| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Year | 2023 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon matched-pairs signed-rank test |
| Difficulty | Standard +0.3 This is a straightforward application of the Wilcoxon matched-pairs signed-rank test with clear data and standard procedure. Students must calculate differences, rank them, sum ranks, and compare to tables—all routine steps for Further Maths Statistics. The context is accessible and no novel insight is required, making it slightly easier than average for an A-level question. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank |
| Contestant | A | B | C | D | E | F | G | H | 1 |
| Before | 480 | 1008 | 0 | 344 | 351 | 781 | 876 | 741 | 457 |
| After | 841 | 998 | 751 | 344 | 954 | 542 | 820 | 1011 | 644 |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) | \(H_0\): The population median difference between the number of social media followers before and after appearing on the television show is 0. | B1 |
| \(H_1\): The population median difference, when subtracting the number of social media followers before appearing on the show from the number of social media followers after appearing on the show, is positive. | ||
| OR \(H_0: \eta_d = 0\) and \(H_1: \eta_d > 0\) where \(\eta_d\) is the median difference in numbers of followers before and after appearing on the show, \(\eta_d = \eta_{\text{after}} - \eta_{\text{before}}\) | ||
| Contestant | A | B |
| Difference | 361 | -10 |
| Ranks | ||
| Contestant | A | B |
| Ranks | 6 | 1 |
| \(W^+ = \) Sum of positive ranks \((W^- = \) Sum of negative ranks\()\) | M1 A1 | |
| \(= 6 + 8 + 7 + 5 + 3 = 29\) | ( \(= 1 + 4 + 2 = 7\)) | |
| \(n = 8\) | ||
| Upper CV \(= 28\) | ||
| Since \(29 > 28\) (OR \(7 < 8\)) there is sufficient evidence to reject \(H_0\). | m1 | |
| There is evidence to suggest that appearing on the show may increase the number of social media followers Lil'r has. | A1 | cso |
| (ii) | Valid comment e.g. He should apply to appear on the show if he wants more social media followers. e.g. He should not apply to appear on the show if he doesn't want more social media followers. | E1 |
| (b) | The underlying distribution of the differences may not be normally distributed. | E1 |
| Data are paired. | E1 | |
| Total [12] |
(a)(i) | $H_0$: The population median difference between the number of social media followers before and after appearing on the television show is 0. | B1 | Both oe |
| $H_1$: The population median difference, when subtracting the number of social media followers before appearing on the show from the number of social media followers after appearing on the show, is positive. | | |
| OR $H_0: \eta_d = 0$ and $H_1: \eta_d > 0$ where $\eta_d$ is the median difference in numbers of followers before and after appearing on the show, $\eta_d = \eta_{\text{after}} - \eta_{\text{before}}$ | | |
| | | |
| Contestant | A | B | C | D | E | F | G | H | I | B1 | Accept differences with opposite signs. |
| Difference | 361 | -10 | 751 | 0 | 603 | -239 | -56 | 270 | 187 | | |
| | | | | | | | | | | | |
| Ranks | | | | | | | | | | M1 A1 | M1 either attempt at ranks. FT one slip in difference for A1 |
| Contestant | A | B | C | D | E | F | G | H | I | | |
| Ranks | 6 | 1 | 8 | - | 7 | 4 | 2 | 5 | 3 | | |
| | | | | | | | | | | | |
| $W^+ = $ Sum of positive ranks $(W^- = $ Sum of negative ranks$)$ | M1 A1 | |
| $= 6 + 8 + 7 + 5 + 3 = 29$ | | ( $= 1 + 4 + 2 = 7$) | | |
| $n = 8$ | | | | | | | | | | | |
| Upper CV $= 28$ | | | ( Lower CV $= 8$) | B1 | Must match direction of $H_1$ |
| | | | | | | | | | | | |
| Since $29 > 28$ (OR $7 < 8$) there is sufficient evidence to reject $H_0$. | m1 | |
| There is evidence to suggest that appearing on the show may increase the number of social media followers Lil'r has. | A1 | cso |
(ii) | Valid comment e.g. He should apply to appear on the show if he wants more social media followers. e.g. He should not apply to appear on the show if he doesn't want more social media followers. | E1 | |
(b) | The underlying distribution of the differences may not be normally distributed. | E1 | Or equivalent statement |
| Data are paired. | E1 | |
| **Total [12]** | | |
---4. Llŷr believes that he will have more social media followers by appearing on a certain Welsh television show. To investigate his belief, he collects data on 9 randomly selected contestants who have appeared on the show. Llŷr records the number of social media followers one week before and one week after the contestants appeared on the show. The data he collects are shown in the table below.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
Contestant & A & B & C & D & E & F & G & H & 1 \\
\hline
Before & 480 & 1008 & 0 & 344 & 351 & 781 & 876 & 741 & 457 \\
\hline
After & 841 & 998 & 751 & 344 & 954 & 542 & 820 & 1011 & 644 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Carry out a Wilcoxon signed-rank test on this data set, at a significance level as close to 10\% as possible.
\item Suggest a possible course of action that Llŷr might take.
\end{enumerate}\item Give two reasons why the Wilcoxon signed-rank test is appropriate in this case.
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 5 2023 Q4 [12]}}