| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Paired t-test |
| Difficulty | Challenging +1.3 This is a straightforward paired t-test application from S4 (Further Maths Statistics). Students must state an assumption, set up one-sided hypotheses with a specific difference (10 minutes = 1/6 hour), calculate differences, perform the test, and find a critical value. While it requires careful handling of the '10 minutes' detail and is from a Further Maths module, it follows a standard template with clear data and no conceptual surprises, making it moderately above average difficulty. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07d Paired vs two-sample: selection |
| Person | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) |
| Hours of sleep with drug | 10.8 | 7.2 | 8.7 | 6.8 | 9.4 | 10.9 | 11.1 | 7.6 |
| Hours of sleep with placebo | 10.0 | 6.5 | 9.0 | 5.6 | 8.7 | 8.0 | 9.8 | 6.8 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| The differences (drug − placebo) are normally distributed | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Differences \(d\) (drug − placebo): \(0.8, 0.7, -0.3, 1.2, 0.7, 2.9, 1.3, 0.8\) | M1 | |
| \(\bar{d} = \frac{8.1}{8} = 1.0125\) | A1 | |
| \(s_d^2 = \frac{1}{7}\left(\sum d^2 - \frac{(\sum d)^2}{8}\right)\), \(\sum d^2 = 12.75\) | M1 | |
| \(s_d^2 = \frac{1}{7}\left(12.75 - \frac{65.61}{8}\right) = \frac{1}{7}(12.75 - 8.20125) = 0.6498\) | A1 | |
| \(H_0: \mu_d = \frac{10}{60} = \frac{1}{6}\), \(H_1: \mu_d > \frac{1}{6}\) | B1 | (i.e. \(\approx 0.1\overline{6}\) hours) |
| \(t = \frac{1.0125 - \frac{1}{6}}{\frac{\sqrt{0.6498}}{\sqrt{8}}} = \frac{0.8458}{0.2851} = 2.967\) | M1 A1 | |
| Critical value: \(t_7 = 2.998\) (one-tailed, 1%) | B1 | |
| Since \(2.967 < 2.998\), do not reject \(H_0\). Insufficient evidence that drug increases sleep by more than 10 minutes | A1 | Correct contextual conclusion |
# Question 4:
**(a)**
| Answer | Mark | Guidance |
|--------|------|----------|
| The differences (drug − placebo) are normally distributed | B1 | |
**(b)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Differences $d$ (drug − placebo): $0.8, 0.7, -0.3, 1.2, 0.7, 2.9, 1.3, 0.8$ | M1 | |
| $\bar{d} = \frac{8.1}{8} = 1.0125$ | A1 | |
| $s_d^2 = \frac{1}{7}\left(\sum d^2 - \frac{(\sum d)^2}{8}\right)$, $\sum d^2 = 12.75$ | M1 | |
| $s_d^2 = \frac{1}{7}\left(12.75 - \frac{65.61}{8}\right) = \frac{1}{7}(12.75 - 8.20125) = 0.6498$ | A1 | |
| $H_0: \mu_d = \frac{10}{60} = \frac{1}{6}$, $H_1: \mu_d > \frac{1}{6}$ | B1 | (i.e. $\approx 0.1\overline{6}$ hours) |
| $t = \frac{1.0125 - \frac{1}{6}}{\frac{\sqrt{0.6498}}{\sqrt{8}}} = \frac{0.8458}{0.2851} = 2.967$ | M1 A1 | |
| Critical value: $t_7 = 2.998$ (one-tailed, 1%) | B1 | |
| Since $2.967 < 2.998$, do not reject $H_0$. Insufficient evidence that drug increases sleep by more than 10 minutes | A1 | Correct contextual conclusion |\begin{enumerate}
\item A random sample of 8 people were given a new drug designed to help people sleep.
\end{enumerate}
In a two-week period the drug was given for one week and a placebo (a tablet that contained no drug) was given for one week.
In the first week 4 people, selected at random, were given the drug and the other 4 people were given the placebo. Those who were given the drug in the first week were given the placebo in the second week. Those who were given the placebo in the first week were given the drug in the second week.
The mean numbers of hours of sleep per night for each of the people are shown in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | }
\hline
Person & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ \\
\hline
Hours of sleep with drug & 10.8 & 7.2 & 8.7 & 6.8 & 9.4 & 10.9 & 11.1 & 7.6 \\
\hline
Hours of sleep with placebo & 10.0 & 6.5 & 9.0 & 5.6 & 8.7 & 8.0 & 9.8 & 6.8 \\
\hline
\end{tabular}
\end{center}
(a) State one assumption that needs to be made in order to carry out a paired $t$-test.\\
(b) Stating your hypotheses clearly, test, at the $1 \%$ level of significance, whether or not the drug increases the mean number of hours of sleep per night by more than 10 minutes. State the critical value for this test.\\
\hfill \mbox{\textit{Edexcel S4 2014 Q4 [9]}}