| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon rank-sum test (Mann-Whitney U test) |
| Difficulty | Standard +0.3 This is a straightforward application of the Wilcoxon rank-sum test with clear data and standard hypothesis. Students must rank combined data, sum ranks for one group, and compare to critical values. While tedious with 23 values, it requires only methodical execution of a learned procedure with no conceptual challenges or novel insights. |
| Spec | 5.07d Paired vs two-sample: selection |
| Women | 51 | 55 | 242 | 167 | 152 | 256 | 75 | 137 | 98 | 238 | 235 | |
| Men | 311 | 262 | 170 | 302 | 175 | 320 | 220 | 260 | 72 | 351 | 86 | 333 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Ranked table: \(x\)-values ranked 1–16, \(y\)-values ranked 3–23 (as shown) | M1 | Attempt at ranking |
| Total ranks: 95 | A1 | |
| \(H_0: m_x = m_y\) and \(H_1: m_x \neq m_y\) | B1 | Allow words, must include 'population' median |
| Use normal approximation with attempts at mean and variance | M1 | |
| Mean \(=132\), variance \(=264\) | A1 | |
| \(\frac{95.5-132}{\sqrt{264}}\) | M1 | Allow no or wrong continuity correction for M1 only |
| \(-2.246\) | A1 | CAO |
| Critical value is \(-1.96\); \(-2.246 < -1.96\), reject \(H_0\) | M1 | Compare their value with \(-1.96\), or area comparison \(0.0123\) or \(0.0124\) with \(0.025\) and FT conclusion |
| There is sufficient evidence of a difference in levels | A1 | Correct conclusion, in context, following correct work. Level of uncertainty in language is used |
| Total: 9 |
## Question 6:
| Answer | Mark | Guidance |
|--------|------|----------|
| Ranked table: $x$-values ranked 1–16, $y$-values ranked 3–23 (as shown) | M1 | Attempt at ranking |
| Total ranks: 95 | A1 | |
| $H_0: m_x = m_y$ and $H_1: m_x \neq m_y$ | B1 | Allow words, must include 'population' median |
| Use normal approximation with attempts at mean and variance | M1 | |
| Mean $=132$, variance $=264$ | A1 | |
| $\frac{95.5-132}{\sqrt{264}}$ | M1 | Allow no or wrong continuity correction for M1 only |
| $-2.246$ | A1 | CAO |
| Critical value is $-1.96$; $-2.246 < -1.96$, reject $H_0$ | M1 | Compare their value with $-1.96$, or area comparison $0.0123$ or $0.0124$ with $0.025$ and FT conclusion |
| There is sufficient evidence of a difference in levels | A1 | Correct conclusion, in context, following correct work. Level of uncertainty in language is used |
| **Total: 9** | | |6 The blood cholesterol levels, measured in suitable units, of a random sample of 11 women and a random sample of 12 men are shown below.
\begin{center}
\begin{tabular}{ l r r r r r r r r r r r r }
Women & 51 & 55 & 242 & 167 & 152 & 256 & 75 & 137 & 98 & 238 & 235 & \\
Men & 311 & 262 & 170 & 302 & 175 & 320 & 220 & 260 & 72 & 351 & 86 & 333 \\
\end{tabular}
\end{center}
Carry out a Wilcoxon rank-sum test, at the $5 \%$ significance level, to test whether, on average, there is a difference in cholesterol levels between women and men.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2021 Q6 [9]}}