Question 6 - A-Level Maths

CAIE S1 2015 June — Question 6 11 marks

Exam Board	CAIE
Module	S1 (Statistics 1)
Year	2015
Session	June
Marks	11
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Data representation
Type	Draw cumulative frequency graph from cumulative frequency table
Difficulty	Easy -1.8 This is a routine statistics question requiring only standard procedures: plotting given cumulative frequency points, reading values from the graph, and converting to a frequency table. All steps are mechanical with no problem-solving or conceptual insight required—significantly easier than average A-level questions.
Spec	2.02a Interpret single variable data: tables and diagrams 2.02b Histogram: area represents frequency

6 Seventy samples of fertiliser were collected and the nitrogen content was measured for each sample. The cumulative frequency distribution is shown in the table below.

Nitrogen content	\(\leqslant 3.5\)	\(\leqslant 3.8\)	\(\leqslant 4.0\)	\(\leqslant 4.2\)	\(\leqslant 4.5\)	\(\leqslant 4.8\)
Cumulative frequency	0	6	18	41	62	70

On graph paper draw a cumulative frequency graph to represent the data.
Estimate the percentage of samples with a nitrogen content greater than 4.4.
Estimate the median.
Construct the frequency table for these results and draw a histogram on graph paper.

Show mark scheme Show mark scheme source

Question 6:

Part (i):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
Cumulative frequency graph drawn	B1	Uniform axes cf and nitrogen content labelled, at least 0 to 70 and 3.5 to 4.8 seen
Points: \((3.5,0)\), \((3.8,6)\), \((4.0,18)\), \((4.2,41)\), \((4.5,62)\), \((4.8,70)\)	M1	5 points plotted correctly on graph paper
A1 [3]	All points correct and a reasonable curve (condone 1 missed point) or line segments

Part (ii):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
\(70 - \text{their } 55 = 15 = 21.4\%\)	M1	Subt a value \(> 41\) from 70 (or \(n/70\), \(n<29\))
A1 [2]	Correct ans, accept \(18.5 - 22\)

Part (iii):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
median \(= 4.15\)	B1 [1]	Accept \(4.1 <\) median \(< 4.2\), nfww

Part (iv):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
nit cont: \(3.5-\), \(3.8-\), \(4.0-\), \(4.2-\), \(4.5-\); fr: 6, 12, 23, 21, 8; fd: 20, 60, 115, 70, 26.7	M1	Attempt at freqs, at least 3 correct, ignore labelling
M1	Attempt at fd as f/cw only at least 3 correct FT (Accept f/cw \(\times k\))
Histogram with correct heights	A1	Correct heights seen on graph (plot at 4.8, 27 A0); graph paper must be used (3 correct relative heights implies M1M1)
B1	Correct bar ends seen on graph — graph paper used
B1 [5]	Correct linear scale and labels

## Question 6:

### Part (i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Cumulative frequency graph drawn | B1 | Uniform axes cf and nitrogen content labelled, at least 0 to 70 and 3.5 to 4.8 seen |
| Points: $(3.5,0)$, $(3.8,6)$, $(4.0,18)$, $(4.2,41)$, $(4.5,62)$, $(4.8,70)$ | M1 | 5 points plotted correctly on graph paper |
| | A1 **[3]** | All points correct and a reasonable curve (condone 1 missed point) or line segments |

### Part (ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $70 - \text{their } 55 = 15 = 21.4\%$ | M1 | Subt a value $> 41$ from 70 (or $n/70$, $n<29$) |
| | A1 **[2]** | Correct ans, accept $18.5 - 22$ |

### Part (iii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| median $= 4.15$ | B1 **[1]** | Accept $4.1 <$ median $< 4.2$, nfww |

### Part (iv):

| Answer/Working | Mark | Guidance |
|---|---|---|
| nit cont: $3.5-$, $3.8-$, $4.0-$, $4.2-$, $4.5-$; fr: 6, 12, 23, 21, 8; fd: 20, 60, 115, 70, 26.7 | M1 | Attempt at freqs, at least 3 correct, ignore labelling |
| | M1 | Attempt at fd as f/cw only at least 3 correct FT (Accept f/cw $\times k$) |
| Histogram with correct heights | A1 | Correct heights seen on graph (plot at 4.8, 27 A0); graph paper must be used (3 correct relative heights implies M1M1) |
| | B1 | Correct bar ends seen on graph — graph paper used |
| | B1 **[5]** | Correct linear scale and labels |

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Show LaTeX source

6 Seventy samples of fertiliser were collected and the nitrogen content was measured for each sample. The cumulative frequency distribution is shown in the table below.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Nitrogen content & $\leqslant 3.5$ & $\leqslant 3.8$ & $\leqslant 4.0$ & $\leqslant 4.2$ & $\leqslant 4.5$ & $\leqslant 4.8$ \\
\hline
Cumulative frequency & 0 & 6 & 18 & 41 & 62 & 70 \\
\hline
\end{tabular}
\end{center}

(i) On graph paper draw a cumulative frequency graph to represent the data.\\
(ii) Estimate the percentage of samples with a nitrogen content greater than 4.4.\\
(iii) Estimate the median.\\
(iv) Construct the frequency table for these results and draw a histogram on graph paper.

\hfill \mbox{\textit{CAIE S1 2015 Q6 [11]}}

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