Question 3 - A-Level Maths

OCR MEI S1 — Question 3 19 marks

Exam Board	OCR MEI
Module	S1 (Statistics 1)
Marks	19
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Data representation
Type	Outliers from cumulative frequency diagram
Difficulty	Moderate -0.3 This is a standard S1 statistics question requiring routine interpretation of cumulative frequency diagrams and box plots. Parts (i)-(ii) involve straightforward reading from graphs, (iii) applies the standard 1.5×IQR outlier rule, (iv) requires basic comparison of summary statistics, and (v)-(vi) involve simple reasoning about data changes and probability. All techniques are textbook exercises with no novel problem-solving required, making it slightly easier than average.
Spec	2.02a Interpret single variable data: tables and diagrams 2.02f Measures of average and spread 2.02h Recognize outliers 2.03a Mutually exclusive and independent events

3 The birth weights of 200 lambs from crossbred sheep are illustrated by the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-3_919_1144_430_476}

Estimate the percentage of lambs with birth weight over 6 kg .
Estimate the median and interquartile range of the data.
Use your answers to part (ii) to show that there are very few, if any, outliers. Comment briefly on whether any outliers should be disregarded in analysing these data. The box and whisker plot shows the birth weights of 100 lambs from Welsh Mountain sheep. \includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-3_321_1610_1818_293}
Use appropriate measures to compare briefly the central tendencies and variations of the weights of the two types of lamb.
The weight of the largest Welsh Mountain lamb was originally recorded as 6.5 kg , but then corrected. If this error had not been corrected, how would this have affected your answers to part (iv)? Briefly explain your answer.
One lamb of each type is selected at random. Estimate the probability that the birth weight of both lambs is at least 3.9 kg .

Show mark scheme Show mark scheme source

Question 3:

Part (i)

Answer	Marks	Guidance
Answer	Marks	Guidance
\(\text{Percentage} = \frac{40}{200} \times 100 = 20\)	M1	For 40 seen or implied
A1	CAO
[2]

Part (ii)

Answer	Marks	Guidance
Answer	Marks	Guidance
Median \(= 5.2\) kg	B1
\(Q1 = 4.2\), \(Q3 = 5.8\)	B1	For Q1 or Q3; Allow 4.2 to 4.3 for Q1
Inter-quartile range \(= 5.8 - 4.2 = 1.6\)	B1	For IQR; Dep on both quartiles correct
[3]

Part (iii)

Answer	Marks	Guidance
Answer	Marks	Guidance
Lower limit \(= 4.2 - (1.5 \times 1.6) = 1.8\)	B1	For 1.8; Any use of median \(\pm 1.5\) IQR scores B0 B0
Upper limit \(= 5.8 + (1.5 \times 1.6) = 8.2\)	B1	For 8.2; E0 if say some outliers at bottom end unless lower limit \(> 2.0\)
So there are one or more outliers (if any lamb weighs more than 8.2 kg)	E1	Dep on their 1.8 and 8.2; Allow any number of outliers \(\leq 5\)
Should not be disregarded because: 'Nothing to suggest they are not genuine items of data'; Allow other convincing reasons e.g. very few so will not make much difference	E1	Indep; Must give reason
[4]

Part (iv)

Answer	Marks	Guidance
Answer	Marks	Guidance
Median for Welsh Mountain \(= 3.6\)	B1	FT their medians
IQR for Welsh Mountain \(= 0.8\)	B1	FT their IQRs
Welsh Mountain lambs have lower average weight than crossbred	E1 indep	Must imply average or CT, not just median; Allow generally lighter
Welsh Mountain lambs also have lower variation in weight than crossbred	E1 indep	Must imply spread or variation, not just IQR or range; Allow correct comment on consistency
[4]

Part (v)

Answer	Marks	Guidance
Answer	Marks	Guidance
Median unchanged	E1	Even if used IQR in (iv); E2 for 'Both comparisons remain the same'
IQR unchanged OR range or spread increased	E1	E1 for 'the range remains smaller'
[2]

Part (vi)

Answer	Marks	Guidance
Answer	Marks	Guidance
\(P(\text{Crossbred} > 3.9) = \frac{165}{200}\)	B1	Allow 162 to 165 out of 200
\(P(\text{Welsh Mountain} > 3.9) = \frac{1}{4}\)	B1
\(P(\text{Both} > 3.9) = \frac{165}{200} \times \frac{1}{4} = \frac{165}{800} = \frac{33}{160} = 0.206\)	M1	For product of their probabilities, provided one is correct
A1	CAO; Allow answers in range 0.2025 to 0.20625 with correct working
[4]

## Question 3:

### Part (i)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Percentage} = \frac{40}{200} \times 100 = 20$ | M1 | For 40 seen or implied |
| | A1 | CAO |
| | **[2]** | |

---

### Part (ii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| Median $= 5.2$ kg | B1 | |
| $Q1 = 4.2$, $Q3 = 5.8$ | B1 | For Q1 or Q3; Allow 4.2 to **4.3** for Q1 |
| Inter-quartile range $= 5.8 - 4.2 = 1.6$ | B1 | For IQR; Dep on both quartiles correct |
| | **[3]** | |

---

### Part (iii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| Lower limit $= 4.2 - (1.5 \times 1.6) = 1.8$ | B1 | For 1.8; Any use of median $\pm 1.5$ IQR scores B0 B0 |
| Upper limit $= 5.8 + (1.5 \times 1.6) = 8.2$ | B1 | For 8.2; E0 if say some outliers at bottom end unless lower limit $> 2.0$ |
| So there are one or more outliers (if any lamb weighs more than 8.2 kg) | E1 | Dep on their 1.8 and 8.2; Allow any number of outliers $\leq 5$ |
| Should not be disregarded because: 'Nothing to suggest they are not genuine items of data'; Allow other convincing reasons e.g. very few so will not make much difference | E1 | Indep; Must give reason |
| | **[4]** | |

---

### Part (iv)

| Answer | Marks | Guidance |
|--------|-------|----------|
| Median for Welsh Mountain $= 3.6$ | B1 | FT their medians |
| IQR for Welsh Mountain $= 0.8$ | B1 | FT their IQRs |
| Welsh Mountain lambs have lower average weight than crossbred | E1 indep | Must imply average or CT, not just median; Allow generally lighter |
| Welsh Mountain lambs also have lower variation in weight than crossbred | E1 indep | Must imply spread or variation, not just IQR or range; Allow correct comment on consistency |
| | **[4]** | |

---

### Part (v)

| Answer | Marks | Guidance |
|--------|-------|----------|
| Median unchanged | E1 | Even if used IQR in (iv); E2 for 'Both comparisons remain the same' |
| IQR unchanged OR range or spread increased | E1 | E1 for 'the range remains smaller' |
| | **[2]** | |

---

### Part (vi)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{Crossbred} > 3.9) = \frac{165}{200}$ | B1 | Allow 162 to 165 out of 200 |
| $P(\text{Welsh Mountain} > 3.9) = \frac{1}{4}$ | B1 | |
| $P(\text{Both} > 3.9) = \frac{165}{200} \times \frac{1}{4} = \frac{165}{800} = \frac{33}{160} = 0.206$ | M1 | For product of their probabilities, provided one is correct |
| | A1 | CAO; Allow answers in range 0.2025 to 0.20625 with correct working |
| | **[4]** | |

Show LaTeX source

3 The birth weights of 200 lambs from crossbred sheep are illustrated by the cumulative frequency diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-3_919_1144_430_476}\\
(i) Estimate the percentage of lambs with birth weight over 6 kg .\\
(ii) Estimate the median and interquartile range of the data.\\
(iii) Use your answers to part (ii) to show that there are very few, if any, outliers. Comment briefly on whether any outliers should be disregarded in analysing these data.

The box and whisker plot shows the birth weights of 100 lambs from Welsh Mountain sheep.\\
\includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-3_321_1610_1818_293}\\
(iv) Use appropriate measures to compare briefly the central tendencies and variations of the weights of the two types of lamb.\\
(v) The weight of the largest Welsh Mountain lamb was originally recorded as 6.5 kg , but then corrected. If this error had not been corrected, how would this have affected your answers to part (iv)? Briefly explain your answer.\\
(vi) One lamb of each type is selected at random. Estimate the probability that the birth weight of both lambs is at least 3.9 kg .

\hfill \mbox{\textit{OCR MEI S1  Q3 [19]}}

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