| Exam Board | WJEC |
|---|---|
| Module | Further Unit 2 (Further Unit 2) |
| Year | 2022 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | Two-tailed test for any correlation |
| Difficulty | Standard +0.3 This is a standard hypothesis test for correlation with routine calculations using given summary statistics, followed by straightforward interpretation questions. Part (a) requires the PMCC formula (textbook application), part (b) is a standard two-tailed test at 5% level, parts (c) and (d) test basic understanding of statistical concepts. While it's a multi-part question worth several marks, each component follows standard procedures with no novel problem-solving required, making it slightly easier than average for Further Maths Statistics. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Answer | Marks | Guidance |
|---|---|---|
| 2(a) | \(S_{xy} = 113.16 - \frac{62.8 \times 19.4}{10}\) | B1 |
| \(S_{xy} = -8.672\) | B1 | |
| \(S_{xx} = 413.44 - \frac{62.8^2}{10}\) | B1 | |
| \(S_{xx} = 19.056\) | B1 | |
| \(S_{yy} = 46.16 - \frac{19.4^2}{10}\) | B1 | |
| \(S_{yy} = 8.524\) | B1 | |
| \(r = \frac{-8.672}{\sqrt{19.056 \times 8.524}}\) | B1 | B1 for \(r\) |
| \(r = -0.68(0427 \ldots)\) | B1 | |
| 2(b) | \(H_0: \rho = 0\) \(H_1: \rho \neq 0\) | B1 |
| 5% two tail critical value \(= -0.6319\) | B1 | |
| Since \(-0.6804 < -0.6319\) reject \(H_0\). | B1 | |
| It suggests that the rate of unemployment and the rate of wages inflation are not independent. | E1 | Only award E1 if previous three B1 awarded E0 for categorical statements |
| 2(c) | Valid comment. e.g. This should cast doubt on Amy's opinion based on her answer in (b) Valid suggestion. e.g. She could look at more countries. She could come to different conclusions for different countries. She could consider more regions within each country | E1 |
| E1 | ||
| 2(d) | The underlying distribution is bivariate normal. The data come from a bivariate normal distribution. | E1 |
| Total [11] |
2(a) | $S_{xy} = 113.16 - \frac{62.8 \times 19.4}{10}$ | B1 | B1 for each of $S_{xy}, S_{xx}$ and $S_{yy}$
| $S_{xy} = -8.672$ | B1 |
| $S_{xx} = 413.44 - \frac{62.8^2}{10}$ | B1 |
| $S_{xx} = 19.056$ | B1 |
| $S_{yy} = 46.16 - \frac{19.4^2}{10}$ | B1 |
| $S_{yy} = 8.524$ | B1 |
| $r = \frac{-8.672}{\sqrt{19.056 \times 8.524}}$ | B1 | B1 for $r$
| $r = -0.68(0427 \ldots)$ | B1 |
2(b) | $H_0: \rho = 0$ $H_1: \rho \neq 0$ | B1 | FT their $r$ Accept in context Or CV = 0.6319 Or 0.6804 > 0.6319
| 5% two tail critical value $= -0.6319$ | B1 |
| Since $-0.6804 < -0.6319$ reject $H_0$. | B1 |
| It suggests that the rate of unemployment and the rate of wages inflation are not independent. | E1 | Only award E1 if previous three B1 awarded E0 for categorical statements
2(c) | Valid comment. e.g. This should cast doubt on Amy's opinion based on her answer in (b) Valid suggestion. e.g. She could look at more countries. She could come to different conclusions for different countries. She could consider more regions within each country | E1 | FT their conclusion from (b)
| | E1 |
2(d) | The underlying distribution is bivariate normal. The data come from a bivariate normal distribution. | E1 |
| **Total [11]** |
---2. An economist suggested the rate of unemployment and the rate of wage inflation are independent. Amy sets about investigating this suggestion. She collects unemployment data and wage inflation data from a random sample of regions in the UK and decides that it is appropriate to carry out a significance test on Pearson's product moment correlation coefficient.
Amy's summary statistics for percentage unemployment, $x$, and percentage wage inflation, $y$, are shown below.
$$\begin{array} { l l l }
\sum x = 62 \cdot 8 & \sum y = 19 \cdot 4 & n = 10 \\
\sum x ^ { 2 } = 413 \cdot 44 & \sum y ^ { 2 } = 46 \cdot 16 & \sum x y = 113 \cdot 16
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Calculate Pearson's product moment correlation coefficient for these data.
\item Carry out Amy's test at the $5 \%$ level of significance and state whether the economist's suggestion is reasonable.
Amy also collects unemployment data and wage inflation data from a random sample of 10 regions in Spain and calculates Pearson's product moment correlation coefficient to be - 0.2525 .
\item Should this change Amy's opinion on the economist's suggestion above? What could she do to improve her investigation?
\item What assumption has Amy made in deciding that it is appropriate to carry out a significance test on Pearson's product moment correlation coefficient?
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 2 2022 Q2 [11]}}