| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2006 |
| 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 paired t-test application with clear data and standard hypotheses. Students must calculate differences, perform the test procedure, and state a normality assumption. While it requires multiple steps (8 marks), all are routine S4 techniques with no novel problem-solving or conceptual challenges beyond standard textbook exercises. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Week | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Without solar heating | 19 | 19 | 18 | 14 | 6 | 7 | 5 | 31 | 43 |
| With solar heating | 13 | 22 | 11 | 16 | 14 | 1 | 0 | 20 | 38 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \mu_D = 0\), \(H_1: \mu_D > 0\) | B1 | |
| \(d: 6, -3, 7, -2, -8, 6, 5, 11, 5\) | M1 | Attempt differences |
| \(\bar{d} = 3\), \(s_d = 6\) \(\left(=\sqrt{\frac{369-9\times3^2}{8}}\right)\) \(\left(\frac{\Sigma d}{9}\right)\) | M1, M1 | |
| \(t_8 = \frac{3-0}{\frac{6}{\sqrt{9}}} = 1.5\) | M1A1 cao | \((\pm)\) |
| \(t_8\) (5% 1-tail c.v.) \(= 1.860\) | B1 | |
| Not significant – insufficient evidence that solar heating has decreased weekly fuel consumption | A1ft | (8) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Difference in weekly fuel consumption is normally distributed | B1 | (1) |
# Question 3:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \mu_D = 0$, $H_1: \mu_D > 0$ | B1 | |
| $d: 6, -3, 7, -2, -8, 6, 5, 11, 5$ | M1 | Attempt differences |
| $\bar{d} = 3$, $s_d = 6$ $\left(=\sqrt{\frac{369-9\times3^2}{8}}\right)$ $\left(\frac{\Sigma d}{9}\right)$ | M1, M1 | |
| $t_8 = \frac{3-0}{\frac{6}{\sqrt{9}}} = 1.5$ | M1A1 cao | $(\pm)$ |
| $t_8$ (5% 1-tail c.v.) $= 1.860$ | B1 | |
| Not significant – insufficient evidence that solar heating has decreased weekly fuel consumption | A1ft | (8) |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Difference in weekly fuel consumption is normally distributed | B1 | (1) |
---3. As part of an investigation into the effectiveness of solar heating, a pair of houses was identified where the mean weekly fuel consumption was the same. One of the houses was then fitted with solar heating and the other was not. Following the fitting of the solar heating, a random sample of 9 weeks was taken and the table below shows the weekly fuel consumption for each house.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | }
\hline
Week & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
Without solar heating & 19 & 19 & 18 & 14 & 6 & 7 & 5 & 31 & 43 \\
\hline
With solar heating & 13 & 22 & 11 & 16 & 14 & 1 & 0 & 20 & 38 \\
\hline
\end{tabular}
\end{center}
Units of fuel used per week
\begin{enumerate}[label=(\alph*)]
\item Stating your hypotheses clearly, test, at the $5 \%$ level of significance, whether or not there is evidence that the solar heating reduces the mean weekly fuel consumption.\\
(8)
\item State an assumption about weekly fuel consumption that is required to carry out this test.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2006 Q3 [9]}}