| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics B AS (Further Statistics B AS) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon signed-rank test (single sample) |
| Difficulty | Standard +0.3 This is a straightforward application of the Wilcoxon signed-rank test with standard interpretation of software output. Students need to read confidence intervals, check normality assumptions, and calculate standard error—all routine procedures for Further Statistics. The question is methodical but requires no novel insight or complex multi-step reasoning. |
| Spec | 5.05d Confidence intervals: using normal distribution5.06b Fit prescribed distribution: chi-squared test |
| 11.12 | 11.29 | 11.42 | 11.43 | 11.20 | 11.25 | 11.65 |
| 11.33 | 11.56 | 11.34 | 11.44 | 11.24 | 11.60 | 11.52 |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (i) | Normal probability plot is roughly a straight line |
| Answer | Marks |
|---|---|
| distribution. | E1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 2.2b | N |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (ii) | 11.2568(cid:31)(cid:80)(cid:31)11.5132 |
| [1] | 1.1 | E |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (iii) | (A) |
| needs to be Normally distributed. | E1 | |
| [1] | 1.2 | M |
| 5 | (iii) | (B) |
| reasonable. | E1 | |
| [1] | I |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (iv) | E |
| Answer | Marks |
|---|---|
| …since the interval contains 11.5 | C |
| Answer | Marks |
|---|---|
| [2] | 3.4 |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (v) | (A) |
| Answer | Marks |
|---|---|
| mean | B1 |
| [1] | 1.2 |
| (B) | S |
| Answer | Marks | Guidance |
|---|---|---|
| 14 | E1 | |
| [1] | 1.1 | |
| 5 | (vi) | Use a larger sample or use a lower confidence level |
| [1] | 3.5c |
Question 5:
5 | (i) | Normal probability plot is roughly a straight line
high p-value
suggests that the data may be from a Normal
distribution. | E1
E1
E1
[3] | 1.1
1.1
2.2b | N
Dep on one previous E mark
5 | (ii) | 11.2568(cid:31)(cid:80)(cid:31)11.5132 | B1
[1] | 1.1 | E
Numbers may be rounded to 2 or
3d.p.
5 | (iii) | (A) | Underlying distribution (of percentage of copper)
needs to be Normally distributed. | E1
[1] | 1.2 | M
5 | (iii) | (B) | Results from (i) show that this assumption is
reasonable. | E1
[1] | I
2.3
5 | (iv) | E
Confidence interval does not suggest that the mean is
different from 11.5% …
…since the interval contains 11.5 | C
B1
B1
[2] | 3.4
2.2b
5 | (v) | (A) | P
Standard error is the standard deviation of the sample
mean | B1
[1] | 1.2
(B) | S
0.1592
(= 0.0425)
14 | E1
[1] | 1.1
5 | (vi) | Use a larger sample or use a lower confidence level | E1
[1] | 3.5c5 A particular alloy of bronze is specified as containing $11.5 \%$ copper on average. A researcher takes a random sample of 14 specimens of this bronze and undertakes an analysis of each of them. The percentages of copper are found to be as follows.
\begin{center}
\begin{tabular}{ l l l l l l l }
11.12 & 11.29 & 11.42 & 11.43 & 11.20 & 11.25 & 11.65 \\
11.33 & 11.56 & 11.34 & 11.44 & 11.24 & 11.60 & 11.52 \\
\end{tabular}
\end{center}
The researcher uses software to draw a Normal probability plot for these data and to conduct a Kolmogorov-Smirnov test for Normality. The output is shown in Fig 5.1.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0de8222f-7df5-4e17-ab68-0f9d84fc615d-4_428_1550_1434_299}
\captionsetup{labelformat=empty}
\caption{Fig 5.1}
\end{center}
\end{figure}
\begin{enumerate}[label=(\roman*)]
\item Comment on what the Normal probability plot and the $p$-value of the test suggest about the data.
The researcher uses software to produce a $99 \%$ confidence interval for the mean percentage of copper in the alloy, based on the $t$ distribution. The output from the software is shown in Fig 5.2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0de8222f-7df5-4e17-ab68-0f9d84fc615d-5_1058_615_434_726}
\captionsetup{labelformat=empty}
\caption{Fig 5.2}
\end{center}
\end{figure}
\item State the confidence interval which the software gives, in the form $a < \mu < b$.
\item (A) State an assumption necessary for the use of the $t$ distribution in the construction of this confidence interval.\\
(B) State whether the assumption in part (iii) (A) seems reasonable.
\item Does the confidence interval suggest that the copper content is different from $11.5 \%$, on average? Explain your answer.
\item In the output from the software shown in Fig 5.2, SE stands for 'standard error'.\\
(A) Explain what a standard error is.\\
(B) Show how the standard error was calculated in this case.
\item Suggest a way in which the researcher could produce a narrower confidence interval.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics B AS Q5 [11]}}