| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw cumulative frequency graph from frequency table (unequal class widths) |
| Difficulty | Easy -1.3 This is a routine S1 statistics question requiring standard procedures: calculating cumulative frequencies, plotting points, and reading values from a graph. All techniques are straightforward textbook exercises with no problem-solving or novel insight required—significantly easier than average A-level maths. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Amount spent \(( \\) x )\( | \)0 < x \leqslant 30\( | \)30 < x \leqslant 50\( | \)50 < x \leqslant 70\( | \)70 < x \leqslant 90\( | \)90 < x \leqslant 140$ |
| Number of shoppers | 16 | 40 | 48 | 26 | 30 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| cf: 16, 56, 104, 130, 160 | M1 | Attempt at cf table (up to 160); no graph needed; accept %cf but give final |
| Linear scale minimum 0 to 160 and 0 to 120 (graph) | B1 | |
| Attempt to plot points at \((30,16)\), \((50,56)\), \((70,104)\), \((90,130)\), \((140,160)\) up to 2 errors; can have a polygon | M1 | |
| All points correct from their scale and joined up, with \((0,0)\) as well | A1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| median \(= \\)59\( | B1\)\checkmark$ | Accept 57–60 or ft their graph if used lb, midpts instead of ub or assume linear interpolation |
| \(IQR = 82 - 43 = \\)39\( | M1, A1\)\checkmark$ [3] | Subtract a (sensible) LQ from a sensible UQ (generous); Ans ft; need a cf graph and UQ 80–84, LQ |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(160 - 149 = 11\) | M1 | 41–46 |
| OR 115 is mid pt of last interval so number of shoppers is \(30/2 = 15\) (can be implied) | A1 [2] | Subtracting from 160 can be implied; Correct answer accept 9–16 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{mean} = \frac{15\times16 + 40\times40 + 60\times48 + 80\times26 + 115\times30}{160} = \frac{10250}{160} = \\)64.1\( | M1, A1 [2] | Using \)\Sigma xf/160$ with mid-points |
## Question 7:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| cf: 16, 56, 104, 130, 160 | M1 | Attempt at cf table (up to 160); no graph needed; accept %cf but give final |
| Linear scale minimum 0 to 160 and 0 to 120 (graph) | B1 | |
| Attempt to plot points at $(30,16)$, $(50,56)$, $(70,104)$, $(90,130)$, $(140,160)$ up to 2 errors; can have a polygon | M1 | |
| All points correct from their scale and joined up, with $(0,0)$ as well | A1 [4] | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| median $= \$59$ | B1$\checkmark$ | Accept 57–60 or ft their graph if used lb, midpts instead of ub or assume linear interpolation |
| $IQR = 82 - 43 = \$39$ | M1, A1$\checkmark$ [3] | Subtract a (sensible) LQ from a sensible UQ (generous); Ans ft; need a cf graph and UQ 80–84, LQ |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $160 - 149 = 11$ | M1 | 41–46 |
| OR 115 is mid pt of last interval so number of shoppers is $30/2 = 15$ (can be implied) | A1 [2] | Subtracting from 160 can be implied; Correct answer accept 9–16 |
### Part (iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{mean} = \frac{15\times16 + 40\times40 + 60\times48 + 80\times26 + 115\times30}{160} = \frac{10250}{160} = \$64.1$ | M1, A1 [2] | Using $\Sigma xf/160$ with mid-points |7 The amounts spent by 160 shoppers at a supermarket are summarised in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Amount spent $( \$ x )$ & $0 < x \leqslant 30$ & $30 < x \leqslant 50$ & $50 < x \leqslant 70$ & $70 < x \leqslant 90$ & $90 < x \leqslant 140$ \\
\hline
Number of shoppers & 16 & 40 & 48 & 26 & 30 \\
\hline
\end{tabular}
\end{center}
(i) Draw a cumulative frequency graph of this distribution.\\
(ii) Estimate the median and the interquartile range of the amount spent.\\
(iii) Estimate the number of shoppers who spent more than $\$ 115$.\\
(iv) Calculate an estimate of the mean amount spent.
\hfill \mbox{\textit{CAIE S1 2016 Q7 [11]}}