Standard +0.3 This is a straightforward application of the Wilcoxon signed-rank test with clear hypotheses (testing against a stated median of 30 minutes). Students must calculate differences, rank absolute values, sum ranks, and compare to critical values from tables. While it requires careful execution of the procedure and stating the symmetry assumption, it's a standard textbook exercise with no conceptual surprises—slightly easier than average for Further Maths S4 material.
2 The manufacturer of a painkiller, designed to relieve headaches, claims that people taking the painkiller feel relief in at most 30 minutes, on average. A random sample of eight users of the painkiller recorded the times it took for them to feel relief from their headaches. These times, in minutes, were as follows:
$$\begin{array} { l l l l l l l l }
33 & 39 & 29 & 35 & 40 & 32 & 26 & 37
\end{array}$$
Use a Wilcoxon single-sample signed-rank test at the \(5 \%\) significance level to test the manufacturer's claim, stating a necessary assumption.
Assumption: The differences are symmetrically distributed. Method: Calculate differences from hypothesized median of 30: \(3, 9, -1, 5, 10, 2, -4, 7\). Rank by absolute value: \(
| Wilcoxon signed-rank test performed correctly. | B1 M1 M1 M1 A1 A1 A1 A1 | **Assumption:** The differences are symmetrically distributed. **Method:** Calculate differences from hypothesized median of 30: $3, 9, -1, 5, 10, 2, -4, 7$. Rank by absolute value: $|-1| = 1, |2| = 2, |3| = 3, |4| = 4, |5| = 5, |7| = 6, |9| = 7, |10| = 8$. Sum of positive ranks: $3 + 9 + 5 + 10 + 2 + 7 = 36$. Sum of negative ranks: $1 + 4 = 5$. Test statistic $T = 5$ (the smaller). For $n = 8$ at 5% significance, critical value is 3. Since $5 > 3$, we fail to reject the null hypothesis. There is insufficient evidence to reject the manufacturer's claim that the mean time to relief is at most 30 minutes. |
2 The manufacturer of a painkiller, designed to relieve headaches, claims that people taking the painkiller feel relief in at most 30 minutes, on average. A random sample of eight users of the painkiller recorded the times it took for them to feel relief from their headaches. These times, in minutes, were as follows:
$$\begin{array} { l l l l l l l l }
33 & 39 & 29 & 35 & 40 & 32 & 26 & 37
\end{array}$$
Use a Wilcoxon single-sample signed-rank test at the $5 \%$ significance level to test the manufacturer's claim, stating a necessary assumption.
\hfill \mbox{\textit{OCR S4 2015 Q2 [8]}}