| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2005 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Paired t-test |
| Difficulty | Standard +0.3 This is a straightforward S4 question testing understanding of when paired tests are appropriate (parts a-c require simple conceptual recall) and execution of a standard paired t-test procedure (part d). The calculations are routine and the question structure is typical of textbook exercises, making it slightly easier than average for A-level standard. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Diet \(A\) | 5 | 6 | 7 | 4.6 | 6.1 | 5.7 | 6.2 | 7.4 | 5 | 3 |
| Diet \(B\) | 7 | 7.2 | 8 | 6.4 | 5.1 | 7.9 | 8.2 | 6.2 | 6.1 | 5.8 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (a) The data were not collected in pairs. | B1 (i) | |
| (b) Use data from two lambs. | B1 (i) | |
| (c) Age, weight, gender | B1; B1 (a) | Any two-variable factors |
| (d) \(d = B - A\) | M1 | |
| \(d\): 2, 1.2, 1, 1.8, -1, 2.2, 2, -1.2, 1.1, 2.8 | ||
| \(\sum d = 11.9\); \(\sum d^2 = 30.01\) | ||
| \(\bar{d} = 1.19\); \(s^2 = 1.761\) (\(s = 1.327\)) | A1; A1 | |
| \(H_0: \bar{\delta} = 0\); \(H_1: \bar{\delta} \neq 0\) | B1 | Allow \(\mu_0\) for \(\bar{\delta}\); both |
| \(t = \frac{1.19 - 0}{\sqrt{1.761/10}} = 2.835...\) | M1 A1 | AWRT 2.84 |
| \(\nu = 9\); CV: \(t = 2.262\) | B1 | |
| Since 2.835... is in the critical region (\(t > 2.262\)), there is evidence to reject \(H_0\). The (mean) weight gained by the lambs is different for each diet. | A1N (a) | |
| (e) Diet B - it has the higher mean | B1 (i) | |
| (f) Using non-paired t-test | B1 | |
| \(H_0: \mu_A = \mu_B\); \(H_1: \mu_A \neq \mu_B\) | B1 | |
| \(t = \frac{\mu_A - \mu_B}{\sqrt{s_p^2(\frac{1}{n_1} + \frac{1}{n_2})}} = -2.30\) | B1 | AWRT -2.3 = |
| CV: \(\ | t\ | = 2.101\) |
| Conclusion: Mean weight gained is different. NB \(\mu_A = 5.6\); \(\mu_B = 6.79\); \(s_p^2 = 1.342722...\) | B1 (4) |
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(a) The data were not collected in pairs.** | B1 (i) | |
| **(b) Use data from two lambs.** | B1 (i) | |
| **(c) Age, weight, gender** | B1; B1 (a) | Any two-variable factors |
| **(d)** $d = B - A$ | M1 | |
| $d$: 2, 1.2, 1, 1.8, -1, 2.2, 2, -1.2, 1.1, 2.8 | | |
| $\sum d = 11.9$; $\sum d^2 = 30.01$ | | |
| $\bar{d} = 1.19$; $s^2 = 1.761$ ($s = 1.327$) | A1; A1 | |
| $H_0: \bar{\delta} = 0$; $H_1: \bar{\delta} \neq 0$ | B1 | Allow $\mu_0$ for $\bar{\delta}$; both |
| $t = \frac{1.19 - 0}{\sqrt{1.761/10}} = 2.835...$ | M1 A1 | AWRT 2.84 |
| $\nu = 9$; CV: $t = 2.262$ | B1 | |
| Since 2.835... is in the critical region ($t > 2.262$), there is evidence to reject $H_0$. The (mean) weight gained by the lambs is different for each diet. | A1N (a) | |
| **(e) Diet B - it has the higher mean** | B1 (i) | |
| **(f) Using non-paired t-test** | B1 | |
| $H_0: \mu_A = \mu_B$; $H_1: \mu_A \neq \mu_B$ | B1 | |
| $t = \frac{\mu_A - \mu_B}{\sqrt{s_p^2(\frac{1}{n_1} + \frac{1}{n_2})}} = -2.30$ | B1 | AWRT -2.3 = |
| CV: $\|t\| = 2.101$ | B1 | |
| Conclusion: Mean weight gained is different. NB $\mu_A = 5.6$; $\mu_B = 6.79$; $s_p^2 = 1.342722...$ | B1 (4) | |
---4. A farmer set up a trial to assess the effect of two different diets on the increase in the weight of his lambs. He randomly selected 20 lambs. Ten of the lambs were given $\operatorname { diet } A$ and the other 10 lambs were given diet $B$. The gain in weight, in kg , of each lamb over the period of the trial was recorded.
\begin{enumerate}[label=(\alph*)]
\item State why a paired $t$-test is not suitable for use with these data.
\item Suggest an alternative method for selecting the sample which would make the use of a paired $t$-test valid.
\item Suggest two other factors that the farmer might consider when selecting the sample.
The following paired data were collected.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | }
\hline
Diet $A$ & 5 & 6 & 7 & 4.6 & 6.1 & 5.7 & 6.2 & 7.4 & 5 & 3 \\
\hline
Diet $B$ & 7 & 7.2 & 8 & 6.4 & 5.1 & 7.9 & 8.2 & 6.2 & 6.1 & 5.8 \\
\hline
\end{tabular}
\end{center}
\item Using a paired $t$-test, at the $5 \%$ significance level, test whether or not there is evidence of a difference in the weight gained by the lambs using $\operatorname { diet } A$ compared with those using $\operatorname { diet } B$.
\item State, giving a reason, which diet you would recommend the farmer to use for his lambs.\\
(Total 13 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2005 Q4 [13]}}