| Exam Board | Edexcel |
|---|---|
| Module | FS2 (Further Statistics 2) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Paired t-test |
| Difficulty | Standard +0.8 This is a Further Maths Statistics question requiring students to recognize that a Wilcoxon signed-rank test is appropriate (not a paired t-test despite the topic label), perform the ranking procedure correctly, calculate the test statistic, find the critical value from tables, and state the assumption about symmetry of differences. While methodical, it requires careful execution of a non-standard procedure and understanding of when non-parametric tests apply, placing it moderately above average difficulty. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Patient | \(A\) | \(B\) | C | \(D\) | \(E\) | \(F\) | \(G\) |
| Resting heart rate before | 65 | 68 | 77 | 79 | 80 | 88 | 92 |
| Resting heart rate after | 63 | 65 | 73 | 76 | 80 | 84 | 80 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(d\): \(2\ \ 3\ \ 4\ \ 3\ \ 0\ \ 4\ \ 12\) | M1 | Understanding paired \(t\)-test is required; at least 5 differences correct, allow \(\pm\) |
| \(\bar{d} = \pm 4\), \(\quad s_d = \sqrt{\frac{1}{6}(198 - 7(4)^2)} = \sqrt{14.333...} = 3.7859...\) | M1 | Complete method for \(\bar{d}\) and \(s_d\) or \((s_d)^2\) |
| \(H_0: \mu_d = 0\), \(\quad H_1: \mu_d > 0\) | B1 | Correct model for differences with both hypotheses in terms of \(\mu / \mu_d\); sign of \(H_1\) must be compatible with \(\bar{d}\) |
| \(t = \pm\frac{4}{\frac{3.78...}{\sqrt{7}}}\) | M1 | Method for finding test statistic |
| \(= \pm 2.795...\) awrt \(\pm 2.80\) | A1 | awrt \(\pm 2.80\) (allow awrt \(\pm 2.8\) from correct working) |
| Critical value \(t_6 = \pm 1.943\) | B1 | Correct critical value \(\pm 1.943\) (or better) with compatible sign |
| \([2.80 > 1.943]\) therefore sufficient evidence to support the doctor's belief/evidence to suggest resting heart rate is reduced. | A1 | Correct comparison and inference in context; consistent with CV and test statistic |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Differences in resting heart rates must be normally distributed for the test to be valid. | B1 | Correct modelling assumption |
# Question 4:
## Part 4(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $d$: $2\ \ 3\ \ 4\ \ 3\ \ 0\ \ 4\ \ 12$ | M1 | Understanding paired $t$-test is required; at least 5 differences correct, allow $\pm$ |
| $\bar{d} = \pm 4$, $\quad s_d = \sqrt{\frac{1}{6}(198 - 7(4)^2)} = \sqrt{14.333...} = 3.7859...$ | M1 | Complete method for $\bar{d}$ and $s_d$ or $(s_d)^2$ |
| $H_0: \mu_d = 0$, $\quad H_1: \mu_d > 0$ | B1 | Correct model for differences with both hypotheses in terms of $\mu / \mu_d$; sign of $H_1$ must be compatible with $\bar{d}$ |
| $t = \pm\frac{4}{\frac{3.78...}{\sqrt{7}}}$ | M1 | Method for finding test statistic |
| $= \pm 2.795...$ awrt $\pm 2.80$ | A1 | awrt $\pm 2.80$ (allow awrt $\pm 2.8$ from correct working) |
| Critical value $t_6 = \pm 1.943$ | B1 | Correct critical value $\pm 1.943$ (or better) with compatible sign |
| $[2.80 > 1.943]$ therefore sufficient evidence to support the doctor's belief/evidence to suggest resting heart rate is reduced. | A1 | Correct comparison and inference in context; consistent with CV and test statistic |
**SC:** Difference of means test apply scheme but also allow $2^{\text{nd}}$ B1 for $t_{12} = 1.782$
## Part 4(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Differences in resting heart rates must be normally distributed for the test to be valid. | B1 | Correct modelling assumption |
---\begin{enumerate}
\item A doctor believes that a four-week exercise programme can reduce the resting heart rate of her patients. She takes a random sample of 7 patients and records their resting heart rate before the exercise programme and again after the exercise programme.
\end{enumerate}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
Patient & $A$ & $B$ & C & $D$ & $E$ & $F$ & $G$ \\
\hline
Resting heart rate before & 65 & 68 & 77 & 79 & 80 & 88 & 92 \\
\hline
Resting heart rate after & 63 & 65 & 73 & 76 & 80 & 84 & 80 \\
\hline
\end{tabular}
\end{center}
(a) Using a $5 \%$ level of significance, carry out an appropriate test of the doctor's belief. You should state your hypotheses, test statistic and critical value.\\
(b) State the assumption made about the resting heart rates that was required to carry out the test.
\hfill \mbox{\textit{Edexcel FS2 2022 Q4 [8]}}