| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2006 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Paired t-test |
| Difficulty | Standard +0.3 This is a straightforward application of the paired t-test procedure with standard hypothesis testing steps. While it requires calculating differences, finding mean and standard deviation, computing the test statistic, and comparing to critical values, these are all routine S4 techniques with no novel problem-solving required. The follow-up parts about assumptions and one-tailed vs two-tailed tests are standard bookwork questions that test basic understanding rather than mathematical sophistication. |
| Spec | 5.07d Paired vs two-sample: selection |
| Soil | A | B | C | D | E | F | G | ||
| 39 | 40 | 43 | 32 | 42 | 33 | 36 | ||
| 41 | 36 | 61 | 48 | 42 | 48 | 45 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| \(d = U - C\): values \(2, -4, 18, 16, 0, 15, 9\) | M1 | |
| \(\bar{d} = \frac{56}{7} = 8\) | B1 | |
| \(s_d^2 = \frac{906 - 7 \times 8^2}{6} = 76\frac{1}{2}\) | M1, A1 | |
| \(H_0: \mu_d = 0\), \(H_1: \mu_d \neq 0\) | B1 | |
| \(t = \frac{8}{\sqrt{\frac{76.\overline{3}}{7}}} = 2.4226\ldots\) | M1, A1 | awrt 2.42 |
| \(t_6(2.5\%) = 2.447\) | B1 | |
| Insufficient evidence to reject \(H_0\). No evidence of a difference between the mean amount of corrosion on coated and uncoated pipes. | A1 | (9) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| Differences are normally distributed | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| Values do not appear to be normally distributed | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| \(t_6(5\%) = 1.943\). There is evidence to reject \(H_0\). There is evidence to suggest there is a greater corrosion on coated pipes. | B1, B1 | (2) |
| TOTAL | 13 |
## Question 5:
### Part (a):
| Answer/Working | Marks | Notes |
|---|---|---|
| $d = U - C$: values $2, -4, 18, 16, 0, 15, 9$ | M1 | |
| $\bar{d} = \frac{56}{7} = 8$ | B1 | |
| $s_d^2 = \frac{906 - 7 \times 8^2}{6} = 76\frac{1}{2}$ | M1, A1 | |
| $H_0: \mu_d = 0$, $H_1: \mu_d \neq 0$ | B1 | |
| $t = \frac{8}{\sqrt{\frac{76.\overline{3}}{7}}} = 2.4226\ldots$ | M1, A1 | awrt 2.42 |
| $t_6(2.5\%) = 2.447$ | B1 | |
| Insufficient evidence to reject $H_0$. No evidence of a difference between the mean amount of corrosion on coated and uncoated pipes. | A1 | (9) |
### Part (b)(i):
| Answer/Working | Marks | Notes |
|---|---|---|
| Differences are normally distributed | B1 | (1) |
### Part (b)(ii):
| Answer/Working | Marks | Notes |
|---|---|---|
| Values do not appear to be normally distributed | B1 | (1) |
### Part (c):
| Answer/Working | Marks | Notes |
|---|---|---|
| $t_6(5\%) = 1.943$. There is evidence to reject $H_0$. There is evidence to suggest there is a greater corrosion on coated pipes. | B1, B1 | (2) |
| **TOTAL** | **13** | |
---5. Seven pipes of equal length are selected at random. Each pipe is cut in half. One piece of each pipe is coated with protective paint and the other is left uncoated. All of the pieces of pipe are buried to the same depth in various soils for 6 months.
The table gives the percentage area of the pieces of pipe in the various soils that are subject to corrosion.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | }
\hline
Soil & A & B & C & D & E & F & G \\
\hline
\begin{tabular}{ l }
\% Corrosion \\
coated pipe \\
\end{tabular} & 39 & 40 & 43 & 32 & 42 & 33 & 36 \\
\hline
\begin{tabular}{ l }
\% Corrosion \\
uncoated pipe \\
\end{tabular} & 41 & 36 & 61 & 48 & 42 & 48 & 45 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Stating your hypotheses clearly and using a $5 \%$ significance level, carry out a paired $t$-test to assess whether or not there is a difference between the mean percentage of corrosion on the coated pipes and the mean percentage of corrosion on the uncoated pipes.
\item \begin{enumerate}[label=(\roman*)]
\item State an assumption that has been made in order to carry out this test.
\item Comment on the validity of this assumption.
\end{enumerate}\item State what difference would be made to the conclusion in part (a) if the test had been to determine whether or not the percentage of corrosion on the uncoated pipes was higher than the mean percentage of corrosion on the coated pipes. Justify your answer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2006 Q5 [13]}}