| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2009 |
| Session | June |
| Marks | 12 |
| 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 steps. Part (a) requires recalling the normality assumption, (b) tests understanding of paired vs unpaired designs, (c) is routine calculation (differences, mean, SD, test statistic, conclusion), and (d) requires knowing Type II error definition. All parts are textbook-standard for S4 with no novel problem-solving required, making it slightly easier than average A-level maths. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Car | A | \(B\) | C | D | E | \(F\) |
| Emissions without device | 151.4 | 164.3 | 168.5 | 148.2 | 139.4 | 151.2 |
| Emissions with device | 148.9 | 162.7 | 166.9 | 150.1 | 140.0 | 146.7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (a) The differences are normally distributed | B1 | |
| (b) The data is collected in pairs or small sample size and variance unknown or samples not independent | B1 | Allow: the same car has been used |
| (c) \(d\): 2.5, 1.6, 1.6, −1.9, −0.6, 4.5 | M1 | at least 2 correct |
| \(\Sigma d = 7.7\), \(\Sigma d^2 = 35.59\); \(\bar{d} = \pm 1.2833\), sd \(= 2.2675\) (Var \(= 5.141\)) | A1, A1 | awrt \(\pm 1.28\), 2.27 |
| \(H_0: \mu_d = 0\), \(H_1: \mu_d > 0\) (or \(H_1: \mu_d < 0\) if \(d\) = −2.5, −1.6, −1.6 etc) | B1 | both depend on their \(d\)'s |
| \(t = \frac{\pm 1.2833\sqrt{6}}{2.2675} = \pm 1.386\ldots\) | M1, A1 | formula and substitution, 1.38 – 1.39 |
| Critical value \(t_5(5\%) = 2.015\) (1 tail) | B1 | |
| Not significant. Insufficient evidence to support that the device reduces \(\text{CO}_2\) emissions. | A1ft | |
| (d) The idea that the device reduces \(\text{CO}_2\) emissions has been rejected when in fact it does reduce emissions. OR Concluding that the device does not reduce emissions when in fact it does | B1 B1 | if not in context can get B1 only |
# Question 2:
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(a)** The differences are normally distributed | B1 | |
| **(b)** The data is collected in pairs or small sample size and variance unknown or samples not independent | B1 | Allow: the same car has been used |
| **(c)** $d$: 2.5, 1.6, 1.6, −1.9, −0.6, 4.5 | M1 | at least 2 correct |
| $\Sigma d = 7.7$, $\Sigma d^2 = 35.59$; $\bar{d} = \pm 1.2833$, sd $= 2.2675$ (Var $= 5.141$) | A1, A1 | awrt $\pm 1.28$, 2.27 |
| $H_0: \mu_d = 0$, $H_1: \mu_d > 0$ (or $H_1: \mu_d < 0$ if $d$ = −2.5, −1.6, −1.6 etc) | B1 | both depend on their $d$'s |
| $t = \frac{\pm 1.2833\sqrt{6}}{2.2675} = \pm 1.386\ldots$ | M1, A1 | formula and substitution, 1.38 – 1.39 |
| Critical value $t_5(5\%) = 2.015$ (1 tail) | B1 | |
| Not significant. Insufficient evidence to support that the device reduces $\text{CO}_2$ emissions. | A1ft | |
| **(d)** The idea that the device reduces $\text{CO}_2$ emissions has been rejected when in fact it does reduce emissions. OR Concluding that the device does not reduce emissions when in fact it does | B1 B1 | if not in context can get B1 only |
---2. An emission-control device is tested to see if it reduces $\mathrm { CO } _ { 2 }$ emissions from cars. The emissions from 6 randomly selected cars are measured with and without the device. The results are as follows.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
Car & A & $B$ & C & D & E & $F$ \\
\hline
Emissions without device & 151.4 & 164.3 & 168.5 & 148.2 & 139.4 & 151.2 \\
\hline
Emissions with device & 148.9 & 162.7 & 166.9 & 150.1 & 140.0 & 146.7 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item State an assumption that needs to be made in order to carry out a $t$-test in this case.
\item State why a paired $t$-test is suitable for use with these data.
\item Using a $5 \%$ level of significance, test whether or not there is evidence that the device reduces $\mathrm { CO } _ { 2 }$ emissions from cars.
\item Explain, in context, what a type II error would be in this case.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2009 Q2 [12]}}