Question 2 - A-Level Maths

OCR MEI S3 2012 June — Question 2 18 marks

Exam Board	OCR MEI
Module	S3 (Statistics 3)
Year	2012
Session	June
Marks	18
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Wilcoxon tests
Type	Wilcoxon signed-rank test (single sample)
Difficulty	Easy -1.8 This question tests basic sampling theory and a straightforward application of the Wilcoxon signed-rank test with clear instructions, small dataset, and standard procedure. The theoretical parts require only recall of definitions, while the hypothesis test is a routine textbook exercise with no conceptual challenges—significantly easier than average A-level questions.
Spec	2.01a Population and sample: terminology 2.01d Select/critique sampling: in context 5.01a Permutations and combinations: evaluate probabilities 5.07b Sign test: and Wilcoxon signed-rank

1. Give two reasons why an investigator might need to take a sample in order to obtain information about a population.
2. State two requirements of a sample.
3. Discuss briefly the advantage of the sampling being random.
1. Under what circumstances might one use a Wilcoxon single sample test in order to test a hypothesis about the median of a population? What distributional assumption is needed for the test?
2. On a stretch of road leading out of the centre of a town, highways officials have been monitoring the speed of the traffic in case it has increased. Previously it was known that the median speed on this stretch was 28.7 miles per hour. For a random sample of 12 vehicles on the stretch, the following speeds were recorded. $$\begin{array} { l l l l l l l l l l l l } 32.0 & 29.1 & 26.1 & 35.2 & 34.4 & 28.6 & 32.3 & 28.5 & 27.0 & 33.3 & 28.2 & 31.9 \end{array}$$ Carry out a test, with a $5 \%$ significance level, to see whether the speed of the traffic on this stretch of road seems to have increased on the whole.
  [0pt] [10]

Show mark scheme Show mark scheme source

Question 2:

Part (a)(i)

Answer	Marks	Guidance
Answer	Marks	Guidance
Population might be too large for it to be sensible to take a complete census.	E1	Reward 1 mark each for any two distinct, sensible points
Sampling process might be destructive.	E1
[2]

Part (a)(ii)

Answer	Marks	Guidance
Answer	Marks	Guidance
Sample should be unbiased.	E1	Reward 1 mark each for any two distinct, sensible points that the sample/data should be fit for purpose
Sample should be representative (of the population).	E1	Further examples: data should not be distorted by the act of sampling; data should be relevant
[2]

Part (a)(iii)

Answer	Marks	Guidance
Answer	Marks	Guidance
A random sample enables proper statistical inference to be undertaken because we know the probability basis on which it has been selected	E2	Award E2, 1, 0 depending on quality of response
[2]

Part (b)(i)

Answer	Marks	Guidance
Answer	Marks	Guidance
A Wilcoxon signed rank test might be used when nothing is known about the distribution of the background population.	E1
Must assume symmetry (about the median).	E1	Do not allow "sample" or "data" unless it clearly refers to the population. Do not allow if "Normality" forms part of the assumption.
[2]

Part (b)(ii)

Answer	Marks	Guidance
Answer	Marks	Guidance
$H_0: m = 28.7 \quad H_1: m > 28.7$ where $m$ is the population median	B1 B1	Both. Accept hypotheses in words. Adequate definition of $m$ to include "population".
Subtract 28.7 from speeds to get differences; rank \(	\)diff\(	\) giving ranks: 8, 3, 6, 12, 11, 1, 9, 2, 5, 10, 4, 7
$W_- = 1 + 2 + 4 + 5 + 6 = 18$	B1	$(W_+ = 3+7+8+9+10+11+12 = 60)$
Refer to Wilcoxon single sample tables for $n = 12$. Lower 5% point is 17 (or upper is 61 if 60 used).	M1 A1	No ft from here if wrong. i.e. a 1-tail test. No ft from here if wrong.
Result is not significant.	A1	ft only c's test statistic
No evidence to suggest that the median speed has increased.	A1	ft only c's test statistic. "Non-assertive" conclusion in context to include "on average" oe.
[10]

# Question 2:

## Part (a)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Population might be too large for it to be sensible to take a complete census. | E1 | Reward 1 mark each for any two distinct, sensible points |
| Sampling process might be destructive. | E1 | |
| **[2]** | | |

## Part (a)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sample should be unbiased. | E1 | Reward 1 mark each for any two distinct, sensible points that the sample/data should be fit for purpose |
| Sample should be representative (of the population). | E1 | Further examples: data should not be distorted by the act of sampling; data should be relevant |
| **[2]** | | |

## Part (a)(iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| A random sample enables proper statistical inference to be undertaken because we know the probability basis on which it has been selected | E2 | Award E2, 1, 0 depending on quality of response |
| **[2]** | | |

## Part (b)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| A Wilcoxon signed rank test might be used when nothing is known about the distribution of the background population. | E1 | |
| Must assume symmetry (about the median). | E1 | Do not allow "sample" or "data" unless it clearly refers to the population. Do not allow if "Normality" forms part of the assumption. |
| **[2]** | | |

## Part (b)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: m = 28.7 \quad H_1: m > 28.7$ where $m$ is the population median | B1 B1 | Both. Accept hypotheses in words. Adequate definition of $m$ to include "population". |
| Subtract 28.7 from speeds to get differences; rank $|$diff$|$ giving ranks: 8, 3, 6, 12, 11, 1, 9, 2, 5, 10, 4, 7 | M1 M1 A1 | M1 for subtracting 28.7. M1 for ranks. A1 ft if ranks wrong. If candidate has tied ranks then penalise A0 here but ft from here. |
| $W_- = 1 + 2 + 4 + 5 + 6 = 18$ | B1 | $(W_+ = 3+7+8+9+10+11+12 = 60)$ |
| Refer to Wilcoxon single sample tables for $n = 12$. Lower 5% point is 17 (or upper is 61 if 60 used). | M1 A1 | No ft from here if wrong. i.e. a 1-tail test. No ft from here if wrong. |
| Result is not significant. | A1 | ft only c's test statistic |
| No evidence to suggest that the median speed has increased. | A1 | ft only c's test statistic. "Non-assertive" conclusion in context to include "on average" oe. |
| **[10]** | | |

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Show LaTeX source

2
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Give two reasons why an investigator might need to take a sample in order to obtain information about a population.
\item State two requirements of a sample.
\item Discuss briefly the advantage of the sampling being random.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Under what circumstances might one use a Wilcoxon single sample test in order to test a hypothesis about the median of a population? What distributional assumption is needed for the test?
\item On a stretch of road leading out of the centre of a town, highways officials have been monitoring the speed of the traffic in case it has increased. Previously it was known that the median speed on this stretch was 28.7 miles per hour. For a random sample of 12 vehicles on the stretch, the following speeds were recorded.

$$\begin{array} { l l l l l l l l l l l l } 
32.0 & 29.1 & 26.1 & 35.2 & 34.4 & 28.6 & 32.3 & 28.5 & 27.0 & 33.3 & 28.2 & 31.9
\end{array}$$

Carry out a test, with a $5 \%$ significance level, to see whether the speed of the traffic on this stretch of road seems to have increased on the whole.\\[0pt]
[10]
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR MEI S3 2012 Q2 [18]}}

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