| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2016 |
| Session | June |
| Marks | 4 |
| Topic | Bivariate data |
| Type | Calculate r from raw bivariate data |
| Difficulty | Moderate -0.8 This is a straightforward calculation of the product moment correlation coefficient (PMCC) using given data. It requires systematic application of the standard formula with no conceptual difficulty—students simply need to compute sums, means, and apply the correlation formula. The answer is provided for verification, making it purely computational rather than requiring interpretation or problem-solving insight. |
| Spec | 5.08a Pearson correlation: calculate pmcc |
| Inflation rate (\%) | 0.9 | 1.2 | 1.6 | 1.5 | 1.7 | 3.0 | 4.1 | 3.7 | 2.8 | 4.2 |
| Pay increase (\%) | 4.8 | 4.7 | 3.8 | 4.4 | 5.6 | 5.5 | 2.4 | 0.4 | 0.6 | 1.7 |
| Answer | Marks | Guidance |
|---|---|---|
| \(S_{xy} = 68.8 - \frac{24.7 \times 33.9}{10} = -14.933\) | M1 | Use of formula for numerator. |
| \(S_{xx} = 74.93 - \frac{24.7^2}{10} = 13.921\) | M1 | Use of formula for either term in denominator. |
| \(S_{yy} = 149.71 - \frac{33.9^2}{10} = 34.789\) | M1 | Use of formula for either term in denominator. |
| \(r = \frac{-14.933}{\sqrt{13.921 \times 34.789}} = -0.678(56...) \approx -0.679\) | M1, A1 | Use of formula for \(r\). c.a.o. Must see unrounded value to at least 4sf first. |
$S_{xy} = 68.8 - \frac{24.7 \times 33.9}{10} = -14.933$ | M1 | Use of formula for numerator.
$S_{xx} = 74.93 - \frac{24.7^2}{10} = 13.921$ | M1 | Use of formula for either term in denominator.
$S_{yy} = 149.71 - \frac{33.9^2}{10} = 34.789$ | M1 | Use of formula for either term in denominator.
$r = \frac{-14.933}{\sqrt{13.921 \times 34.789}} = -0.678(56...) \approx -0.679$ | M1, A1 | Use of formula for $r$. c.a.o. Must see unrounded value to at least 4sf first. | [4]
With no working shown allow only correct answers.
---The following data refer to the annual rate of inflation and the annual percentage pay increase measured on 10 randomly chosen occasions.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Inflation rate (\%) & 0.9 & 1.2 & 1.6 & 1.5 & 1.7 & 3.0 & 4.1 & 3.7 & 2.8 & 4.2 \\
\hline
Pay increase (\%) & 4.8 & 4.7 & 3.8 & 4.4 & 5.6 & 5.5 & 2.4 & 0.4 & 0.6 & 1.7 \\
\hline
\end{tabular}
\end{center}
Show that, for these data, the product moment correlation coefficient between the rate of inflation and the annual pay increase is $-0.679$, correct to 3 significant figures. [4]
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2016 Q1 [4]}}