1) | |
| $r\sqrt{\frac{n-2}{1-r^2}} = 2.913$ | (B1) | |
| **Part (c)** |
|---|
| (Strong) evidence of a positive correlation between best performances of junior athletes in the long jump and in the high jump | B1↑ | ft on (b); or equivalent; 1 mark |
**Total: 7 marks**
---1 The best performances of a random sample of 20 junior athletes in the long jump, $x$ metres, and in the high jump, $y$ metres, were recorded. The following statistics were calculated from the results.
$$S _ { x x } = 7.0036 \quad S _ { y y } = 0.8464 \quad S _ { x y } = 1.3781$$
\begin{enumerate}[label=(\alph*)]
\item Calculate the value of the product moment correlation coefficient between $x$ and $y$.\\
(2 marks)
\item Assuming that these data come from a bivariate normal distribution, investigate, at the $1 \%$ level of significance, the claim that for junior athletes there is a positive correlation between $x$ and $y$.
\item Interpret your conclusion in the context of this question.
\end{enumerate}
\hfill \mbox{\textit{AQA S3 2008 Q1 [7]}}