| Exam Board | Edexcel |
|---|---|
| Module | FS2 (Further Statistics 2) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Paired t-test |
| Difficulty | Standard +0.3 This is a straightforward application of a paired t-test with clear context. Part (a) requires recognizing paired data structure, (b) needs stating the normality assumption, and (c) involves standard hypothesis testing procedure with given data. The calculations are routine (finding differences, mean, standard deviation, t-statistic) and the question provides all necessary information. Slightly easier than average because it's a textbook application with no conceptual surprises, though it does require proper statistical reasoning and multiple steps. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Person | A | \(B\) | C | \(D\) | E | \(F\) | G | \(H\) |
| Percentage of words recalled before training | 24 | 33 | 33 | 39 | 30 | 38 | 32 | 34 |
| Percentage of words recalled after training | 28 | 30 | 37 | 41 | 32 | 44 | 35 | 34 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| For each person the scores before and after are not independent | B1 | Correct explanation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The differences in scores are normally distributed. | B1 | Correct modelling assumption |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(d\): 4 −3 4 2 2 6 3 0 | M1 | For setting up paired \(t\)-test with at least 5 correct differences consistent in direction |
| \(\bar{d} = \pm 2.25\), \(s_d = \sqrt{7.642857...} = 2.76457...\) | M1 | Complete method for \(\bar{d}\) and \(s_d\) or \(s_d^2\) |
| \(H_0: \mu_d = 0\), \(H_1: \mu_d > 0\) | B1 | Use of \(t\)-distribution for differences with both hypotheses correct in terms of \(\mu/\mu_d\). Allow \(\mu_d < 0\) if candidate does test with negative test statistic |
| \(t = \pm\frac{\text{"}\pm2.25\text{"}}{\frac{\text{"2.76..."}}{\sqrt{8}}}\) | M1 | Method for finding test statistic with their values |
| \(= \pm 2.3019...\) awrt \(\pm \mathbf{2.30}\) | A1 | awrt \(\pm\)2.30 |
| Critical value \(t_7 = \pm 1.895\) | B1 | Correct critical \(\pm\)1.895 (or better) with compatible sign |
| \(2.30 > 1.895\), therefore (reject \(H_0\)) there is sufficient evidence to support an increase in the percentage of words recalled / support psychologist's claim | A1ft | Drawing correct inference in context. Must mention "percentage" and "words" or "psychologist's claim". Must be consistent with CV and standardised test statistic from \(t\)-distribution |
# Question 5:
## Part 5(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| For each person the scores before and after are not independent | B1 | Correct explanation |
## Part 5(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The differences in scores are normally distributed. | B1 | Correct modelling assumption |
## Part 5(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $d$: 4 −3 4 2 2 6 3 0 | M1 | For setting up paired $t$-test with at least 5 correct differences consistent in direction |
| $\bar{d} = \pm 2.25$, $s_d = \sqrt{7.642857...} = 2.76457...$ | M1 | Complete method for $\bar{d}$ and $s_d$ or $s_d^2$ |
| $H_0: \mu_d = 0$, $H_1: \mu_d > 0$ | B1 | Use of $t$-distribution for differences with both hypotheses correct in terms of $\mu/\mu_d$. Allow $\mu_d < 0$ if candidate does test with negative test statistic |
| $t = \pm\frac{\text{"}\pm2.25\text{"}}{\frac{\text{"2.76..."}}{\sqrt{8}}}$ | M1 | Method for finding test statistic with their values |
| $= \pm 2.3019...$ awrt $\pm \mathbf{2.30}$ | A1 | awrt $\pm$**2.30** |
| Critical value $t_7 = \pm 1.895$ | B1 | Correct critical $\pm$**1.895** (or better) with compatible sign |
| $2.30 > 1.895$, therefore (reject $H_0$) there is sufficient evidence to support an increase in the percentage of words recalled / support psychologist's claim | A1ft | Drawing correct inference in context. Must mention "percentage" and "words" or "psychologist's claim". Must be consistent with CV and standardised test statistic from $t$-distribution |\begin{enumerate}
\item A psychologist claims to have developed a technique to improve a person's memory.
\end{enumerate}
A random sample of 8 people are each given the same list of words to memorise and recall.
Each person then receives memory training from the psychologist. After the training, each person is given the same list of new words to memorise and recall.
The table shows the percentage of words recalled by each person before and after the training.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
Person & A & $B$ & C & $D$ & E & $F$ & G & $H$ \\
\hline
Percentage of words recalled before training & 24 & 33 & 33 & 39 & 30 & 38 & 32 & 34 \\
\hline
Percentage of words recalled after training & 28 & 30 & 37 & 41 & 32 & 44 & 35 & 34 \\
\hline
\end{tabular}
\end{center}
(a) State why a paired $t$-test is suitable for these data.\\
(b) State an assumption that needs to be made in order to carry out a paired $t$-test in this case.\\
(c) Test, at the $5 \%$ level of significance, whether or not there is evidence of an increase in the percentage of words recalled after receiving the psychologist's training. State your hypotheses, test statistic and critical value used for this test.
\hfill \mbox{\textit{Edexcel FS2 2023 Q5 [9]}}